research article estimation of the basic reproductive

Research ArticleEstimation of the Basic Reproductive Ratio for Dengue Fever atthe Take-Off Period of Dengue Infection

Jafaruddin12 Sapto W Indratno1 Nuning Nuraini1 Asep K Supriatna3 and Edy Soewono1

1Departemen Matematika FMIPA Institut Teknologi Bandung Bandung Indonesia2Jurusan Matematika FST Universitas Nusa Cendana Kupang Indonesia3Jurusan Matematika FMIPA Universitas Padjadjaran Bandung Indonesia

Correspondence should be addressed to Jafaruddin jafaruddinhamidyahoocom

Received 22 December 2014 Revised 6 July 2015 Accepted 7 July 2015

Academic Editor Chung-Min Liao

Copyright copy 2015 Jafaruddin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Estimating the basic reproductive ratioR0of dengue fever has continued to be an ever-increasing challenge among epidemiologists

In this paperwe propose two different constructions to estimateR0which is derived fromadynamical systemof host-vector dengue

transmissionmodelThe construction is based on the original assumption that in the early states of an epidemic the infected humancompartment increases exponentially at the same rate as the infectedmosquito compartment (previous work) In the first proposedconstruction we modify previous works by assuming that the rates of infection for mosquito and human compartments might bedifferent In the second construction we add an improvement by including more realistic conditions in which the dynamics of aninfected human compartments are intervened by the dynamics of an infected mosquito compartment and vice versa We applyour construction to the real dengue epidemic data from SB Hospital Bandung Indonesia during the period of outbreak Nov 252008ndashDec 2012 We also propose two scenarios to determine the take-off rate of infection at the beginning of a dengue epidemicfor construction of the estimates ofR

0 scenario I from equation of new cases of dengue with respect to time (daily) and scenario

II from equation of new cases of dengue with respect to cumulative number of new cases of dengue The results show that our firstconstruction ofR

0accommodates the take-off rate differences between mosquitoes and humans Our second construction of the

R0estimation takes into account the presence of infective mosquitoes in the early growth rate of infective humans and vice versa

We conclude that the second approach is more realistic compared with our first approach and the previous work

1 Introduction

Dengue is amosquito-borne viral disease found inmore than100 countries around the world which are mostly located intropical and subtropical countriesThis disease is transmittedthrough the bites of female Aedes aegypti In recent yearsdengue transmission has increased predominantly in urbanand semiurban areas and has become a major internationalpublic health concern Controlling of dengue fever has beenconducted continuously but the spread of dengue virus isstill increasing in many countries Many efforts have beenconducted to control the spread of the virus for instance areduction in the population of Aedes aegypti in the field [1]Fumigation methods have been used to reduce mosquitoesand the use of temephos has beenutilized to reduce the larvaeUntil recently there was no vaccine against any of the four

virus serotypes (DEN-1 DEN-2 DEN-3 and DEN-4) Tocure patients treatment in hospitals is usually given in theform of supportive care which includes bed rest antipyreticsand analgesics [2] Mathematical models have proved to beuseful tools in the understanding of dengue transmission [3ndash5] So far the dynamics of the dengue transmission are still aninteresting issue in epidemiological modeling

The first model of dengue transmission and the stabilityanalysis of equilibrium points are shown in [6] in which thebasic reproductive ratio R

0is constructed from the stability

condition of the disease-free equilibrium The general con-cept ofR

0can be seen in [7 8] which was adapted to various

infectious disease models [9 10] The difficulty in using R0

for measuring the level of dengue endemicity in the fieldis that R

0often depends on several parameters (biological

environmental transmission etc) which are difficult to be

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 206131 14 pageshttpdxdoiorg1011552015206131

2 Computational and Mathematical Methods in Medicine

obtained from the field With limited information aboutmosquitos such as the mosquito population size the estimateofR0can not be computed from the dynamical model only

With daily human incidence being the only available data itis natural to ask how to extract relevant parameters from thedata to fit in the model A simple approach has been achievedby assuming that both a human andmosquito undergo lineargrowth at the early state of infection As shown in [11ndash14] theestimation does not distinguish between the growth rate of aninfected human and the growth rate of infected mosquitoesand does not yet consider how the interaction betweeninfectedmosquitoes and infected humans influences the earlygrowth of a dengue epidemic There are other problems thatare treated in a recent paper [15] which examines two modelsvector-borne infections namely dengue transmission In thispaper we constructR

0estimation by taking into account the

difference between the growth rate of an infected human andinfected mosquito and the effect of the interaction betweenthem and the characteristics of dengue incidence data

The existing data do not specify the infection status (ageof infection) of each patient Hence we assume that peoplewho come to the hospital should be identified as denguepatient at the end of incubation period (approximately atthe 7th day after contact with infected mosquito) Also weassume that there are no available alternative hosts as bloodsources and there are no death and recovery in the earlydays of dengue infection The main purpose of this paperis to construct R

0estimation of the real conditions in the

field in which only the daily incidence data are availableBased on the dengue transmissionmodel in [6] we build twodifferent constructions forR

0estimation and used them for

estimating the value ofR0for dengue incidence data between

the datesNov 25 2008 and 2012 fromSBHospital IndonesiaThe organization of the paper is given as follows In

Section 2 a dynamical system of a host-vector transmis-sion model is introduced Based on this dynamical systemwe derive the basic reproductive ratio of the model Theconstructions of the proposed R

0estimation are presented

in Section 3 Formulations of the take-off rate of dengueinfection are presented in Section 4 We apply the formula tothe real data of dengue incidence from SBHospital BandungIndonesia All the numerical results are given in Section 5alongwith the insights and interpretation from the numericalresults We provide conclusions in Section 6

2 Basic Reproductive Ratio of the DengueTransmission Model

Generally for a vector-borne diseaseR0was understood as

the number of persons who would be infected from a sin-gle person initially infected by a mosquito [3 16] In thehost-vector system the basic reproductive ratioR

0is defined

as the expected number of secondary infections resultingdirectly from a single infected individual in a virgin popula-tion during the infection period [7] The general host-vectordengue transmission model was conducted in [6] Here weassume that there are no alternative hosts available as bloodsources 119898 = 0 for system in [6] Modification of the dengue

Ah

Sh

120583hSh

bphShNh

I

Ih

I

120583I

bpSIhNh

120583hIh

120574Ih

Rh

120583hRh

AvS

120583S

Figure 1 The diagram of dengue transmission model

transmissionmodel in [6] is schematically represented by thediagram in Figure 1 where

119860ℎ recruitment rate of host population

119873ℎ number of host populations

119878ℎ susceptible host population size

119868ℎ infected host population size

119877ℎ recoveredimmunes host population size

119901ℎ successful transmission probability within host

population120583ℎ birthdeath rate of host population

120574 recovery rate of host population119860V recruitment rate of vector population119873V number of vector populations119878V susceptible vector population size119868V infected vector population size119901V transmission probability within the vector popu-lation120583V death rate of vector population119887 biting rate of vector population

The correspondingmodel of dengue transmission in the shortperiod dengue infection is given in [17] as follows

119889119878ℎ

119889119905

= 119860ℎminus

119887119901ℎ119878ℎ119868V

119873ℎ

minus 120583ℎ119878ℎ

119889119868ℎ

119889119905

=

119887119901ℎ119878ℎ119868V

119873ℎ

minus 120574119868ℎminus 120583ℎ119868ℎ

119889119877ℎ

119889119905

= 120574119868ℎminus 120583ℎ119877ℎ

119889119878V

119889119905

= 119860V minus119887119901V119878V119868ℎ

119873ℎ

minus 120583V119878V

119889119868V

119889119905

=

119887119901V119878V119868ℎ

119873ℎ

minus 120583V119868V

(1)

Here we give a short explanation for system (1) as followsThe effective contact rate to human 119887119901

ℎis the average number

Computational and Mathematical Methods in Medicine 3

of contacts per day that would result in infection if thevector is infectious Furthermore the effective contact rate119887119901V defines the average number of contacts per day thateffectively transmit the infection to vectors

First of all we determine equilibrium points for thesystem (1) Because 119873

ℎand 119873V are assumed to be constant

then from system (1) ones obtain 119878ℎ+ 119868ℎ+ 119877ℎ= 119873ℎ= 119860ℎ120583ℎ

and 119878V+119868V = 119873V = 119860V120583V Consequently we obtain a disease-free equilibrium point of system (1) which is given by

1198640= (119878ℎ0

=

119860ℎ

120583ℎ

119868ℎ0

= 0 119877ℎ0

= 0 119878V0 =119860V

120583V 119868V0 = 0) (2)

and the endemic equilibrium point in the form of

1198641= (119878lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119868lowast

V ) (3)

where

119878lowast

ℎ=

120583V (120574 + 120583ℎ)1198732

ℎ+ 120583V119873ℎ119873

2

V

119901ℎ120583ℎ119873ℎ+ 119887119901ℎ119873V

119868lowast

ℎ=

120583V120583ℎ119873ℎ ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901V (120583ℎ + 119887119901ℎ120588)

119877lowast

ℎ=

120583V119873ℎ120574 ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901V (120583ℎ + 119887119901ℎ120588)

119878lowast

V =

119901ℎ120583ℎ119873ℎ+ 119887119901V119901ℎ119873V

119887119901ℎ(1 + 119887119901V120583ℎ)

119868lowast

V

=

120583V120583ℎ119873ℎ (120574 + 120583ℎ) ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901ℎ(120583V (120574 + 120583

ℎ) + 120583ℎ119887119901V)

(4)

Next we derive the basic reproductive ratio R0 R0is

often conveniently found using a next generation method[7 8 18 19] We obtain the next generation matrix of system(1) at 119864

0 as follows

NGM =

[

[

[

[

0

119887119901ℎ

120583V119887119901V

(120574 + 120583ℎ)

119873V

119873ℎ

0

]

]

]

]

(5)

Based on the definition in [20] NGM represents thatthe number of 119887119901

ℎ120583V is the generation factor of dengue

transmission from mosquito to human It means that onemosquito infects 119887119901

ℎhumans per unit of time during

its infection period 1120583V and the number of (119887119901V(120574 +

120583ℎ))(119873V119873ℎ) is the generation factor of dengue transmission

from human to mosquito This represents that one humaninfects 119887119901V(119873V119873ℎ)mosquitoes during onersquos infection period1(120583ℎ

+ 120574) NGM in (5) have two eigen values that areplusmnradic(1198872119901ℎ119901V120583V(120574 + 120583

ℎ))(119873V119873ℎ) Therefore the basic repro-

ductive ratio of system (1) is the largest eigen value of NGMthat is

R0= radic

1198872119901ℎ119901V

120583V (120574 + 120583ℎ)

119873V

119873ℎ

(6)

It is clear that the endemic equilibrium1198641in (3) exists ifR

0gt

1 In the next section we propose two constructions of anR0

estimation at the initial growth of dengue infection based onincidence data

3 Construction for R0

Estimation

Predicting a basic reproductive ratio is not easily done in thefield In the real-life situation information about the numberof mosquitoes precisely the ratio between mosquito andhumanpopulation is limited Based on the incidence data wewill estimate some related parameters for the construction ofthe basic reproductive ratioR

0 from dengue incidence data

This construction is motivated by previous work in [12]which assumes that the infective growth rate is linear atthe early state of infection The exponential growth of theinfective compartment at the early state of infection iscommonly found in the estimation of the basic reproductiveratio All estimation of R

0 in previous works [11 12 14

21 22] have been conducted based on the assumption that119870(119905) prop 119890

120582119905 varies where 119870(119905) is the cumulative numberof dengue cases and 120582 is called the force of infection [1214] Chowell et al [11] called 120582 the initial growth rate ofdengue epidemic Here we will call 120582 the take-off rate ofdengue infection In the following section we build twoconstructions to estimate R

0in (6) which is derived from

a dynamical system of the host-vector dengue transmissionmodel in system (1)

First of all we assume the construction of R0estimation

from equation (3) in [12] The construction ofR0estimation

at1198640is based on the assumptions that the numbers of infected

human population 119868ℎand infected mosquito population 119868V

grow exponentially at the same rate at the short time periodrelative to each other

119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890120582119905

(7)

where 119868ℎ0

and 119868V0 are constant and 120582 is the take-off rate ofthe initial growth dengue epidemic Moreover the numberof nonsusceptible hosts and vectors can be assumed to benegligible and by assuming (7) is ldquolike solutionrdquo of linearizedsystem (1) for R

0gt 1 These assumptions are also used in

[11 14] for calculating the force of infection from the spatialdata of a dengue epidemic Next by substituting (7) with (1)and assuming that 119878

ℎsim 119873ℎand 119881V sim 119873

119881at the early state of

an epidemic we get

(

120582

120574 + 120583ℎ

+ 1) 119868ℎ0

=

119887119901ℎ

120574 + 120583ℎ

119868V0 (8)

(

120582

120583V+ 1) 119868V0 =

119887119901V

120583V

119873V

119873ℎ

119868ℎ0 (9)

Multiplying (8) and (9) and using (6) one gets the construc-tion of theR

0estimation as follows

R2

0119865= (

120582

120583ℎ+ 120574

+ 1)(

120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(10)


which is equivalent to

R0119865

= radic1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1 (11)

The value ofR0119865

in (11) is used to estimateR0in (6) where 120582

is estimated from the dengue incidence data Note that since120582 gt 0 then R

0119865gt 1 Note also that a small increase of the

value 120582 in the interval 0 lt 120582 lt 1 causes a significant increaseof the valueR

0119865

Next we introduce our construction of R0estimation

The first construction is derived under the assumption thatat the beginning of dengue infection the take-off rate of thehost and vector varies differently that is

119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890119896120582119905

(12)

where 119896 gt 0 is the rate of infected mosquitoes per humanindex Using the same process as in the derivation of R

0119865

one obtains

R2

0119872119865= (

120582

120583ℎ+ 120574

+ 1)(

119896120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(13)

or equivalently

R0119872119865

= radic1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1 (14)

Note that R0119865

is a special case from R0119872119865

that is if 119896 = 1

then R0119865

= R0119872119865

In reality R0119872119865

is more realistic thanR0119865 From (11) and (14) we obtain

R20119872119865

R20119865

= 1 +

(119896 minus 1) 120582

120582 + 120583V (15)

The comparison betweenR0119872119865

andR0119865is shown as the level

set of (15) in Figure 2Figure 2 shows that the variation in take-off rates might

significantly change the values of R0119865

and R0119872119865

Bothestimations are derived under the assumption that the pureexponential growth phase of an epidemic increases or hasa fluctuating increase Accordingly the estimates of R

0

obtained from R0119865

and R0119872119865

are affected by biasnessthat is the derivations of R

0119865and R

0119872119865use the extreme

assumption in the early dengue epidemic that the dynamicsof an infected human are not yet intervened by the presenceof an infected mosquitorsquos influence and vice versa

The second construction of R0estimation is done in

order to revise the first construction We assume that theexponential growth of the infective host at the early state ofinfection is slightly affected by the growth of the infectivevector and vice versa The main problem here is to find thebest fit interval (here we use the terminology take-off period)where exponential growth occurs Statistically this situationcan be related to the linear phase fit in a short time period

k

120582

0 005 010 015 020

20

15

10

5

0

ℛ20MFℛ

20F = 1

ℛ20MFℛ

20F = 035

ℛ20MFℛ

20F = 075

ℛ20MFℛ

20F = 125

ℛ20MFℛ

20F = 3

ℛ20MFℛ

20F = 5

ℛ20MFℛ

20F = 7

Figure 2 Level set forR20119872119865

R20119865in (15) for data set 120583V = 130 120582 isin

[0 020] 119896 isin [0 20]

We define the initial growth in a period of time as theinitial value problem that is

119868ℎ(119905) = 119868V0 (119886119890

120578119905+ 119887119890minus120578119905

)

+ 119868ℎ0

(119888119890120578119905

+ 119889119890minus120578119905

)

119868V (119905) = 119868ℎ0

(119901119890120578119905

+ 119902119890minus120578119905

)

+ 119868V0 (119903119890120578119905

+ 119904119890minus120578119905

)

(119868ℎ(0) 119868V (0)) = (119868

ℎ0 119868V0)

(1198681015840

ℎ(0) 1198681015840

V (0)) = (120578119868V0 120578119868ℎ0)

(16)

We obtain

119886 =

1

2

119887 =

1

2

119888 =

1

2

119889 =

1

2

119901 =

1

2

119902 =

1

2

119903 =

1

2

119904

=

1

2

(17)

such that the solution of (16) is obtained namely

119868ℎ (

119905) = 119868ℎ0cosh (120578119905) + 119868V0sinh (120578119905)

119868V (119905) = 119868V0cosh (120578119905) + 119868ℎ0sinh (120578119905)

(18)

The main idea from (18) is to extract the relevant param-eter(s) during the early state of infection in which lineargrowth rate of infection is assumed to take place Next wesubstitute (18) with the expression for the derivative of 119868

ℎand

119868V in (1) by simple algebraic manipulation we then obtain


a homogeneous system of equations in cosh(120578119905) and sinh(120578119905)namely

119860[

cosh (120578119905)

sinh (120578119905)

] = [

0

0

] (19)

where coefficient matrix is

119860

= [

119868ℎ0120578 minus 119887119901

ℎ119868ℎ0

+ (120574 + 120583ℎ) 119868V0 119868V0120578 minus 119887119901

ℎ119868V0 + (120574 + 120583

ℎ) 119868ℎ0

119868V0120578 minus 119887119901V120588119868V0 + 120583V119868ℎ0 119868ℎ0120578 minus 119887119901V120588119868ℎ0 + 120583V119868V0

]

( 20 )

A nontrivial solution from (19) exists if |119860| = 0 which gives

1205782119873ℎminus (119887119901V119873V + 119873

ℎ119887119901ℎ) 120578 minus 120583V (120574 + 120583

ℎ)119873ℎ

+ 1198872119901V119873V119901ℎ

(21)

as the coefficient of 1198682ℎ0

and 1198682

V0 must be equal to zero Equiv-alently from (21) we obtain

R2

0119860= minus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V (119873V119873ℎ))

120583V (120574 + 120583ℎ)

120578 + 1

=

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(22)

Therefore we have

R0119860

= radicminus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120578 + 1 (23)

where 120588 = 119873V119873ℎ is the knownmosquitoes per person indexWe obtain (23) as a new construction for estimation R

0 as

a function of 120578 The construction ofR0in (23) is related to a

short time interval where linear growth may still take placeHere we will estimate the value of 120578 during the early state ofinfection in which a linear growth of infection is assumed totake place

Note that R0119860

increases with respect to 120578 0 le 120578 le

119887(119901ℎ+119901V120588)2 Based on the definition that 120578 is probability per

unit of time a susceptible becomes infected at the beginningof a dengue epidemic (see [7]) with assumption that 119887(119901

ℎ+

119901V120588)2 lt 1 This assumption means that the maximum take-off rate of dengue virus infection is defending on the indexmosquito per human This condition implies

2120578 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (24)

R0119860

R0119865

ratio in the early phase of dengue epidemicscould be significantly larger or smaller than one dependingon the value of 120588 in (24) and the value of 120582 In Section 4we determine 120578 as a function of 120582 at the beginning of adengue epidemic We will further analyze the level set of theratioR

0119860R0119865 Considering that the value ofR

0estimation

depends on the value of take-off rate 120582 we discuss a methodto formulate the take-off rate in the following section

4 Formulation of the Take-Off Rate

Herewe derive two scenarios for the derivation of the take-offrate (tor) at the beginning of every take-off period (top)of dengue infection Both scenarios are based on the sameassumptions but with a slightly different implementation forreal-life dengue epidemicsTherefore the main problem hereis to find the best fit interval (here we use the terminologyldquotake-off periodrdquo) where exponential growth occurs

The first scenario is done under the assumption thatduring early infection recovery and natural death do not yetoccur Note that the original data taken from the hospital isin the form of daily new cases Let 119868(119905) be the number of newcases of infection we then have 119868(119905) = 119889119868

ℎ119889119905 (see in [14])

Therefore from (7) it is

119868 (119905) = 120582119868ℎ0119890120582119905 with 119868

0= 119868 (0) = 120582119868

ℎ0(25)

and from (12) it is

119868 (119905) = 120578 (119868V0 cosh (120578119905) + 119868ℎ0sinh (120578119905))

with 1198680= 119868 (0) = 120578119868V0

(26)

By using the Taylor expansion around 120582 = 0 for (25) andaround 120578 = 0 for (26) we obtain

119868 (119905) asymp 1205821198680119905 + 1198680

119868 (119905) asymp

1205782

120582

1198680119905 + 1198680

(27)

respectively Consequently we have a relation that 120582 = 120578Thus 120578 at R

0119860in (23) can be replaced by 120582 By comparing

(11) and (23) we have

R20119860

R20119865

= 1 +

120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus 120582

120582 + 120583ℎ+ 120574

minus 1) (28)

Also 120578 in (24) can be replaced by 120582 such that we obtain

2120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (29)

The ratio R20119860

R20119865

ge 1 if and only if 120588 ge (2120582 + 120583ℎ+ 120574 minus

119887119901ℎ)119887119901V andR

2

0119860R20119865

lt 1 if and only if 120588 lt (2120582 + 120583ℎ+ 120574 minus

119887119901ℎ)119887119901V Because of (2120582minus119887119901

ℎ)119887119901V lt (2120582+120583

ℎ+120574minus119887119901

ℎ)119887119901V

we have (2120582 + 120583ℎ+ 120574 minus 119887119901

ℎ)119887119901V which belongs to the range

of 120588 for small 120582The second scenario is conducted under the same

assumption as the first scenario But here we regress betweenthe number of new cases in a day 119868(119905) = 119889119868

ℎ119889119905 and cumula-

tive number of new cases119870(119905) where we have the initial value119870(0) = 119870

0= 1198680= 119868(0) = 120582119868

ℎ0from (7) and119870(0) = 119870

0= 1198680=

119868(0) = 120578119868V0 from (18) Therefore by using integral equation119870(119905) minus 119870(0) = int

119905

0119868(120591)119889120591 with 119868(120591) from the first equation in

(7) and 119868(120591) from the first equation in (18) we obtain

119868 (119905) = 120582119870 (119905) + (1 minus 120582)1198700

119868 (119905) = 1205782119870 (119905) + (1 minus 120578

2)1198700

(30)


120582

0 005 010 015 020

20

15

5

0

10120588

ℛ20Aℛ

20F = 1

ℛ20Aℛ

20F = 7

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

P = 5

ℛ20Aℛ

20F = 125

Figure 3 Level setR20119860

R20119865

in (28) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 20]

respectively It is clear that 1205782 = 120582 so that 120578 in R0119860

at (23)can be replaced byradic120582 If (23) is divided by (11) we obtain

R20119860

R20119865

= 1 +

radic120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus radic120582

120582 + 120583ℎ+ 120574

minus radic120582) (31)

Also 120578 in (24) can be replaced byradic120582 such that we obtain

2radic120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (32)

We note that the ratioR20119860

R20119865

ge 1 if and only if 120588 ge (12058232

+

12058212

(120583V +120583ℎ+120574+1)minus119887119901

ℎ)119887119901V = 120588

0andR2

0119860R20119865

lt 1 if andonly if 120588 lt 120588

0 Note also that 120588

0lt (2 minus 119887119901

ℎ)119887119901V for small 120582

Figures 3 and 4 show the level sets for some values ofthe ratio R2

0119860R20119865 for both scenarios of construction of 120582

It shows that for small 120582 R20119860

R20119865

increases faster withrespect to 120588 in the second scenario (Figure 4) than in thefirst scenario (Figure 3) Summary of the construction ofR

0

estimation for the first and the second scenario of 120582 is givenin Table 1

The magnitude of R0119860

in Figure 5 indicates that thesecond scenario is rising faster thanR

0119860of the first scenario

at the beginning of the growing epidemic of dengue feveralthough the rate of infection is very small This means thatthe presence of mosquitoes in early dengue infection directlyaffects the human dengue infection Furthermore Figure 5shows that the second scenario of construction ofR

0 which

is summarized in Table 1 is increasing faster than the firstscenario Therefore we obtain that the second scenario is amore realistic construction of R

0than the first one at the

beginning of the short period of dengue virus infection

120582

00

2

4

6

8

10

005 010 015 020

120588

ℛ20Aℛ

20F = 1 ℛ2

0Aℛ20F = 7

ℛ20Aℛ

20F = 5

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

ℛ20Aℛ

20F = 125

Figure 4 Level set R20119860

R20119865

in (31) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 10]

0 02 04 06 08 1Proportion for take-off rate 120582

10

20

30

40

50

ℛ0

estim

atio

n

ℛ0F

ℛ0A for scenario I 120588 = 48 74ℛ0MF for k = 10

ℛ0A for scenario II 120588 = 48 74

Figure 5 The comparison ofR0estimation

In the next section we apply least square method todetermine estimation of 120582 119868

0 and119870

0from dengue incidence

data based on (27) and (30) We calculated 120582 1198680 and 119870

0by

using least square method which differs from the estimationthat has been conducted in [14]They assume that the numberof cases of dengue at 119905 = 0 is equal to one but here weestimate from the dengue incidence data In the next sectionwe examine the dependence of the basic reproductive ratioon the take-off rate at the beginning of the take-off period ofdengue infection for daily cases of dengue from SB Hospitalin Bandung Indonesia


Table 1 Summary of the construction ofR0and 120582 estimation for the first scenario

The first scenario for 120582 R0estimation

Based on [12] assumption R20119865

=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1

The first construction R20119872119865

=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1

The second construction R20119860

= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680

The second scenario for 120582 R0estimation


=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1

The tor 120582 equation 119868(119905) = 120582119870(119905) + (1 minus 120582)1198700

top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070

Figure 6 A time series of the number of daily new cases of dengueinfection in the SB Hospital is from Nov 05 2008 to Dec 2012 Thecut off period of new cases of dengue is inside the solid box Everytop contains itop of dengue infection

5 Application of R0

Estimation tothe Real-Life Dengue Epidemic

This section presents the application of themethod to the dataof dengue incidence from SB hospital the estimation valueof the tor 120582 the value basic reproductive ratio and theirimplication Calculation of 120582 is based on the assumption thatat the beginning of the infection natural death and recoveryhave not yet occurred This assumption can be taken beforethe eighth day of incidence Data is divided into five take-off periods (top) of infections which is represented insidethe solid box in Figure 6 Each top contains an initial take-off period (itop) of dengue infection Here we define theitop as being within the range of the fourth and the seventhdays of the dengue incidence The correspondents of thebioepidemiological parameter fromhuman andmosquito aregiven in Table 2

Figure 6 shows the five possibilities for the topThe topis defined by the minimum to maximum values of 119903-square

Table 2 Bioepidemiological data of human and mosquito

Symbol Definition Value

119887 Biting rate (dayminus1) 1

119901ℎ

Effective contact rate to human 001

119901V Effective contact rate to mosquito 025

120583ℎ

Human mortality rate (dayminus1) 1

365 times 70

120583V Mosquito mortality rate (dayminus1) 1

30

120574 Recovering rate of human (dayminus1) 1

8

120588 Mosquito per human index 0ndash796

119896 Proportion of mosquito per human index 0ndash2

(1199032) The number of days for every itop of incidence datadepends on the value of 1199032 criteria (approximately at the 4ndash7th day after contact with the infectedmosquito)Meanwhilethe itop is defined by the minimum to maximum values of1199032 before the eighth day Next we determine the rate of theitop of dengue infection (tor) 120582 for every top The valueof 120582 is calculated at intervals of four to seven days for thedengue of dengue infection by using two different scenariosin respect to the incidence data and the value of parameter inTable 2

In the first scenario we calculate the values of 120582 and 1198680

by using least square method for (27) and in the secondscenario we calculate the values of 120582 and 119870

0by using least

square method for (30) for every itopThose values of 120582 1198680

and1198700are summarized in Table 3

Using take-off rate values of infection 120582 in Table 3 wecalculate the value of 120588 R

0119865 R0119872119865

and R0119860 which are

given in Tables 4ndash7 as well as the intervals of 120588 R0119872119865

R0


Table 3 The value of 120582 and 1198680from the first scenario and the value of 120582 and 119870

0for the second scenario respectively

Take-off period Initial take-off period The values of 120582 and 1198680for

the first scenarioThe values of 120582 and 119870

0for

the second scenarioI 103 days I 5 days 120582 = 0093 119868

0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31

Table 4 The values of R0119865

for a given value of 120582 at the itop and the corresponding values of R0119872119865

for different values of 119896 for the firstscenario

The 119899 days at the itop 120582 R0119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558

Table 5 The values ofR0119865

for a given value of 120582 at the itop and the corresponding values ofR0119872119865

for different values of 119896 for the secondscenario


R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403

Table 6The values ofR0119865for every value of 120582 at each initial top and the values ofR

0119860for every change in the value of 120582 and 120588 for the first

scenario for calculation of 120582


R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]


for every value of 120582 at each initial top and the values of R0119860

for every change in the value of 120582 and 120588 for thesecond scenario for calculating of 120582


R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359

I 5 days 0091 [254] mdash mdash [254] 296 337

V 7 days 0105 [276] mdash mdash 227 [276] 323

III 6 days 0121 [302] mdash mdash 187 248 [302]


Table 8 Summary of the R0and R

0119872119865for the first scenario of the value of 120582 and 120588 in (28)

The 119899 days at the itop 120582

120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865

Note The value ofR0119865 from Table 6


0119872119865for the second scenario of the value of 120582 and 120588 in (31)


120588R0

R0119872119865 Important notes

from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865


Table 10 Summary of theR0andR



120588R0

R0119860

Important notesfrom (28)

II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860

and their important notes which are given in Tables8ndash13

The interpretation of the result in Tables 9ndash13 forinstance on R

0119865at top III is that each infective person

infected 389 other persons via the mosquito vector at thestart of the epidemic when the population was susceptible Itis similar meaning for other constructions ofR

0estimation

The maximum basic reproductive ratioR0119865

found was 410R0119872119865

found was 558 and R0119860

found was 913 whichshould be compared with the maximum value which wasobtained 103 in [12] Furthermore R

0119865is a special case for

both R0119872119865

and R0119860 in terms of magnitude Furthermore

R0119872119865

magnitude is a special case of R0119860

magnitude R0119860

magnitude therefore generalized both R0119865

and R0119872119865

mag-nitudes Table 4 shows that the value of R

0119865increases with

an increasing 120582 AlsoR0119872119865

increases with increasing 120582 and119896 The special cases areR

0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865

for 119896 gt 1 Thus if we have thevalue of 119896 from the real-life situation then the value ofR

0119872119865

is more realistic for estimatingR0rather thanR

0119865

From Tables 4 and 5 with respect toR0119865 we obtain that

the average secondary case of dengue infection is at most


Table 12 The value ofR0119865

is related to R0119872119865

R0 and R

0119860for the first scenario



for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860

Note R0119860 and R0 are from Table 10



R0 and R

0119860for the second scenario



for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860

R0119860 and R0 are from Table 11

four persons during the time period of dengue infectionMeanwhile the value ofR

0119872119865depends on the value of 119896 We

show that for 119896 = 1 then R0119865

= R0119872119865

(values in bracket)The increases in the value of 120582 and 119896 cause an increase inthe value of R

0119872119865 While the value in the square brackets

in Tables 5 and 7 shows the relationship between the valueof 120582 and 120588 so that R

0119865= R0119860 We also found that an

increase in the value of 120582 for a particular value of 120588 doesnot always cause an increase in the value of R

0119860 Particular

information for the four top in Tables 6 and 7 shows that toobtain the four secondary infection cases from one primaryinfection the number of mosquitoes should be at least twiceas large as the human population whereas in Tables 4 and 5we do not obtain information about the ratio of the numberof mosquitoes and humans Therefore R

0119860provides more

complete information thanR0119865

andR0119872119865

Next FromTables 6 and 7 there is the value of 120588 such that

R0119865

= R0119860

for every dengue infection itop (see the valuein brackets of Tables 6 and 7) For 119896 = 1 there are a certainpair of 120588 and 120582 for every itop such that R

0119860= R0119872119865

=

R0119865

(Table 4) For the first scenario if 120588 is relatively smallthen the value of R

0119860is inversely proportional to the value

of 120582 For instance if 120588 = 067 (see top II Table 6) thenR0119860

increases or decreases with an increasing 120582 The magnituderelation between 120582 and R

0is not the case in other values of

120588 Also we obtained that R0119865

is a special case of R0119860

for acertain pair of 120588 and120582Thus if we have the value of 120588 from thereal-life situation then the value ofR

0119860is more realistic than

R0119865

when estimatingR0from the real-life dengue epidemic

Next in the first scenario the value of R0119865

is not onlycontained in R

0119872119865and R

0119860but also contained within R

0

Also the value of R0119865

belongs to the intersection of R0119872119865

and R0119860 This is also the case in the first scenario where

R0119865

have good precision but not as good a predictor when

compared with R0119872119865

and R0119860 Whilst R

0119872119865and R

0119860are

good predictors they do not have a good precision (seeTables 7ndash13) In the second scenario the value of R

0119865is

contained within the range of R0119872119865

and R0values but not

contained in R0119860

range Furthermore the value of R0119865

islower than rangersquos lower end of values of R

0119860(see Tables

10ndash13) Meanwhile it is not enough just to pay attention toR0value of the model in (1) Therefore from the aspect

of the efficiency and effectiveness of dengue control thegovernment should pay attention to the basic reproductiveratio estimation of R

0119860that is reported here in order to

improve the quality control of the spread of dengue feverThe important notes in Tables 10ndash13 show thatR

0119865value

(see Table 4 or Tables 5ndash7) is included in the intervals R0119872119865

R0119860 and R

0for the first scenario It means that the value of

R0119865

can still be used as a standard for the control of denguevirus infection However to obtain the maximum value ofR0119860

= 2 minus 12 then the number of mosquitoes would be2ndash8 times greater than the number of people with the take-off rate of 0121 see Table 11 Therefore the use of value of 120588120582 and R

0119860as a standard value for the control of a dengue

endemicity is more realistic than the othersFrom Figures 7ndash14 not only do we obtain the value of

the basic reproductive ratio estimation of the R0119860

which isgreater than one but also we do obtain the value which isless than one for instance from the first scenario at 120582 =

0190 and 120588 = 067 we have R0119860

= 067 lt 1 (seeTable 7) Also we can determine the intersection range ofa basic reproductive ratio from a different construction ofvalue of 120582 Both proposed constructions of R

0are the same

when using assumption in [12] for these particular cases Butincreasing the number of secondary cases for every topwill depend on 119896 for the first construction and depend on120588 for the second construction These results indicate that


Proportion for take-off rate k0

0

1

2

3

4

5

05 1 15 2

120582 from scenario I

ℛ0MF 120582 = 0009ℛ0MF 120582 = 0171Threshold ℛ0MF = 1

ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF

for itop II withfor itop V with

for itop IV withfor itop III withfor itop I with 120582 = 0093

ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)

Figure 7 The dynamic of R0119872119865

for the dengue outbreak at thebeginning of the top as a function of the infected mosquito perhuman index for 120582 from the first scenario

k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F

ℛ0MFℛ0F 120582 = 0190ℛ0MFℛ0F 120582 = 0038Threshold ℛ0MF = 1

ℛ0MFℛ0F 120582 = 0171ℛ0MFℛ0F 120582 = 0009ℛ0MFℛ0F

for itop III withfor itop IV with

for itop V withfor itop II withfor itop I with 120582 = 0093

Figure 8 The ratio of R0119872119865

R0119865

for the dengue outbreak at thebeginning of the top as a function of the ratio of the rate of theinfected mosquito per human index 119896 for 120582 from the first scenario

the greater the number of mosquitoes means the higher thenumber of new cases of dengue and the shorter the period ofthe itop of dengue infection Furthermore R

0119865estimates

of R0are too low when compared to R

0119860 This is very

dangerous in application since the intervention to controldengue that results from R

0119865estimation will not be able

to stop the disease once it exists Therefore prevention ofdengue incidences is more appropriate if it is done early andexactly how early is dependent onR

0119860estimation


Mosquito per human index 120588

12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A

for itop II withfor itop I withfor itop III withfor itop IV withfor itop V with 120582 = 0171

Threshold ℛ0A = 1

Figure 9The dynamic ofR0119860

for the dengue outbreak at the begin-ning of the top as a function of the mosquito per person index for120582 from the first scenario

120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038

Threshold ℛ0Aℛ0F = 1

ℛ0Aℛ0F 120582 = 0171

for itop II withfor itop I with

for itop IV withfor itop V with

for itop III with 120582 = 0190


R0119865

for the dengue outbreak at thebeginning of the top as a function of the infected mosquito perhuman index 120588 for 120582 from the first scenario

We used real dengue epidemic data from a dengue epi-demic between the dates Nov 25 2008 and 2012 from SBHospital This includes the estimation of the basic reproduc-tive ratio at the beginning of top of dengue infection byusingR

0119865R0119872119865

andR0119860

forR0estimationR

0magnitude

estimation under the same assumptionwas conducted in [12]resulting inR

0119865 We proposeR

0119872119865andR

0119860for comparing

withR0119865forR

0estimation fromdengue incidence data from

SB Hospital As a results we get R0119872119865

gt 159 from the first


120582 from scenario II

1

2

3

4

Proportion for take-off rate k0 05 1 15 2

ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)

ℛ0MF 120582 = 0019ℛ0MF 120582 = 0091ℛ0MF 120582 = 0121ℛ0MF 120582 = 0048ℛ0MF


Threshold ℛ0MF = 1


for the dengue outbreak at thebeginning of the top as a function of the infected mosquito perhuman index for 120582 from the first Scenario


k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold

ℛ0MFℛ0F 120582 = 0121ℛ0MFℛ0F 120582 = 0091ℛ0MFℛ0F 120582 = 0019ℛ0MFℛ0F 120582 = 0105ℛ0MFℛ0F

for itop III withfor itop I withfor itop II withfor itop V withfor itop IV with 120582 = 0048

ℛ0M

Fℛ

0F


R0119865

for the dengue outbreak at thebeginning of the top as a function of ratio of the rate of the infectedmosquito per human index 119896 for 120582 from the second scenario

scenario (Table 4 Figures 7 and 9) andR0119872119865

gt 140 from thesecond scenario respectively (Table 6 Figures 7 and 9) Fromthe first scenario we obtained the the basic reproductive ratiowas lowest in the top II [101233) and largest in the topIII [310913) (Table 10 Figures 8 and 10) Furthermore fromthe second scenario the basic reproductive ratio was lowestin the top II [2 37793) and largest in the top III [5481178)(Table 11 Figures 8 and 10) We believe that R

0119872119865and R

0119860

are a good R0estimation at the beginning of the take-off

period of infectionThus our results can be considered in thecontrol of a dengue fever epidemic in the dengue endemicregions of Bandung Indonesia But the effects of dengue


8

2

4

6

10

Mosquito per human index 1205880 1 2 3 4 5 6 7

ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A


for the dengue outbreak at thebeginning of the top as a function of themosquito per person indexfor 120582 from the second scenario

120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0019ℛ0Aℛ0F 120582 = 0091ℛ0Aℛ0F 120582 = 0121ℛ0Aℛ0F 120582 = 0048ℛ0Aℛ0F


Threshold


R0119865

for the dengue outbreak at thebeginning of the top as a function of the infected mosquito perhuman index 120588 for 120582 from the second scenario

incidence data were obtained from SB Hospital Bandung tothe value ofR

0estimation and describe the transmission of

dengue fever because only the dengue incidence data of thesame serotype were considered

Some of the results of the basic reproductive ratioestimation for dengue fever in compartmental model werereported in several publications Favier et al in [12] estimateda basic reproductive ratio which ranged widely from 20 to103 for dengue epidemics occurring during the years 1996 to2003 in 9 Brazilian regions The upper bound of their rangein estimating the basic reproductive ratio is significantlyhigher than the upper bound estimate reported here Thelower bound of their range estimate of basic reproductive


ratio is relatively smaller than the lower bound estimatereported here for some take-off period but the others arerelatively higher than the lower bound estimate reportedhere (Tables 4 and 6) Chowell et al in [11] estimated thebasic reproductive ratio by using the relationship betweenthe basic reproductive ratio and the growth rate in the earlyphase of dengue fever transmission in Toceman MexicoTheir mean estimate of the basic reproductive ratio is 422which is significantly higher than themean estimate reportedhere for every take-off period which is obtained by R

0119865

The weekly number of dengue cases in Peru for the period1994ndash2006 was analyzed in [21] which obtained that the basicreproductive ratio had amedian 176 (083ndash446) where thesevalues inside the value of R

0are reported here see Table 7

During the 2000 epidemic occurring in 12 cities of the stateof Sao Paulo Brazil [4] estimated that the basic reproductiveratio was in the range [36 129] More recently [14] estimatedthe reproductive ratio by using the relationship between thebasic reproductive ratio and the force of infection of denguefever transmission in Salvador from periods 1996 to 1997 and2002 They obtain an estimation of R

0= 285(277 370)

in the periods 1996 to 1997 and R0

= 265(250 330)

for 2002 The newest report of R0estimation is in [22]

for dengue epidemics in 2002 dengue outbreak in EasterIsland Chile Their mean of the basic reproductive ratioR0at 272 is significantly higher than every value of R

0

estimation reported here for five initial take-off periods ofdengue infection from two scenarios for estimating of 120582 Ourestimation results for the basic reproductive ratio by usingR0119865

range widely are from 117 to 389 for the first scenarioand from 135 to 302 for the second scenario which belongsto the range of the first scenario Our R

0119872119865estimation

accommodates for the difference of the take-off rate 120582

between mosquito and human If there are no differences inthe take-off rate then [12] estimates are exactly the same asours Otherwise R

0119865could overestimate or underestimate

depending on the value of 119896 Tables 4 and 6 show that R0119860

accommodates differences in 120588 R0119865

is exactly the same asour second estimates for a certain pair of 120588 and 120582 OtherwiseR0119865

could overestimate or underestimate depending on thevalue of 120588 and so forth Therefore our R

0119872119865and R

0119860

constructions are more realistic for estimating the value ofthe basic reproductive ratio than theR

0119865

In the first scenario the control of dengue fever whichis based on the standard valueR

0119865 is still realistic Further-

more better control of dengue fever when based on themax-imum value of R

0119872119865or maximum value of R

0119860 is shown

in Table 6 and Figures 7ndash10 While for the second scenariowhich is shown by Table 6 and Figures 11ndash14 it is better tochoose one value from the interval as a standard value usedto control dengue fever Therefore the second constructionis better choice for dengue control strategies in BandungIndonesia Nevertheless the estimation ofR

0has limitations

because it is only done from the fourth day until the seventhday so the effect of the increase in the incidence of dengueR

0

is no longer considered Furthermore the dengue incidencedata used from SB Hospital do not distinguish between thelatent population and the dengue infected human populationso the estimate ofR

0is limited by the data characteristics In

reality nobody can distinguish between the latent populationand the dengue infected population Furthermore the latentpopulation has a short incubation time period

6 Conclusion

We have conducted two different constructions of basicreproductive ratio estimations based on incidence dataTheseconstructions modify R

0119865model by allowing the take-off

rates of human andmosquito to vary In the first constructionwe assumed that the take-off rate of an infected mosquitois proportional to the take-off rate of a human This con-struction leads to a new form of modified basic reproductiveratio estimation for R

0119872119865was a function of the take-off

rate of a human (120582) and the take-off rate of a mosquito(119896120582) A sensitivity analysis indicated that a small variationof a mosquito take-off rate could affect significant change inthe basic reproductive ratio In the second construction weassumed that during the take-off period the initial growthof infected humans is slightly affected by the growth ofinfected mosquitoes and vice versa This construction leadsto a one-parameter coupling of the initial growth of infectedhuman and mosquitoes The modified basic reproductiveratio estimation forR

0119860resulted from this construction and

it depends on the mosquitoes per person index 120588 This ratioincreases for small take-off rate and decreases for larger take-off rate A numerical analysis of R

0119872119865R0119865

and R0119860

R0119865

ratios shows variation in a relatively large range of values forR0119865 in respect to (120582 119896) and (120582 120588) for the first and second

scenario for determined 120582 respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Hibah Stretegis Nasionalunder the DIKTI Indonesia scheme The authors wouldlike to thank to anonymous reviewers for their very helpfulsuggestions and comments that helped to improve the paper

References

[1] M N Burattini M Chen A Chow et al ldquoModelling thecontrol strategies against dengue in Singaporerdquo Epidemiologyand Infection vol 136 no 3 pp 309ndash319 2008

[2] WHO ldquoDengue guidelines for diagnosis treatment preventionand control 2012rdquo New Edition Sheet World Health Organi-zation Geneva Switzerland 2012 httpwwwwhointdengue-controlresourcesen

[3] C A Marques O P Forattini and E Massad ldquoThe basic repro-duction number for dengue feverrdquo Transactions of the RoyalSociety of Tropical Medicine and Hygiene vol 10 no 8 p 58591994

[4] E Massad F A B Coutinho M N Burattini and L F LopezldquoThe risk of yellow fever in a dengue-infested areardquoTransactions


of the Royal Society of TropicalMedicine andHygiene vol 95 no4 pp 370ndash374 2001

[5] E-E Ooi K-T Goh and D J Gubler ldquoDengue prevention and35 years of vector control in Singaporerdquo Emerging InfectiousDiseases vol 12 no 6 pp 887ndash893 2006

[6] L Esteva and C Vargas ldquoAnalysis of a dengue disease transmis-sion modelrdquo Mathematical Biosciences vol 150 no 2 pp 131ndash151 1998

[7] O Diekmann and J A P Heesterbeek Mathematical Epidemi-ologi of Infectious Diseases Model Building Analisis and Inter-pretation Wiley Series in Mathematical and ComputationalBiology Editor in chief Simon Levin Princeton UniversityPrinceton NJ USA 2000

[8] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[9] J M Haffernan R J Smith and L M Wahl ldquoPerspectives onthe reproductive ratiordquo Journal of The Royal Society Interfacevol 2 pp 281ndash293 2005

[10] C C Chaves Z Feng and W Huang ldquoOn computation of R0

and its rule on global stabilityrdquo 2009 httpsecommonscornelledubitstream1813321461BU-1553-Mpdf

[11] G Chowell P D Duenas J C Miller et al ldquoEstimation of thereproduction number of dengue fever from spatial epidemicdatardquo Mathematical Biosciences vol 208 no 2 pp 571ndash5892007

[12] C Favier N Degallier M G Rosa-Freitas et al ldquoEarlydetermination of the reproductive number for vector-bornediseases the case of dengue in Brazilrdquo Tropical Medicine andInternational Health vol 11 no 3 pp 332ndash340 2006

[13] A K Supriatna ldquoEstimating the basic reproduction number ofdengue transmission during 2002ndash2007 outbreaks in BandungIndonesiardquo Dengue Bulletin vol 33 no 1 pp 21ndash32 2009

[14] S T R Pinho C P Ferreira L Esteva F R Barreto V C M eSilva andM G L Teixeira ldquoModelling the dynamics of denguereal epidemicsrdquo Philosophical Transactions of the Royal SocietyAMathematical Physical and Engineering Sciences vol 368 no1933 pp 5679ndash5693 2010

[15] M Amaku F AzevedoaM N Burattini F A B Coutinho L FLopez and EMassad ldquoInterpretations and pitfalls inmodellingvector-transmitted infectionsrdquo Epidemiology and Infection vol9 pp 1803ndash1815 2014

[16] GMacdonald ldquoThe analysis of equilibrium inmalariardquoTropicalDiseases Bulletin vol 49 no 9 pp 813ndash829 1952

[17] L Esteva and C Vargas ldquoA model for dengue disease withvariable human populationrdquo Journal of Mathematical Biologyvol 38 no 3 pp 220ndash240 1999

[18] N Nuraini E Soewono and K A Sidarta ldquoMathematicalmodel of dengue disease transmission with severe DHF con-partmentrdquo Bulletin of the Malaysian Mathematical SciencesSociety vol 30 no 2 pp 143ndash157 2007

[19] P Pongsumpun and I M Tang ldquoA realistic age structuredtransmissionmodel for dengue hemorrhagic fever inThailandrdquoSoutheast Asian Journal of Tropical Medicine and Public Healthvol 32 no 2 pp 336ndash340 2001

[20] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[21] G Chowell C A Torre C Munayco-Escate et al ldquoSpatialand temporal dynamics of dengue fever in Peru 1994ndash2006rdquoEpidemiology and Infection vol 136 no 12 pp 1667ndash1677 2008

[22] G Chowell R Fuentes A Olea X Aguilera H Nesse andJ M Hyman ldquoThe basic reproduction number 119877

0and effec-

tiveness of reactive interventions during dengue epidemics the2002 dengue outbreak in Easter Island Chilerdquo MathematicalBiosciences and Engineering vol 10 no 5-6 pp 1155ndash1174 2013

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


obtained from the field With limited information aboutmosquitos such as the mosquito population size the estimateofR0can not be computed from the dynamical model only

With daily human incidence being the only available data itis natural to ask how to extract relevant parameters from thedata to fit in the model A simple approach has been achievedby assuming that both a human andmosquito undergo lineargrowth at the early state of infection As shown in [11ndash14] theestimation does not distinguish between the growth rate of aninfected human and the growth rate of infected mosquitoesand does not yet consider how the interaction betweeninfectedmosquitoes and infected humans influences the earlygrowth of a dengue epidemic There are other problems thatare treated in a recent paper [15] which examines two modelsvector-borne infections namely dengue transmission In thispaper we constructR

0estimation by taking into account the

difference between the growth rate of an infected human andinfected mosquito and the effect of the interaction betweenthem and the characteristics of dengue incidence data

The existing data do not specify the infection status (ageof infection) of each patient Hence we assume that peoplewho come to the hospital should be identified as denguepatient at the end of incubation period (approximately atthe 7th day after contact with infected mosquito) Also weassume that there are no available alternative hosts as bloodsources and there are no death and recovery in the earlydays of dengue infection The main purpose of this paperis to construct R

0estimation of the real conditions in the

field in which only the daily incidence data are availableBased on the dengue transmissionmodel in [6] we build twodifferent constructions forR

0estimation and used them for

estimating the value ofR0for dengue incidence data between

the datesNov 25 2008 and 2012 fromSBHospital IndonesiaThe organization of the paper is given as follows In

Section 2 a dynamical system of a host-vector transmis-sion model is introduced Based on this dynamical systemwe derive the basic reproductive ratio of the model Theconstructions of the proposed R

0estimation are presented

in Section 3 Formulations of the take-off rate of dengueinfection are presented in Section 4 We apply the formula tothe real data of dengue incidence from SBHospital BandungIndonesia All the numerical results are given in Section 5alongwith the insights and interpretation from the numericalresults We provide conclusions in Section 6

2 Basic Reproductive Ratio of the DengueTransmission Model

Generally for a vector-borne diseaseR0was understood as

the number of persons who would be infected from a sin-gle person initially infected by a mosquito [3 16] In thehost-vector system the basic reproductive ratioR

0is defined

as the expected number of secondary infections resultingdirectly from a single infected individual in a virgin popula-tion during the infection period [7] The general host-vectordengue transmission model was conducted in [6] Here weassume that there are no alternative hosts available as bloodsources 119898 = 0 for system in [6] Modification of the dengue

Ah

Sh

120583hSh

bphShNh

I

Ih

I

120583I

bpSIhNh

120583hIh

120574Ih

Rh

120583hRh

AvS

120583S

Figure 1 The diagram of dengue transmission model

transmissionmodel in [6] is schematically represented by thediagram in Figure 1 where

119860ℎ recruitment rate of host population

119873ℎ number of host populations

119878ℎ susceptible host population size

119868ℎ infected host population size

119877ℎ recoveredimmunes host population size

119901ℎ successful transmission probability within host

population120583ℎ birthdeath rate of host population

120574 recovery rate of host population119860V recruitment rate of vector population119873V number of vector populations119878V susceptible vector population size119868V infected vector population size119901V transmission probability within the vector popu-lation120583V death rate of vector population119887 biting rate of vector population

The correspondingmodel of dengue transmission in the shortperiod dengue infection is given in [17] as follows

119889119878ℎ

119889119905

= 119860ℎminus

119887119901ℎ119878ℎ119868V

119873ℎ

minus 120583ℎ119878ℎ

119889119868ℎ

119889119905

=

119887119901ℎ119878ℎ119868V

119873ℎ

minus 120574119868ℎminus 120583ℎ119868ℎ

119889119877ℎ

119889119905

= 120574119868ℎminus 120583ℎ119877ℎ

119889119878V

119889119905

= 119860V minus119887119901V119878V119868ℎ

119873ℎ

minus 120583V119878V

119889119868V

119889119905

=

119887119901V119878V119868ℎ

119873ℎ

minus 120583V119868V

(1)

Here we give a short explanation for system (1) as followsThe effective contact rate to human 119887119901

ℎis the average number







1198640= (119878ℎ0

=

119860ℎ

120583ℎ

119868ℎ0

= 0 119877ℎ0

= 0 119878V0 =119860V

120583V 119868V0 = 0) (2)


1198641= (119878lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119868lowast

V ) (3)

where

119878lowast

ℎ=

120583V (120574 + 120583ℎ)1198732

ℎ+ 120583V119873ℎ119873

2

V

119901ℎ120583ℎ119873ℎ+ 119887119901ℎ119873V

119868lowast

ℎ=

120583V120583ℎ119873ℎ ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901V (120583ℎ + 119887119901ℎ120588)

119877lowast

ℎ=

120583V119873ℎ120574 ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901V (120583ℎ + 119887119901ℎ120588)

119878lowast

V =

119901ℎ120583ℎ119873ℎ+ 119887119901V119901ℎ119873V

119887119901ℎ(1 + 119887119901V120583ℎ)

119868lowast

V

=

120583V120583ℎ119873ℎ (120574 + 120583ℎ) ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901ℎ(120583V (120574 + 120583

ℎ) + 120583ℎ119887119901V)

(4)



0 as follows

NGM =

[

[

[

[

0

119887119901ℎ

120583V119887119901V

(120574 + 120583ℎ)

119873V

119873ℎ

0

]

]

]

]

(5)











R0= radic

1198872119901ℎ119901V

120583V (120574 + 120583ℎ)

119873V

119873ℎ

(6)


0gt




Estimation














119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890120582119905

(7)

where 119868ℎ0




ℎsim 119873ℎand 119881V sim 119873


an epidemic we get

(

120582

120574 + 120583ℎ

+ 1) 119868ℎ0

=

119887119901ℎ

120574 + 120583ℎ

119868V0 (8)

(

120582

120583V+ 1) 119868V0 =

119887119901V

120583V

119873V

119873ℎ

119868ℎ0 (9)



R2

0119865= (

120582

120583ℎ+ 120574

+ 1)(

120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(10)



R0119865

= radic1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1 (11)






0119865



119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890119896120582119905

(12)


0119865

one obtains

R2

0119872119865= (

120582

120583ℎ+ 120574

+ 1)(

119896120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(13)

or equivalently

R0119872119865

= radic1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1 (14)

Note that R0119865


that is if 119896 = 1

then R0119865

= R0119872119865



R20119872119865

R20119865

= 1 +

(119896 minus 1) 120582

120582 + 120583V (15)





and R0119872119865


0


and R0119872119865


0119865and R





k

120582

0 005 010 015 020

20

15

10

5

0

ℛ20MFℛ

20F = 1

ℛ20MFℛ

20F = 035

ℛ20MFℛ

20F = 075

ℛ20MFℛ

20F = 125

ℛ20MFℛ

20F = 3

ℛ20MFℛ

20F = 5

ℛ20MFℛ

20F = 7



[0 020] 119896 isin [0 20]


119868ℎ(119905) = 119868V0 (119886119890

120578119905+ 119887119890minus120578119905

)

+ 119868ℎ0

(119888119890120578119905

+ 119889119890minus120578119905

)

119868V (119905) = 119868ℎ0

(119901119890120578119905

+ 119902119890minus120578119905

)

+ 119868V0 (119903119890120578119905

+ 119904119890minus120578119905

)

(119868ℎ(0) 119868V (0)) = (119868

ℎ0 119868V0)

(1198681015840

ℎ(0) 1198681015840

V (0)) = (120578119868V0 120578119868ℎ0)

(16)

We obtain

119886 =

1

2

119887 =

1

2

119888 =

1

2

119889 =

1

2

119901 =

1

2

119902 =

1

2

119903 =

1

2

119904

=

1

2

(17)


119868ℎ (

119905) = 119868ℎ0cosh (120578119905) + 119868V0sinh (120578119905)

119868V (119905) = 119868V0cosh (120578119905) + 119868ℎ0sinh (120578119905)

(18)


ℎand




119860[

cosh (120578119905)

sinh (120578119905)

] = [

0

0

] (19)


119860

= [

119868ℎ0120578 minus 119887119901

ℎ119868ℎ0

+ (120574 + 120583ℎ) 119868V0 119868V0120578 minus 119887119901

ℎ119868V0 + (120574 + 120583

ℎ) 119868ℎ0


]

( 20 )


1205782119873ℎminus (119887119901V119873V + 119873

ℎ119887119901ℎ) 120578 minus 120583V (120574 + 120583

ℎ)119873ℎ

+ 1198872119901V119873V119901ℎ

(21)


and 1198682


R2

0119860= minus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V (119873V119873ℎ))

120583V (120574 + 120583ℎ)

120578 + 1

=

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(22)

Therefore we have

R0119860

= radicminus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120578 + 1 (23)


0 as



Note that R0119860




ℎ+


2120578 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (24)

R0119860

R0119865



0estimation





ℎ119889119905 (see in [14])


119868 (119905) = 120582119868ℎ0119890120582119905 with 119868

0= 119868 (0) = 120582119868

ℎ0(25)

and from (12) it is

119868 (119905) = 120578 (119868V0 cosh (120578119905) + 119868ℎ0sinh (120578119905))

with 1198680= 119868 (0) = 120578119868V0

(26)


119868 (119905) asymp 1205821198680119905 + 1198680

119868 (119905) asymp

1205782

120582

1198680119905 + 1198680

(27)




R20119860

R20119865

= 1 +

120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus 120582

120582 + 120583ℎ+ 120574

minus 1) (28)


2120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (29)

The ratio R20119860

R20119865


119887119901ℎ)119887119901V andR

2

0119860R20119865



ℎ)119887119901V lt (2120582+120583

ℎ+120574minus119887119901

ℎ)119887119901V

we have (2120582 + 120583ℎ+ 120574 minus 119887119901






0= 1198680= 119868(0) = 120582119868

ℎ0from (7) and119870(0) = 119870

0= 1198680=


119905



119868 (119905) = 120582119870 (119905) + (1 minus 120582)1198700

119868 (119905) = 1205782119870 (119905) + (1 minus 120578

2)1198700

(30)


120582

0 005 010 015 020

20

15

5

0

10120588

ℛ20Aℛ

20F = 1

ℛ20Aℛ

20F = 7

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

P = 5

ℛ20Aℛ

20F = 125


R20119865

in (28) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 20]



R20119860

R20119865

= 1 +

radic120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus radic120582

120582 + 120583ℎ+ 120574



2radic120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (32)


R20119865


+

12058212

(120583V +120583ℎ+120574+1)minus119887119901

ℎ)119887119901V = 120588

0andR2

0119860R20119865



0lt (2 minus 119887119901

ℎ)119887119901V for small 120582




R20119865


0






0 which




120582

00

2

4

6

8

10

005 010 015 020

120588

ℛ20Aℛ

20F = 1 ℛ2

0Aℛ20F = 7

ℛ20Aℛ

20F = 5

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

ℛ20Aℛ

20F = 125


R20119865

in (31) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 10]


10

20

30

40

50

ℛ0

estim

atio

n

ℛ0F





0 and119870



0by






=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680



=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1


top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070


5 Application of R0







119901ℎ



120583ℎ


365 times 70


30


8






0by using least




0119865 R0119872119865



R0






0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















1198640= (119878ℎ0

=

119860ℎ

120583ℎ

119868ℎ0

= 0 119877ℎ0

= 0 119878V0 =119860V

120583V 119868V0 = 0) (2)


1198641= (119878lowast

ℎ 119868lowast

ℎ 119877lowast

ℎ 119878lowast

V 119868lowast

V ) (3)

where

119878lowast

ℎ=

120583V (120574 + 120583ℎ)1198732

ℎ+ 120583V119873ℎ119873

2

V

119901ℎ120583ℎ119873ℎ+ 119887119901ℎ119873V

119868lowast

ℎ=

120583V120583ℎ119873ℎ ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901V (120583ℎ + 119887119901ℎ120588)

119877lowast

ℎ=

120583V119873ℎ120574 ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901V (120583ℎ + 119887119901ℎ120588)

119878lowast

V =

119901ℎ120583ℎ119873ℎ+ 119887119901V119901ℎ119873V

119887119901ℎ(1 + 119887119901V120583ℎ)

119868lowast

V

=

120583V120583ℎ119873ℎ (120574 + 120583ℎ) ((1198872119901ℎ119901V120583V (120574 + 120583

ℎ)) (119873V119873ℎ) minus 1)

119887119901ℎ(120583V (120574 + 120583

ℎ) + 120583ℎ119887119901V)

(4)



0 as follows

NGM =

[

[

[

[

0

119887119901ℎ

120583V119887119901V

(120574 + 120583ℎ)

119873V

119873ℎ

0

]

]

]

]

(5)











R0= radic

1198872119901ℎ119901V

120583V (120574 + 120583ℎ)

119873V

119873ℎ

(6)


0gt




Estimation














119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890120582119905

(7)

where 119868ℎ0




ℎsim 119873ℎand 119881V sim 119873


an epidemic we get

(

120582

120574 + 120583ℎ

+ 1) 119868ℎ0

=

119887119901ℎ

120574 + 120583ℎ

119868V0 (8)

(

120582

120583V+ 1) 119868V0 =

119887119901V

120583V

119873V

119873ℎ

119868ℎ0 (9)



R2

0119865= (

120582

120583ℎ+ 120574

+ 1)(

120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(10)



R0119865

= radic1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1 (11)






0119865



119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890119896120582119905

(12)


0119865

one obtains

R2

0119872119865= (

120582

120583ℎ+ 120574

+ 1)(

119896120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(13)

or equivalently

R0119872119865

= radic1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1 (14)

Note that R0119865


that is if 119896 = 1

then R0119865

= R0119872119865



R20119872119865

R20119865

= 1 +

(119896 minus 1) 120582

120582 + 120583V (15)





and R0119872119865


0


and R0119872119865


0119865and R





k

120582

0 005 010 015 020

20

15

10

5

0

ℛ20MFℛ

20F = 1

ℛ20MFℛ

20F = 035

ℛ20MFℛ

20F = 075

ℛ20MFℛ

20F = 125

ℛ20MFℛ

20F = 3

ℛ20MFℛ

20F = 5

ℛ20MFℛ

20F = 7



[0 020] 119896 isin [0 20]


119868ℎ(119905) = 119868V0 (119886119890

120578119905+ 119887119890minus120578119905

)

+ 119868ℎ0

(119888119890120578119905

+ 119889119890minus120578119905

)

119868V (119905) = 119868ℎ0

(119901119890120578119905

+ 119902119890minus120578119905

)

+ 119868V0 (119903119890120578119905

+ 119904119890minus120578119905

)

(119868ℎ(0) 119868V (0)) = (119868

ℎ0 119868V0)

(1198681015840

ℎ(0) 1198681015840

V (0)) = (120578119868V0 120578119868ℎ0)

(16)

We obtain

119886 =

1

2

119887 =

1

2

119888 =

1

2

119889 =

1

2

119901 =

1

2

119902 =

1

2

119903 =

1

2

119904

=

1

2

(17)


119868ℎ (

119905) = 119868ℎ0cosh (120578119905) + 119868V0sinh (120578119905)

119868V (119905) = 119868V0cosh (120578119905) + 119868ℎ0sinh (120578119905)

(18)


ℎand




119860[

cosh (120578119905)

sinh (120578119905)

] = [

0

0

] (19)


119860

= [

119868ℎ0120578 minus 119887119901

ℎ119868ℎ0

+ (120574 + 120583ℎ) 119868V0 119868V0120578 minus 119887119901

ℎ119868V0 + (120574 + 120583

ℎ) 119868ℎ0


]

( 20 )


1205782119873ℎminus (119887119901V119873V + 119873

ℎ119887119901ℎ) 120578 minus 120583V (120574 + 120583

ℎ)119873ℎ

+ 1198872119901V119873V119901ℎ

(21)


and 1198682


R2

0119860= minus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V (119873V119873ℎ))

120583V (120574 + 120583ℎ)

120578 + 1

=

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(22)

Therefore we have

R0119860

= radicminus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120578 + 1 (23)


0 as



Note that R0119860




ℎ+


2120578 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (24)

R0119860

R0119865



0estimation





ℎ119889119905 (see in [14])


119868 (119905) = 120582119868ℎ0119890120582119905 with 119868

0= 119868 (0) = 120582119868

ℎ0(25)

and from (12) it is

119868 (119905) = 120578 (119868V0 cosh (120578119905) + 119868ℎ0sinh (120578119905))

with 1198680= 119868 (0) = 120578119868V0

(26)


119868 (119905) asymp 1205821198680119905 + 1198680

119868 (119905) asymp

1205782

120582

1198680119905 + 1198680

(27)




R20119860

R20119865

= 1 +

120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus 120582

120582 + 120583ℎ+ 120574

minus 1) (28)


2120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (29)

The ratio R20119860

R20119865


119887119901ℎ)119887119901V andR

2

0119860R20119865



ℎ)119887119901V lt (2120582+120583

ℎ+120574minus119887119901

ℎ)119887119901V

we have (2120582 + 120583ℎ+ 120574 minus 119887119901






0= 1198680= 119868(0) = 120582119868

ℎ0from (7) and119870(0) = 119870

0= 1198680=


119905



119868 (119905) = 120582119870 (119905) + (1 minus 120582)1198700

119868 (119905) = 1205782119870 (119905) + (1 minus 120578

2)1198700

(30)


120582

0 005 010 015 020

20

15

5

0

10120588

ℛ20Aℛ

20F = 1

ℛ20Aℛ

20F = 7

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

P = 5

ℛ20Aℛ

20F = 125


R20119865

in (28) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 20]



R20119860

R20119865

= 1 +

radic120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus radic120582

120582 + 120583ℎ+ 120574



2radic120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (32)


R20119865


+

12058212

(120583V +120583ℎ+120574+1)minus119887119901

ℎ)119887119901V = 120588

0andR2

0119860R20119865



0lt (2 minus 119887119901

ℎ)119887119901V for small 120582




R20119865


0






0 which




120582

00

2

4

6

8

10

005 010 015 020

120588

ℛ20Aℛ

20F = 1 ℛ2

0Aℛ20F = 7

ℛ20Aℛ

20F = 5

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

ℛ20Aℛ

20F = 125


R20119865

in (31) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 10]


10

20

30

40

50

ℛ0

estim

atio

n

ℛ0F





0 and119870



0by






=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680



=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1


top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070


5 Application of R0







119901ℎ



120583ℎ


365 times 70


30


8






0by using least




0119865 R0119872119865



R0






0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















R0119865

= radic1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1 (11)






0119865



119868ℎ(119905) asymp 119868

ℎ0119890120582119905

119868V (119905) asymp 119868V0119890119896120582119905

(12)


0119865

one obtains

R2

0119872119865= (

120582

120583ℎ+ 120574

+ 1)(

119896120582

120583V+ 1) =

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(13)

or equivalently

R0119872119865

= radic1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1 (14)

Note that R0119865


that is if 119896 = 1

then R0119865

= R0119872119865



R20119872119865

R20119865

= 1 +

(119896 minus 1) 120582

120582 + 120583V (15)





and R0119872119865


0


and R0119872119865


0119865and R





k

120582

0 005 010 015 020

20

15

10

5

0

ℛ20MFℛ

20F = 1

ℛ20MFℛ

20F = 035

ℛ20MFℛ

20F = 075

ℛ20MFℛ

20F = 125

ℛ20MFℛ

20F = 3

ℛ20MFℛ

20F = 5

ℛ20MFℛ

20F = 7



[0 020] 119896 isin [0 20]


119868ℎ(119905) = 119868V0 (119886119890

120578119905+ 119887119890minus120578119905

)

+ 119868ℎ0

(119888119890120578119905

+ 119889119890minus120578119905

)

119868V (119905) = 119868ℎ0

(119901119890120578119905

+ 119902119890minus120578119905

)

+ 119868V0 (119903119890120578119905

+ 119904119890minus120578119905

)

(119868ℎ(0) 119868V (0)) = (119868

ℎ0 119868V0)

(1198681015840

ℎ(0) 1198681015840

V (0)) = (120578119868V0 120578119868ℎ0)

(16)

We obtain

119886 =

1

2

119887 =

1

2

119888 =

1

2

119889 =

1

2

119901 =

1

2

119902 =

1

2

119903 =

1

2

119904

=

1

2

(17)


119868ℎ (

119905) = 119868ℎ0cosh (120578119905) + 119868V0sinh (120578119905)

119868V (119905) = 119868V0cosh (120578119905) + 119868ℎ0sinh (120578119905)

(18)


ℎand




119860[

cosh (120578119905)

sinh (120578119905)

] = [

0

0

] (19)


119860

= [

119868ℎ0120578 minus 119887119901

ℎ119868ℎ0

+ (120574 + 120583ℎ) 119868V0 119868V0120578 minus 119887119901

ℎ119868V0 + (120574 + 120583

ℎ) 119868ℎ0


]

( 20 )


1205782119873ℎminus (119887119901V119873V + 119873

ℎ119887119901ℎ) 120578 minus 120583V (120574 + 120583

ℎ)119873ℎ

+ 1198872119901V119873V119901ℎ

(21)


and 1198682


R2

0119860= minus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V (119873V119873ℎ))

120583V (120574 + 120583ℎ)

120578 + 1

=

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(22)

Therefore we have

R0119860

= radicminus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120578 + 1 (23)


0 as



Note that R0119860




ℎ+


2120578 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (24)

R0119860

R0119865



0estimation





ℎ119889119905 (see in [14])


119868 (119905) = 120582119868ℎ0119890120582119905 with 119868

0= 119868 (0) = 120582119868

ℎ0(25)

and from (12) it is

119868 (119905) = 120578 (119868V0 cosh (120578119905) + 119868ℎ0sinh (120578119905))

with 1198680= 119868 (0) = 120578119868V0

(26)


119868 (119905) asymp 1205821198680119905 + 1198680

119868 (119905) asymp

1205782

120582

1198680119905 + 1198680

(27)




R20119860

R20119865

= 1 +

120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus 120582

120582 + 120583ℎ+ 120574

minus 1) (28)


2120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (29)

The ratio R20119860

R20119865


119887119901ℎ)119887119901V andR

2

0119860R20119865



ℎ)119887119901V lt (2120582+120583

ℎ+120574minus119887119901

ℎ)119887119901V

we have (2120582 + 120583ℎ+ 120574 minus 119887119901






0= 1198680= 119868(0) = 120582119868

ℎ0from (7) and119870(0) = 119870

0= 1198680=


119905



119868 (119905) = 120582119870 (119905) + (1 minus 120582)1198700

119868 (119905) = 1205782119870 (119905) + (1 minus 120578

2)1198700

(30)


120582

0 005 010 015 020

20

15

5

0

10120588

ℛ20Aℛ

20F = 1

ℛ20Aℛ

20F = 7

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

P = 5

ℛ20Aℛ

20F = 125


R20119865

in (28) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 20]



R20119860

R20119865

= 1 +

radic120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus radic120582

120582 + 120583ℎ+ 120574



2radic120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (32)


R20119865


+

12058212

(120583V +120583ℎ+120574+1)minus119887119901

ℎ)119887119901V = 120588

0andR2

0119860R20119865



0lt (2 minus 119887119901

ℎ)119887119901V for small 120582




R20119865


0






0 which




120582

00

2

4

6

8

10

005 010 015 020

120588

ℛ20Aℛ

20F = 1 ℛ2

0Aℛ20F = 7

ℛ20Aℛ

20F = 5

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

ℛ20Aℛ

20F = 125


R20119865

in (31) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 10]


10

20

30

40

50

ℛ0

estim

atio

n

ℛ0F





0 and119870



0by






=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680



=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1


top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070


5 Application of R0







119901ℎ



120583ℎ


365 times 70


30


8






0by using least




0119865 R0119872119865



R0






0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119860[

cosh (120578119905)

sinh (120578119905)

] = [

0

0

] (19)


119860

= [

119868ℎ0120578 minus 119887119901

ℎ119868ℎ0

+ (120574 + 120583ℎ) 119868V0 119868V0120578 minus 119887119901

ℎ119868V0 + (120574 + 120583

ℎ) 119868ℎ0


]

( 20 )


1205782119873ℎminus (119887119901V119873V + 119873

ℎ119887119901ℎ) 120578 minus 120583V (120574 + 120583

ℎ)119873ℎ

+ 1198872119901V119873V119901ℎ

(21)


and 1198682


R2

0119860= minus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V (119873V119873ℎ))

120583V (120574 + 120583ℎ)

120578 + 1

=

1198872119901ℎ119901V

120583V (120583ℎ + 120574)

119873V

119873ℎ

= R2

0

(22)

Therefore we have

R0119860

= radicminus

1205782

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120578 + 1 (23)


0 as



Note that R0119860




ℎ+


2120578 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (24)

R0119860

R0119865



0estimation





ℎ119889119905 (see in [14])


119868 (119905) = 120582119868ℎ0119890120582119905 with 119868

0= 119868 (0) = 120582119868

ℎ0(25)

and from (12) it is

119868 (119905) = 120578 (119868V0 cosh (120578119905) + 119868ℎ0sinh (120578119905))

with 1198680= 119868 (0) = 120578119868V0

(26)


119868 (119905) asymp 1205821198680119905 + 1198680

119868 (119905) asymp

1205782

120582

1198680119905 + 1198680

(27)




R20119860

R20119865

= 1 +

120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus 120582

120582 + 120583ℎ+ 120574

minus 1) (28)


2120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (29)

The ratio R20119860

R20119865


119887119901ℎ)119887119901V andR

2

0119860R20119865



ℎ)119887119901V lt (2120582+120583

ℎ+120574minus119887119901

ℎ)119887119901V

we have (2120582 + 120583ℎ+ 120574 minus 119887119901






0= 1198680= 119868(0) = 120582119868

ℎ0from (7) and119870(0) = 119870

0= 1198680=


119905



119868 (119905) = 120582119870 (119905) + (1 minus 120582)1198700

119868 (119905) = 1205782119870 (119905) + (1 minus 120578

2)1198700

(30)


120582

0 005 010 015 020

20

15

5

0

10120588

ℛ20Aℛ

20F = 1

ℛ20Aℛ

20F = 7

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

P = 5

ℛ20Aℛ

20F = 125


R20119865

in (28) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 20]



R20119860

R20119865

= 1 +

radic120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus radic120582

120582 + 120583ℎ+ 120574



2radic120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (32)


R20119865


+

12058212

(120583V +120583ℎ+120574+1)minus119887119901

ℎ)119887119901V = 120588

0andR2

0119860R20119865



0lt (2 minus 119887119901

ℎ)119887119901V for small 120582




R20119865


0






0 which




120582

00

2

4

6

8

10

005 010 015 020

120588

ℛ20Aℛ

20F = 1 ℛ2

0Aℛ20F = 7

ℛ20Aℛ

20F = 5

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

ℛ20Aℛ

20F = 125


R20119865

in (31) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 10]


10

20

30

40

50

ℛ0

estim

atio

n

ℛ0F





0 and119870



0by






=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680



=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1


top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070


5 Application of R0







119901ℎ



120583ℎ


365 times 70


30


8






0by using least




0119865 R0119872119865



R0






0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















120582

0 005 010 015 020

20

15

5

0

10120588

ℛ20Aℛ

20F = 1

ℛ20Aℛ

20F = 7

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

P = 5

ℛ20Aℛ

20F = 125


R20119865

in (28) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 20]



R20119860

R20119865

= 1 +

radic120582

120582 + 120583V(

119887 (119901ℎ+ 119901V120588) minus radic120582

120582 + 120583ℎ+ 120574



2radic120582 minus 119887119901ℎ

119887119901Vle 120588 le

2 minus 119887119901ℎ

119887119901V (32)


R20119865


+

12058212

(120583V +120583ℎ+120574+1)minus119887119901

ℎ)119887119901V = 120588

0andR2

0119860R20119865



0lt (2 minus 119887119901

ℎ)119887119901V for small 120582




R20119865


0






0 which




120582

00

2

4

6

8

10

005 010 015 020

120588

ℛ20Aℛ

20F = 1 ℛ2

0Aℛ20F = 7

ℛ20Aℛ

20F = 5

ℛ20Aℛ

20F = 9

ℛ20Aℛ

20F = 12

ℛ20Aℛ

20F = 075

ℛ20Aℛ

20F = 035

ℛ20Aℛ

20F = 3

ℛ20Aℛ

20F = 125


R20119865

in (31) for data set 119887 = 1 119901ℎ= 001

119901V = 025 120574 = 18 120583ℎ

= 1(3 times 70) 120583V = 130 120582 isin [0 020] 120588 isin

[0 10]


10

20

30

40

50

ℛ0

estim

atio

n

ℛ0F





0 and119870



0by






=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680



=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1


top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070


5 Application of R0







119901ℎ



120583ℎ


365 times 70


30


8






0by using least




0119865 R0119872119865



R0






0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

1205822

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

120582 + 1

The tor 120582 equation 119868(119905) = 1205821198680119905 + 1198680



=

1205822

120583V (120583ℎ + 120574)

+

(120583V + 120583ℎ+ 120574) 120582

120583V (120583ℎ + 120574)

+ 1


=

1198961205822

120583V (120583ℎ + 120574)

+

(120583V + 119896 (120583ℎ+ 120574)) 120582

120583V (120583ℎ + 120574)

+ 1


= minus

120582

120583V (120574 + 120583ℎ)

+

119887 (119901ℎ+ 119901V120588)

120583V (120574 + 120583ℎ)

radic120582 + 1


top Itop II

top IIItop IV

top V

Nov

05

200

8Ja

n 0

5 2

009

Mar

05

200

9M

ay 0

5 2

009

Jul

05 2

009

Sep

05

200

9N

ov 0

5 2

009

Jan

05

201

0M

ar 0

5 2

010

May

05

201

0Ju

l 05

201

0Se

p 0

5 2

010

Nov

05

201

0Ja

n 0

5 2

011

Mar

05

201

1M

ay 0

5 2

011

Jul

05 2

011

Sep

05

201

1N

ov 0

5 2

011

Jan

05

201

2M

ar 0

5 2

012

May

05

201

2Ju

l 05

201

2Se

p 0

5 2

012

Nov

05

201

2

Time (daily)

Num

ber o

f new

case

sof

den

gue

010203040506070


5 Application of R0







119901ℎ



120583ℎ


365 times 70


30


8






0by using least




0119865 R0119872119865



R0






0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















0for


0= 54 120582 = 0091 119870

0= 522

II 187 days II 6 days 120582 = 0009 1198680= 1153 120582 = 0019 119870

0= 1117

III 13 days III 6 days 120582 = 0190 1198680= 362 120582 = 0121 119870

0= 378

IV 17 days IV 5 days 120582 = 0038 1198680= 16 120582 = 0048 119870

0= 1554

V 85 days V 7 days 120582 = 0171 1198680= 293 120582 = 0105 119870

0= 31





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0009 [117] 104 114 [117] 120 129

IV 5 days 0038 [166] 114 155 [166] 177 206

I 5 days 0093 [256] 132 232 [256] 279 338

V 7 days 0171 [380] 154 338 [380] 418 516

III 6 days 0190 [389] 159 364 [389] 452 558





R0119872119865

R0119872119865

R0119872119865

R0119872119865

R0119872119865

119896 = 0 119896 = 075 119896 = 1 119896 = 125 119896 = 2

II 7 days 0019 [135] 107 129 [135] 141 158

IV 5 days 0048 [183] 117 169 [183] 196 230

I 5 days 0091 [254] 132 230 [254] 277 335

V 7 days 0105 [276] 136 249 [276] 301 366

III 6 days 0121 [302] 140 271 [302] 330 403





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 067 120588 = 089 120588 = 133 120588 = 196 120588 = 211

II 7 days 0009 [117] [117] 122 132 145 148

IV 5 days 0038 [166] 150 [166] 194 227 234

I 5 days 0093 [256] 170 203 [256] 317 330

V 7 days 0171 [380] 112 189 284 [380] 400

III 6 days 0190 [410] 067 173 283 389 [410]





R0119860

R0119860

R0119860

R0119860

R0119860

120588 = 061 120588 = 101 120588 = 147 120588 = 160 120588 = 174

II 7 days 0019 [135] [135] 227 299 316 334

IV 5 days 0048 [183] mdash [183] 306 332 359








120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















120588R0119872119865 R

0Important notes

from (28) for 119896 isin (0 2]

II 7 days 0009 (0034 796) (104 129] (002 477) R0119865

isinR0119872119865

subR0

IV 5 days 0038 (026 796) (114 206] (016 477) R0119865

isinR0119872119865

subR0

I 5 days 0093 (0701 796) (132 338] (042 477) R0119865

isinR0119872119865

subR0

V 7 days 0171 (133 796) (154 516] (080 477) R0119865

isinR0cap

R0119872119865

III 6 days 0190 (148 796) (159 558] (089 477) R0119865

isinR0cap

R0119872119865





120588R0


from (31) for 119896 isin (0 2]

II 7 days 0019 (107 796) (064 477) (107 158) R0119865

isinR0119872119865

subR0

IV 5 days 0048 (171 796) (102 477) (117 230) R0119865

isinR0119872119865

subR0

I 5 days 0091 (237 796) (143 477) (132 335) R0119865

isinR0cap

R0119872119865

V 7 days 0105 (255 796) (153 477) (136 366) R0119865

isinR0cap

R0119872119865

III 6 days 0121 (274 796) (165 477) (140 403) R0119865

isinR0cap

R0119872119865





120588R0

R0119860


II 7 days 0009 (0034 796) (002 477) (101 233) R0119865

isinR0119860

subR0

IV 5 days 0038 (026 796) (016 477) (116 432) R0119865

isinR0119860

subR0

I 5 days 0093 (0701 796) (042 477) (175 659) R0119865

isinR0cap

R0119860

V 7 days 0171 (133 796) (080 477) (283 871) R0119865

isinR0cap

R0119860

III 6 days 0190 (148 796) (089 477) (310 913) R0119865

isinR0cap

R0119860





120588R0

R0119860


II 7 days 0019 (107 796) (064 477) (237 793) R0119865

isinR0R0119860

IV 5 days 0048 (1704 796) (102 477) (352 970) R0119865

isinR0R0119860

I 5 days 0091 (237 796) (143 477) (479 1114) R0119865

isinR0R0119860

V 7 days 0105 (255 796) (153 477) (512 1146) R0119865

isinR0R0119860

III 6 days 0121 (274 796) (165 477) (548 1178) R0119865

isinR0R0119860


and R0119860





0estimation


found was 410R0119872119865




both R0119872119865


R0119872119865


magnitude R0119860


and R0119872119865





0119865= R0119872119865

for 119896 = 1R0119872119865

lt R0119865

for 119896 lt 1 and R0119865

lt R0119872119865


0119872119865


0119865






R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















R0 and R




for 119896 isin (0 2]

II 7 days 0009 117 (104 129] R0119865

isinR0119872119865

capR0cap

R0119860

IV 5 days 0038 166 (114 206] R0119865

isinR0119872119865

capR0cap

R0119860

I 5 days 0093 256 (132 338] R0119865

isinR0119872119865

capR0cap

R0119860

V 7 days 0171 380 (154 516] R0119865

isinR0119872119865

capR0cap

R0119860

III 6 days 0190 389 (159 558] R0119865

isinR0119872119865

capR0cap

R0119860




R0 and R




for 119896 isin (0 2]

II 7 days 0019 135 (107 158] R0119865

isin (R0119872119865

capR0)

R0119860

IV 5 days 0048 183 (117 230] R0119865

isin (R0119872119865

capR0)

R0119860

I 5 days 0091 254 (132 335] R0119865

isin (R0119872119865

capR0)

R0119860

V 7 days 0105 276 (136 366] R0119865

isin (R0119872119865

capR0)

R0119860

III 6 days 0121 302 (140 403] R0119865

isin (R0119872119865

capR0)

R0119860





= R0119872119865






0119860 Particular




andR0119872119865


R0119865

= R0119860


0119860= R0119872119865

=

R0119865










R0119865




0119872119865and R


0




R0119865




0119872119865and R

0119860are


0119865is






0119860(see Tables





0119865value


R0119860 and R


R0119865







0190 and 120588 = 067 we have R0119860


0are the same




0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















0

1

2

3

4

5

05 1 15 2



ℛ0MF 120582 = 0038ℛ0MF 120582 = 0190ℛ0MF



ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)



k

14

12

08

06

04

02

00 05 1 15 2


ℛ0M

Fℛ

0F






R0119865



0119865estimates






0119860estimation



12

10

8

6

4

2

00 1 2 3 4 5 6 7

ℛ0A

(the s

econ

d co

nstr

uctio

nof

ℛ0

estim

atio

n)

ℛ0A 120582 = 0009ℛ0A 120582 = 0093ℛ0A 120582 = 0190ℛ0A 120582 = 0038ℛ0A


Threshold ℛ0A = 1



120588

0 1 2 3 4 5 6 7


25

2

15

1

05

0

ℛ0Aℛ

0F

ℛ0Aℛ0F 120582 = 0009ℛ0Aℛ0F 120582 = 0093ℛ0Aℛ0F

ℛ0Aℛ0F 120582 = 0038


ℛ0Aℛ0F 120582 = 0171





R0119865



0119865R0119872119865

andR0119860

forR0estimationR

0magnitude


0119865 We proposeR

0119872119865andR


withR0119865forR






1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















1

2

3

4


ℛ0M

F(th

e firs

t con

struc

tion

ofℛ

0es

timat

ion)







k

0 05 1 15 2

12

08

09

10

11

13

06

07

05

Threshold



ℛ0M

Fℛ

0F


R0119865




0119872119865and R

0119860




8

2

4

6

10


ℛ0A 120582 = 0019ℛ0A 120582 = 0091ℛ0A 120582 = 0121ℛ0A 120582 = 0048ℛ0A


Threshold ℛ0A = 1

ℛ0A



120588

0 1 2 3 4 5 6 7

4

1

2

3

5


ℛ0Aℛ

0F



Threshold


R0119865








0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















0119865




0= 285(277 370)


= 265(250 330)



0











0119872119865and R

0119860


0119865





0119860 is shown


0has limitations


0




6 Conclusion








0119872119865R0119865

and R0119860

R0119865





Acknowledgments


References




























0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of







































0and effec-







of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















research article estimation of the basic reproductive

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