Research Article Estimation of the Basic Reproductive Ratio for Dengue Fever at the Take-Off Period of Dengue Infection

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1 Computational and Matematical Metods in Medicine Volume 15, Article ID 6131, 14 pages ttp://dx.doi.org/1.1155/15/6131 Researc Article Estimation of te Basic Reproductive Ratio for Dengue Fever at te Take-Off Period of Dengue Infection Jafaruddin, 1, Sapto W. Indratno, 1 Nuning Nuraini, 1 Asep K. Supriatna, 3 and Edy Soewono 1 1 Departemen Matematika, FMIPA, Institut Teknologi Bandung, Bandung, Indonesia Jurusan Matematika, FST, Universitas Nusa Cendana, Kupang, Indonesia 3 Jurusan Matematika, FMIPA, Universitas Padjadjaran, Bandung, Indonesia Correspondence sould be addressed to Jafaruddin; jafaruddin.amid@yaoo.com Received December 14; Revised 6 July 15; Accepted 7 July 15 Academic Editor: Cung-Min Liao Copyrigt 15 Jafaruddin et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Estimating te basic reproductive ratio R of dengue fever as continued to be an ever-increasing callenge among epidemiologists. In tis paper we propose two different constructions to estimate R wic is derived from a dynamical system of ost-vector dengue transmission model. Te construction is based on te original assumption tat in te early states of an epidemic te infected uman compartment increases exponentially at te same rate as te infected mosquito compartment (previous work). In te first proposed construction, we modify previous works by assuming tat te rates of infection for mosquito and uman compartments migt be different. In te second construction, we add an improvement by including more realistic conditions in wic te dynamics of an infected uman compartments are intervened by te dynamics of an infected mosquito compartment, and vice versa. We apply our construction to te real dengue epidemic data from SB Hospital, Bandung, Indonesia, during te period of outbreak Nov. 5, 8 Dec. 1. We also propose two scenarios to determine te take-off rate of infection at te beginning of a dengue epidemic forconstructionofteestimatesofr : scenario I from equation of new cases of dengue wit respect to time (daily) and scenario II from equation of new cases of dengue wit respect to cumulative number of new cases of dengue. Te results sow tat our first construction of R accommodates te take-off rate differences between mosquitoes and umans. Our second construction of te R estimation takes into account te presence of infective mosquitoes in te early growt rate of infective umans and vice versa. We conclude tat te second approac is more realistic, compared wit our first approac and te previous work. 1. Introduction Dengue is a mosquito-borne viral disease found in more tan 1 countries around te world wic are mostly located in tropical and subtropical countries. Tis disease is transmitted troug te bites of female Aedes aegypti. Inrecentyears, dengue transmission as increased predominantly in urban and semiurban areas and as become a major international public ealt concern. Controlling of dengue fever as been conducted continuously, but te spread of dengue virus is still increasing in many countries. Many efforts ave been conductedtocontroltespreadoftevirus,forinstance,a reduction in te population of Aedes aegypti in te field [1]. Fumigationmetodsavebeenusedtoreducemosquitoes, and te use of temepos as been utilized to reduce te larvae. Until recently, tere was no vaccine against any of te four virus serotypes (DEN-1, DEN-, DEN-3, and DEN-4). To cure patients, treatment in ospitals is usually given in te form of supportive care, wic includes bed rest, antipyretics, and analgesics []. Matematical models ave proved to be useful tools in te understanding of dengue transmission [3 5]. So far te dynamics of te dengue transmission are still an interesting issue in epidemiological modeling. Te first model of dengue transmission and te stability analysis of equilibrium points are sown in [6] in wic te basic reproductive ratio R is constructed from te stability condition of te disease-free equilibrium. Te general concept of R canbeseenin[7,8]wicwasadaptedtovarious infectious disease models [9, 1]. Te difficulty in using R for measuring te level of dengue endemicity in te field is tat R often depends on several parameters (biological, environmental, transmission, etc), wic are difficult to be

2 Computational and Matematical Metods in Medicine obtained from te field. Wit limited information about mosquitos suc as te mosquito population size te estimate of R can not be computed from te dynamical model only. Wit daily uman incidence being te only available data, it is natural to ask ow to extract relevant parameters from te data to fit in te model. A simple approac as been acieved by assuming tat bot a uman and mosquito undergo linear growt at te early state of infection. As sown in [11 14] te estimation does not distinguis between te growt rate of an infected uman and te growt rate of infected mosquitoes and does not yet consider ow te interaction between infected mosquitoes and infected umans influences te early growt of a dengue epidemic. Tere are oter problems tat are treated in a recent paper [15] wic examines two models vector-borne infections, namely, dengue transmission. In tis paper, we construct R estimation by taking into account te difference between te growt rate of an infected uman and infected mosquito and te effect of te interaction between tem and te caracteristics of dengue incidence data. Te existing data do not specify te infection status (age of infection) of eac patient. Hence we assume tat people wo come to te ospital sould be identified as dengue patient at te end of incubation period (approximately at te 7t day after contact wit infected mosquito). Also, we assume tat tere are no available alternative osts as blood sources and tere are no deat and recovery in te early days of dengue infection. Te main purpose of tis paper is to construct R estimation of te real conditions in te field, in wic only te daily incidence data are available. Based on te dengue transmission model in [6], we build two different constructions for R estimation and used tem for estimating te value of R for dengue incidence data between tedatesnov.5,8,and1fromsbhospital,indonesia. Te organization of te paper is given as follows. In Section a dynamical system of a ost-vector transmission model is introduced. Based on tis dynamical system we derive te basic reproductive ratio of te model. Te constructions of te proposed R estimation are presented in Section 3. Formulations of te take-off rate of dengue infection are presented in Section 4. We apply te formula to te real data of dengue incidence from SB Hospital, Bandung, Indonesia. All te numerical results are given in Section 5 along wit te insigts and interpretation from te numerical results. We provide conclusions in Section 6.. Basic Reproductive Ratio of te Dengue Transmission Model Generally, for a vector-borne disease, R was understood as tenumberofpersonswowouldbeinfectedfromasingle person initially infected by a mosquito [3, 16]. In te ost-vector system, te basic reproductive ratio R is defined as te expected number of secondary infections resulting directly from a single infected individual, in a virgin population, during te infection period [7]. Te general ost-vector dengue transmission model was conducted in [6]. Here, we assume tat tere are no alternative osts available, as blood sources m=for system in [6]. Modification of te dengue A S μ S I bp S N I μ I I bp S I N μ I γi S μ S Figure 1: Te diagram of dengue transmission model. A v R μ R transmission model in [6], is scematically represented by te diagraminfigure1,were A : recruitment rate of ost population; N :numberofostpopulations; S : susceptible ost population size; I : infected ost population size; R : recovered/immunes ost population size; p : successful transmission probability witin ost population; μ : birt/deat rate of ost population; γ: recovery rate of ost population; A V : recruitment rate of vector population; N V :numberofvectorpopulations; S V : susceptible vector population size; I V : infected vector population size; p V : transmission probability witin te vector population; μ V : deat rate of vector population; b: biting rate of vector population. Te corresponding model of dengue transmission in te sort perioddengueinfectionisgivenin[17]asfollows: ds dt =A bp S I V μ N S, di dt = bp S I V γi N μ I, dr =γi dt μ R, ds V dt =A V bp VS V I μ N V S V, di V dt = bp VS V I μ N V I V. Here, we give a sort explanation for system (1) as follows. Te effective contact rate to uman bp is te average number (1)

3 Computational and Matematical Metods in Medicine 3 of contacts per day tat would result in infection, if te vector is infectious. Furtermore, te effective contact rate bp V defines te average number of contacts per day tat effectively transmit te infection to vectors. First of all, we determine equilibrium points for te system (1). Because N and N V are assumed to be constant ten from system (1) ones obtain S +I +R =N =A /μ and S V +I V =N V =A V /μ V. Consequently, we obtain a diseasefree equilibrium point of system (1), wic is given by E =(S = A,I μ =,R =,S V = A V,I μ V =), () V and te endemic equilibrium point in te form of were S = μ V (γ + μ )N +μ VN N V p μ N +bp N V, E 1 =(S,I,R,S V,I V ), (3) I = μ Vμ N ((b p p V /μ V (γ + μ )) (N V /N ) 1), bp V (μ +bp ρ) R = μ VN γ((b p p V /μ V (γ + μ )) (N V /N ) 1), bp V (μ +bp ρ) S V = p μ N +bp V p N V, bp (1 + bp V μ ) I V = μ Vμ N (γ + μ )((b p p V /μ V (γ + μ )) (N V /N ) 1). bp (μ V (γ + μ )+μ bp V ) Next, we derive te basic reproductive ratio R. R is often conveniently found using a next generation metod [7, 8, 18, 19]. We obtain te next generation matrix of system (1), at E,asfollows (4) bp NGM = [ μ V [ bp V N V ]. (5) [(γ + μ ) N ] Based on te definition in [], NGM represents tat te number of bp /μ V is te generation factor of dengue transmission, from mosquito to uman. It means tat one mosquito infects bp umans per unit of time during its infection period 1/μ V, and te number of (bp V /(γ + μ ))(N V /N ) is te generation factor of dengue transmission, from uman to mosquito. Tis represents tat one uman infects bp V (N V /N ) mosquitoes during one s infection period 1/(μ +γ). NGM in (5) ave two eigen values, tat are ± (b p p V /μ V (γ + μ ))(N V /N ). Terefore te basic reproductive ratio of system (1) is te largest eigen value of NGM, tat is N V R = b p p V. (6) μ V (γ + μ ) N It is clear tat te endemic equilibrium E 1 in (3) exists if R > 1.IntenextsectionweproposetwoconstructionsofanR estimation, at te initial growt of dengue infection, based on incidence data. 3. Construction for R Estimation Predicting a basic reproductive ratio is not easily done in te field. In te real-life situation, information about te number of mosquitoes, precisely te ratio between mosquito and umanpopulation,islimited.basedonteincidencedata,we will estimate some related parameters, for te construction of te basic reproductive ratio R, from dengue incidence data. Tis construction is motivated by previous work in [1], wic assumes tat te infective growt rate is linear at te early state of infection. Te exponential growt of te infective compartment, at te early state of infection, is commonly found in te estimation of te basic reproductive ratio. All estimation of R,inpreviousworks[11,1,14, 1,],avebeenconductedbasedonteassumptiontat K(t) e t varies, were K(t) is te cumulative number of dengue cases, and is called te force of infection [1, 14]. Cowell et al. [11] called te initial growt rate of dengue epidemic. Here, we will call te take-off rate of dengueinfection.intefollowingsection,webuildtwo constructions to estimate R in (6), wic is derived from a dynamical system of te ost-vector dengue transmission model in system (1). First of all we assume te construction of R estimation from equation (3) in [1]. Te construction of R estimation at E is based on te assumptions tat te numbers of infected uman population I andinfectedmosquitopopulationi V grow exponentially at te same rate at te sort time period relative to eac oter: I (t) I e t, (7) I V (t) I V e t, were I and I V are constant and is te take-off rate of te initial growt dengue epidemic. Moreover, te number of nonsusceptible osts and vectors can be assumed to be negligible and by assuming (7) is like solution oflinearized system (1) for R >1. Tese assumptions are also used in [11, 14] for calculating te force of infection from te spatial data of a dengue epidemic. Next, by substituting (7) wit (1), and assuming tat S N and V V N V at te early state of an epidemic, we get ( +1)I γ+μ = bp I γ+μ V, (8) ( +1)I μ V = bp V N V I V μ V N. (9) Multiplying (8) and (9), and using (6), one gets te construction of te R estimation as follows: R F =( μ +γ +1)( +1)= b p p V N V μ V μ V (μ +γ) N (1) = R,

4 4 Computational and Matematical Metods in Medicine wic is equivalent to R F = μ V (μ +γ) + (μ V +μ +γ) +1. (11) μ V (μ +γ) Te value of R F in (11) is used to estimate R in (6), were is estimated from te dengue incidence data. Note tat since >,tenr F >1. Note also tat a small increase of te value in te interval <<1causes a significant increase of te value R F. Next we introduce our construction of R estimation. Te first construction is derived under te assumption tat, at te beginning of dengue infection, te take-off rate of te ost and vector varies differently; tat is I (t) I e t, I V (t) I V e kt, (1) were k > isterateofinfectedmosquitoesperuman index. Using te same process as in te derivation of R F, one obtains R MF =( μ +γ +1)(k +1)= b p p V N V μ V μ V (μ +γ) N (13) = R or, equivalently, k R MF = μ V (μ +γ) + (μ V +k(μ +γ)) +1. (14) μ V (μ +γ) Note tat R F is a special case from R MF ;tatis,ifk=1 ten R F = R MF.InrealityR MF is more realistic tan R F. From (11) and (14) we obtain R MF (k 1) R =1+. (15) F +μ V Te comparison between R MF and R F is sown as te level set of (15) in Figure. Figure sows tat te variation in take-off rates migt significantly cange te values of R F and R MF. Bot estimations are derived under te assumption tat te pure exponential growt pase, of an epidemic, increases or as a fluctuating increase. Accordingly te estimates of R, obtained from R F and R MF, are affected by biasness; tat is, te derivations of R F and R MF use te extreme assumption in te early dengue epidemic tat te dynamics of an infected uman are not yet intervened by te presence of an infected mosquito s influence, and vice versa. Te second construction of R estimation is done in order to revise te first construction. We assume tat te exponential growt of te infective ost, at te early state of infection, is sligtly affected by te growt of te infective vector, and vice versa. Te main problem ere is to find te best fit interval (ere we use te terminology take-off period) were exponential growt occurs. Statistically, tis situation canberelatedtotelinearpasefitinasorttimeperiod. 15 k R MF /R F =1 R MF /R F =.35 R MF /R F =.75 R MF /R F = 1.5 R MF /R F =3 R MF /R F =5 R MF /R F =7 Figure : Level set for R MF /R F in (15) for data set {μ V =1/3, [,.], k [, ]}. We define te initial growt in a period of time as te initial value problem; tat is, We obtain I (t) =I V (ae ηt +be ηt ) +I (ce ηt +de ηt ), I V (t) =I (pe ηt +qe ηt ) (I (),I V ()) =(I,I V ), +I V (re ηt +se ηt ), (I (),I V ()) =(ηi V,ηI ). (16) {a = 1,b= 1,c= 1,d= 1,p= 1,q= 1,r= 1,s = 1 (17) } suc tat te solution of (16) is obtained; namely, I (t) =I cos (ηt) + I V sin (ηt), I V (t) =I V cos (ηt) + I sin (ηt). (18) Te main idea from (18) is to extract te relevant parameter(s) during te early state of infection, in wic linear growt rate of infection is assumed to take place. Next we substitute (18) wit te expression for te derivative of I and I V in (1) by simple algebraic manipulation; we ten obtain

5 Computational and Matematical Metods in Medicine 5 a omogeneous system of equations in cos(ηt) and sin(ηt), namely, were coefficient matrix is A cos (ηt) A[ sin (ηt) ]=[ ], (19) =[ I η bp I +(γ+μ )I V I V η bp I V +(γ+μ )I () ]. I V η bp V ρi V +μ V I I η bp V ρi +μ V I V A nontrivial solution from (19) exists if A =,wicgives η N (bp V N V +N bp )η μ V (γ + μ )N +b p V N V p (1) as te coefficient of I and I V must be equal to zero. Equivalently, from (1), we obtain R A = η μ V (γ + μ ) + b(p +p V (N V /N )) η+1 μ V (γ + μ ) N V = b p p V = R μ V (μ +γ) N. Terefore we ave R A = η () μ V (γ + μ ) + b(p +p V ρ) μ V (γ + μ ) η+1, (3) were ρ=n V /N is te known mosquitoes per person index. We obtain (3), as a new construction for estimation R,as afunctionofη.teconstructionofr in (3) is related to a sort time interval were linear growt may still take place. Here,wewillestimatetevalueofη during te early state of infection, in wic a linear growt of infection is assumed to take place. Note tat R A increases wit respect to η, η b(p +p V ρ)/. Based on te definition tat η is probability, per unit of time, a susceptible becomes infected at te beginning of a dengue epidemic (see [7]), wit assumption tat b(p + p V ρ)/ < 1. Tis assumption means tat te maximum takeoff rate of dengue virus infection is defending on te index mosquito per uman. Tis condition implies η bp bp V ρ bp bp V. (4) R A /R F ratio in te early pase of dengue epidemics could be significantly larger or smaller tan one depending on te value of ρ, in(4),andtevalueof. InSection4, we determine η as a function of at te beginning of a dengue epidemic. We will furter analyze te level set of te ratio R A /R F. Considering tat te value of R estimation depends on te value of take-off rate,wediscussametod to formulate te take-off rate in te following section. 4. Formulation of te Take-Off Rate Here we derive two scenarios for te derivation of te take-off rate (t.o.r.) at te beginning of every take-off period (t.o.p.) of dengue infection. Bot scenarios are based on te same assumptions, but wit a sligtly different implementation for real-life dengue epidemics. Terefore, te main problem ere is to find te best fit interval (ere we use te terminology take-off period ) were exponential growt occurs. Te first scenario is done under te assumption tat, during early infection, recovery and natural deat do not yet occur. Note tat te original data taken from te ospital is in te form of daily new cases. Let I(t) be te number of new cases of infection; we ten ave I(t) = di /dt (see in [14]). Terefore, from (7), it is and, from (1), it is I (t) =I e t, wit I =I() =I (5) I (t) =η(i V cos (ηt) + I sin (ηt)), wit I =I() =ηi V. (6) By using te Taylor expansion around = for (5) and around η=for (6), we obtain I (t) I t+i, I (t) η I t+i, (7) respectively. Consequently, we ave a relation tat = η. Tus η at R A in (3) can be replaced by. Bycomparing (11) and (3), we ave R A R F =1+ ( b(p +p V ρ) 1). (8) +μ V +μ +γ Also η in(4)canbereplacedby suc tat we obtain bp bp V ρ bp bp V. (9) Te ratio R A /R F 1,ifandonlyifρ ( + μ +γ bp )/bp V,andR A /R F <1,ifandonlyifρ < ( + μ +γ bp )/bp V.Becauseof( bp )/bp V <(+μ +γ bp )/bp V, we ave ( + μ +γ bp )/bp V, wic belongs to te range of ρ for small. Te second scenario is conducted under te same assumption as te first scenario. But, ere, we regress between te number of new cases in a day I(t) = di /dt and cumulative number of new cases K(t),wereweaveteinitialvalue K() = K =I = I() = I from (7) and K() = K =I = I() = ηi V from (18). Terefore, by using integral equation K(t) K() = t I(τ)dτ wit I(τ) from te first equation in (7) and I(τ) from te first equation in (18), we obtain I (t) =K(t) + (1 ) K, I (t) =η K (t) +(1 η )K, (3)

6 6 Computational and Matematical Metods in Medicine ρ 1 ρ R A /R F =1 R A /R F =7 R A /R F =9 R A /R F =1 R A /R F =.75 R A /R F =.35 R A /R F =3 P=5 R A /R F = 1.5 Figure 3: Level set R A /R F in (8) for data set {b = 1, p =.1, p V =.5, γ = 1/8, μ =1/(3 7),μ V = 1/3, [,.], ρ [, ]}. respectively. It is clear tat η =so tat η in R A at (3) canbereplacedby. If (3) is divided by (11), we obtain R A R F =1+ ( b(p +p V ρ) ). (31) +μ V +μ +γ Also η in(4)canbereplacedby suc tat we obtain bp bp V ρ bp bp V. (3) We note tat te ratio R A /R F 1if and only if ρ (3/ + 1/ (μ V +μ +γ+1) bp )/bp V =ρ and R A /R F <1if and only if ρ<ρ. Note also tat ρ <( bp )/bp V for small. Figures3and4sowtelevelsetsforsomevaluesof te ratio R A /R F, for bot scenarios of construction of. It sows tat for small, R A /R F increases faster, wit respect to ρ, in te second scenario (Figure 4) tan in te first scenario (Figure 3). Summary of te construction of R estimation for te first and te second scenario of is given in Table 1. Te magnitude of R A in Figure 5 indicates tat te second scenario is rising faster tan R A of te first scenario at te beginning of te growing epidemic of dengue fever, altoug te rate of infection is very small. Tis means tat te presence of mosquitoes in early dengue infection directly affects te uman dengue infection. Furtermore, Figure 5 sows tat te second scenario of construction of R,wic is summarized in Table 1, is increasing faster tan te first scenario. Terefore, we obtain tat te second scenario is a more realistic construction of R tan te first one at te beginning of te sort period of dengue virus infection R A /R F =1 R A /R F =7 R A /R F =.35 R A /R F =5 R A /R F =.75 R A /R F =9 R A /R F = 1.5 R A /R F =1 R A /R F =3 Figure 4: Level set R A /R F in (31) for data set {b = 1, p =.1, p V =.5, γ = 1/8, μ =1/(3 7),μ V = 1/3, [,.], ρ [, 1]}. R estimation R F R A for scenario I, ρ=48, 74 Proportion for take-off rate, R MF for k=1 R A for scenario II, ρ=48, 74 Figure 5: Te comparison of R estimation. In te next section, we apply least square metod to determine estimation of, I,andK from dengue incidence data based on (7) and (3). We calculated, I,andK by using least square metod wic differs from te estimation tat as been conducted in [14]. Tey assume tat te number of cases of dengue at t = is equal to one, but, ere, we estimate from te dengue incidence data. In te next section, we examine te dependence of te basic reproductive ratio on te take-off rate, at te beginning of te take-off period of dengue infection, for daily cases of dengue from SB Hospital in Bandung Indonesia.

7 Computational and Matematical Metods in Medicine 7 Table 1: Summary of te construction of R and estimation for te first scenario. Te first scenario for R estimation Based on [1] assumption R F = μ V (μ +γ) + (μ V +μ +γ) +1 μ V (μ +γ) Te first construction R MF = k μ V (μ +γ) + (μ V +k(μ +γ)) +1 μ V (μ +γ) Te second construction R A = μ V (γ + μ ) + b(p +p V ρ) μ V (γ + μ ) +1 Te t.o.r. equation I(t) = I t+i Te second scenario for R estimation Based on [1] assumption R F = μ V (μ +γ) + (μ V +μ +γ) +1 μ V (μ +γ) Te first construction R MF = k μ V (μ +γ) + (μ V +k(μ +γ)) +1 μ V (μ +γ) Te second construction R A = μ V (γ + μ ) + b(p +p V ρ) μ V (γ + μ ) +1 Te t.o.r. equation I(t) = K(t) + (1 )K Number of new cases of dengue t.o.p. I t.o.p. II t.o.p. III t.o.p. IVt.o.p. V Nov. 5, 8 Jan. 5, 9 Mar. 5, 9 May. 5, 9 Jul. 5, 9 Sep. 5, 9 Nov. 5, 9 Jan. 5, 1 Mar. 5, 1 May. 5, 1 Jul. 5, 1 Sep. 5, 1 Nov. 5, 1 Jan. 5, 11 Mar. 5, 11 May. 5, 11 Jul. 5, 11 Sep. 5, 11 Nov. 5, 11 Jan. 5, 1 Mar. 5, 1 May. 5, 1 Jul. 5, 1 Sep. 5, 1 Nov. 5, 1 Time (daily) Figure 6: A time series of te number of daily new cases of dengue infection in te SB Hospital is from Nov. 5, 8, to Dec. 1. Te cutoffperiodofnewcasesofdengueisinsidetesolidbox.every t.o.p. contains i.t.o.p. of dengue infection. Table : Bioepidemiological data of uman and mosquito. Symbol Definition Value b Biting rate (day 1 ) 1 p Effective contact rate to uman.1 p V Effective contact rate to mosquito.5 μ Human mortality rate (day 1 ) μ V Mosquito mortality rate (day 1 ) 1 3 γ Recovering rate of uman (day 1 ) 1 8 ρ Mosquito per uman index 7.96 k Proportion of mosquito per uman index 5. Application of R Estimation to te Real-Life Dengue Epidemic Tis section presents te application of te metod to te data of dengue incidence from SB ospital, te estimation value of te t.o.r., te value basic reproductive ratio, and teir implication. Calculation of is based on te assumption tat at te beginning of te infection natural deat and recovery avenotyetoccurred.tisassumptioncanbetakenbefore te eigt day of incidence. Data is divided into five takeoff periods (t.o.p.) of infections, wic is represented inside te solid box in Figure 6. Eac t.o.p. contains an initial takeoff period (i.t.o.p.) of dengue infection. Here, we define te i.t.o.p. as being witin te range of te fourt and te sevent days of te dengue incidence. Te correspondents of te bioepidemiological parameter from uman and mosquito are given in Table. Figure6sowstefivepossibilitiesfortet.o.p.Tet.o.p. is defined by te minimum to maximum values of r-square (r ). Te number of days for every i.t.o.p. of incidence data depends on te value of r criteria (approximately at te 4 7tdayaftercontactwitteinfectedmosquito).Meanwile, te i.t.o.p. is defined by te minimum to maximum values of r before te eigt day. Next we determine te rate of te i.t.o.p. of dengue infection (t.o.r.) for every t.o.p. Te value of iscalculatedatintervalsoffourtosevendaysforte dengue of dengue infection by using two different scenarios in respect to te incidence data and te value of parameter in Table. In te first scenario we calculate te values of and I by using least square metod for (7), and, in te second scenario, we calculate te values of and K by using least square metod for (3), for every i.t.o.p. Tose values of, I, and K are summarized in Table 3. Using take-off rate values of infection in Table 3, we calculate te value of ρ, R F, R MF,andR A,wicare givenintables4 7aswellasteintervalsof ρ, R MF, R,

8 8 Computational and Matematical Metods in Medicine Take-off period Table 3: Te value of and I from te first scenario and te value of and K for te second scenario, respectively. Initial take-off period Te values of and I for te first scenario Te values of and K for te second scenario I: 13 days I: 5 days =.93, I = 5.4 =.91, K = 5. II: 187 days II: 6 days =.9, I = =.19, K = III: 13 days III: 6 days =.19, I = 3.6 =.11, K = 3.78 IV: 17 days IV: 5 days =.38, I = 16 =.48, K = V: 85 days V: 7 days =.171, I =.93 =.15, K = 3.1 Table 4: Te values of R F for a given value of at te i.t.o.p. and te corresponding values of R MF for different values of k for te first scenario. R Te n days at te i.t.o.p. R MF R MF R MF R MF R MF F k = k =.75 k = 1 k = 1.5 k = II: 7 days.9 [1.17] [1.17] IV: 5 days.38 [1.66] [1.66] I: 5 days.93 [.56] [.56] V: 7 days.171 [3.8] [3.8] III: 6 days.19 [3.89] [3.89] Table 5: Te values of R F for a given value of at te i.t.o.p. and te corresponding values of R MF for different values of k for te second scenario. R Te n days at te i.t.o.p. R MF R MF R MF R MF R MF F k = k =.75 k = 1 k = 1.5 k = II: 7 days.19 [1.35] [1.35] IV: 5 days.48 [1.83] [1.83] I: 5 days.91 [.54] [.54] V: 7 days.15 [.76] [.76] III: 6 days.11 [3.] [3.] Table 6: Te values of R F for every value of at eac initial t.o.p. and te values of R A for every cange in te value of and ρ for te first scenario for calculation of. R Te n days at te i.t.o.p. R A R A R A R A R A F ρ =.67 ρ =.89 ρ = 1.33 ρ = 1.96 ρ =.11 II: 7 days.9 [1.17] [1.17] IV: 5 days.38 [1.66] 1.5 [1.66] I: 5 days.93 [.56] [.56] V: 7 days.171 [3.8] [3.8] 4. III: 6 days.19 [4.1] [4.1] Table 7: Te values of R F for every value of at eac initial t.o.p. and te values of R A for every cange in te value of and ρ for te second scenario for calculating of. R Te n days at te i.t.o.p. R A R A R A R A R A F ρ =.61 ρ = 1.1 ρ = 1.47 ρ = 1.6 ρ = 1.74 II: 7 days.19 [1.35] [1.35] IV: 5 days.48 [1.83] [1.83] I: 5 days.91 [.54] [.54] V: 7 days.15 [.76].7 [.76] 3.3 III: 6 days.11 [3.] [3.]

9 Computational and Matematical Metods in Medicine 9 Table8:Summaryofte R and R MF for te first scenario of te value of and ρ in (8). Te n days at te i.t.o.p. ρ RMF R Important notes from (8) for k (,] II: 7 days.9 (.34, 7.96) (1.4, 1.9] (., 4.77) R F R MF R IV: 5 days.38 (.6, 7.96) (1.14,.6] (.16, 4.77) R F R MF R I: 5 days.93 (.71, 7.96) (1.3, 3.38] (.4, 4.77) R F R MF R V: 7 days.171 (1.33, 7.96) (1.54, 5.16] (.8, 4.77) R F R R MF III: 6 days.19 (1.48, 7.96) (1.59, 5.58] (.89, 4.77) R F R R MF Note.TevalueofR F from Table 6. Table9:Summaryofte R and R MF for te second scenario of te value of and ρ in (31). Te n days at te i.t.o.p. ρ RMF R Important notes from (31) for k (,] II : 7 days.19 (1.7, 7.96) (.64, 4.77) (1.7, 1.58) R F R MF R IV: 5 days.48 (1.71, 7.96) (1., 4.77) (1.17,.3) R F R MF R I: 5 days.91 (.37, 7.96) (1.43, 4.77) (1.3, 3.35) R F R R MF V: 7 days.15 (.55, 7.96) (1.53, 4.77) (1.36, 3.66) R F R R MF III: 6 days.11 (.74, 7.96) (1.65, 4.77) (1.4, 4.3) R F R R MF Note.TevalueofR F from Table 7. Table1:SummaryofteR and R A for te first scenario of te value of and ρ in (8). Te n days at te i.t.o.p. ρ R RA Important notes from (8) II: 7 days.9 (.34, 7.96) (., 4.77) (1.1,.33) R F R A R IV: 5 days.38 (.6, 7.96) (.16, 4.77) (1.16, 4.3) R F R A R I: 5 days.93 (.71, 7.96) (.4, 4.77) (1.75, 6.59) R F R R A V: 7 days.171 (1.33, 7.96) (.8, 4.77) (.83, 8.71) R F R R A III: 6 days.19 (1.48, 7.96) (.89, 4.77) (3.1, 9.13) R F R R A Note.TevalueofR F from Table 6. Table 11: Summary of te R and R A for te second scenario of te value of and ρ in (31). Te n days at te i.t.o.p. ρ R RA Important notes from (31) II: 7 days.19 (1.7, 7.96) (.64, 4.77) (.37, 7.93) R F R \ R A IV: 5 days.48 (1.74, 7.96) (1., 4.77) (3.5, 9.7) R F R \ R A I: 5 days.91 (.37, 7.96) (1.43, 4.77) (4.79, 11.14) R F R \ R A V: 7 days.15 (.55, 7.96) (1.53, 4.77) (5.1, 11.46) R F R \ R A III: 6 days.11 (.74, 7.96) (1.65, 4.77) (5.48, 11.78) R F R \ R A Note.TevalueofR F from Table 7. and R A and teir important notes, wic are given in Tables Te interpretation of te result in Tables 9 13, for instance, on R F at t.o.p. III, is tat eac infective person infected 3.89 oter persons via te mosquito vector at te start of te epidemic wen te population was susceptible. It is similar meaning for oter constructions of R estimation. Te maximum basic reproductive ratio, R F found was 4.1, R MF found was 5.58, and R A found was 9.13, wic sould be compared wit te maximum value wic was obtained 13 in [1]. Furtermore, R F is a special case for bot R MF and R A, in terms of magnitude. Furtermore, R MF magnitude is a special case of R A magnitude. R A magnitude terefore generalized bot R F and R MF magnitudes. Table 4 sows tat te value of R F increases wit an increasing.also,r MF increases wit increasing and k.tespecialcasesarer F = R MF for k=1, R MF < R F for k<1,andr F < R MF for k>1.tus,ifweavete value of k from te real-life situation, ten te value of R MF is more realistic for estimating R rater tan R F. From Tables 4 and 5, wit respect to R F,weobtaintat teaveragesecondarycaseofdengueinfectionisatmost

10 1 Computational and Matematical Metods in Medicine Table 1: Te value of R F is related to R MF, R,and R A for te first scenario. RMF Te n days at te i.t.o.p. R F for k (,] Important notes II: 7 days (1.4, 1.9] R F R MF R R A IV: 5 days (1.14,.6] R F R MF R R A I: 5 days (1.3, 3.38] R F R MF R R A V: 7 days (1.54, 5.16] R F R MF R R A III: 6 days (1.59, 5.58] R F R MF R R A Note. R A and R are from Table 1. Table 13: Te value of R F is related to R MF, R,and R A for te second scenario. RMF Te n days at te i.t.o.p. R F for k (,] Important notes II: 7 days (1.7, 1.58] R F ( R MF R )\ R A IV: 5 days (1.17,.3] R F ( R MF R )\ R A I: 5 days (1.3, 3.35] R F ( R MF R )\ R A V: 7 days (1.36, 3.66] R F ( R MF R )\ R A III: 6 days (1.4, 4.3] R F ( R MF R )\ R A R A and R are from Table 11. four persons during te time period of dengue infection. Meanwile, te value of R MF depends on te value of k.we sow tat for k=1,tenr F = R MF (values in bracket). Te increases in te value of and k causeanincreasein te value of R MF. Wile te value in te square brackets in Tables 5 and 7 sows te relationsip between te value of and ρ, sotatr F = R A.Wealsofoundtatan increase in te value of, for a particular value of ρ, does not always cause an increase in te value of R A.Particular information for te four t.o.p. in Tables 6 and 7 sows tat, to obtain te four secondary infection cases from one primary infection, te number of mosquitoes sould be at least twice aslargeasteumanpopulation,wereasintables4and5 we do not obtain information about te ratio of te number of mosquitoes and umans. Terefore, R A provides more complete information tan R F and R MF. Next,FromTables6and7,tereistevalueofρsuc tat R F = R A for every dengue infection i.t.o.p. (see te value in brackets of Tables 6 and 7). For k=1, tere are a certain pair of ρ and for every i.t.o.p., suc tat R A = R MF = R F (Table 4). For te first scenario, if ρ is relatively small ten te value of R A is inversely proportional to te value of.forinstance,ifρ =.67 (see t.o.p. II, Table 6) ten R A increases or decreases wit an increasing. Temagnitude relation between and R is not te case in oter values of ρ. Also, we obtained tat R F is a special case of R A for a certain pair of ρ and.tus,ifweavetevalueofρfrom te real-life situation ten te value of R A is more realistic tan R F wen estimating R from te real-life dengue epidemic. Next, in te first scenario, te value of R F is not only contained in R MF and R A but also contained witin R. Also, te value of R F belongs to te intersection of R MF and R A. Tis is also te case in te first scenario, were R F ave good precision but not as good a predictor wen compared wit R MF and R A.Wilst R MF and R A are good predictors, tey do not ave a good precision (see Tables 7 13). In te second scenario, te value of R F is contained witin te range of R MF and R values but not contained in R A range. Furtermore, te value of R F is lower tan range s lower end of values of R A (see Tables 1 13). Meanwile, it is not enoug just to pay attention to R value of te model in (1). Terefore, from te aspect of te efficiency and effectiveness of dengue control, te government sould pay attention to te basic reproductive ratio estimation of R A tat is reported ere, in order to improve te quality control of te spread of dengue fever. Te important notes in Tables 1 13 sow tat R F value (see Table 4 or Tables 5 7) is included in te intervals R MF, R A,and R for te first scenario. It means tat te value of R F can still be used as a standard for te control of dengue virus infection. However, to obtain te maximum value of R A = 1, ten te number of mosquitoes would be 8 times greater tan te number of people, wit te takeoff rate of.11; see Table 11. Terefore, te use of value of ρ,, andr A as a standard value for te control of a dengue endemicity is more realistic tan te oters. From Figures 7 14, not only do we obtain te value of te basic reproductive ratio estimation of te R A wic is greater tan one but also we do obtain te value wic is less tan one; for instance, from te first scenario at =.19 and ρ =.67 we ave R A =.67 < 1 (see Table 7). Also, we can determine te intersection range of a basic reproductive ratio from a different construction of value of. Bot proposed constructions of R are te same wen using assumption in [1] for tese particular cases. But increasing te number of secondary cases for every t.o.p. will depend on k for te first construction and depend on ρ for te second construction. Tese results indicate tat

11 Computational and Matematical Metods in Medicine 11 R MF (te first construction of R estimation) from scenario I Proportion for take-off rate, k R MF for i.t.o.p. II wit =.9 R MF for i.t.o.p. V wit =.171 Tresold, R MF =1 R MF for i.t.o.p. IV wit =.38 R MF for i.t.o.p. III wit =.19 R MF for i.t.o.p. I wit =.93 Figure 7: Te dynamic of R MF for te dengue outbreak at te beginning of te t.o.p. as a function of te infected mosquito per uman index for from te first scenario. R MF /R F from scenario I R MF /R F for i.t.o.p. III wit =.19 R MF /R F for i.t.o.p. IV wit =.38 Tresold, R MF =1 R MF /R F for i.t.o.p. V wit =.171 R MF /R F for i.t.o.p. II wit =.9 R MF /R F for i.t.o.p. I wit =.93 Figure 8: Te ratio of R MF /R F for te dengue outbreak at te beginning of te t.o.p. as a function of te ratio of te rate of te infected mosquito per uman index k for from te first scenario. k R A (te second construction of R estimation) from scenario I Mosquito per uman index, ρ R A for i.t.o.p. II wit =.9 R A for i.t.o.p. I wit =.93 R A for i.t.o.p. III wit =.19 R A for i.t.o.p. IV wit =.38 R A for i.t.o.p. V wit =.171 Tresold, R A =1 Figure 9: Te dynamic of R A for te dengue outbreak at te beginning of te t.o.p. as a function of te mosquito per person index for from te first scenario. R A /R F from scenario I ρ R A /R F for i.t.o.p. II wit =.9 R A /R F for i.t.o.p. I wit =.93 R A /R F for i.t.o.p. III wit =.19 R A /R F for i.t.o.p. IV wit =.38 R A /R F for i.t.o.p. V wit =.171 Tresold, R A /R F =1 Figure 1: Te ratio of R A /R F for te dengue outbreak at te beginning of te t.o.p. as a function of te infected mosquito per uman index ρ for from te first scenario. te greater te number of mosquitoes means te iger te number of new cases of dengue and te sorter te period of te i.t.o.p. of dengue infection. Furtermore, R F estimates of R aretoolowwencomparedtor A. Tis is very dangerous in application, since te intervention to control dengue tat results from R F estimation will not be able to stop te disease once it exists. Terefore, prevention of dengue incidences is more appropriate if it is done early, and exactly ow early is dependent on R A estimation. We used real dengue epidemic data from a dengue epidemic, between te dates Nov. 5, 8, and 1 from SB Hospital. Tis includes te estimation of te basic reproductive ratio at te beginning of t.o.p. of dengue infection by using R F, R MF,andR A for R estimation. R magnitude estimation under te same assumption was conducted in [1], resulting in R F.WeproposeR MF and R A for comparing wit R F for R estimation from dengue incidence data from SB Hospital. As a results, we get R MF >1.59from te first

12 1 Computational and Matematical Metods in Medicine R MF (te first construction of R estimation) from scenario II Proportion for take-off rate, k R MF for i.t.o.p. II wit =.19 R MF for i.t.o.p. I wit =.91 R MF for i.t.o.p. III wit =.11 R MF for i.t.o.p. IV wit =.48 R MF for i.t.o.p. V wit =.15 Tresold, R MF =1 Figure 11: Te dynamic of R MF for te dengue outbreak at te beginning of te t.o.p. as a function of te infected mosquito per uman index for from te first Scenario. R A from scenario II Mosquito per uman index, ρ R A for i.t.o.p. II wit =.19 R A for i.t.o.p. I wit =.91 R A for i.t.o.p. III wit R A for i.t.o.p. IV wit R A for i.t.o.p. V wit =.15 Tresold, R A =1 =.11 =.48 Figure 13: Te dynamic of R A for te dengue outbreak at te beginning of te t.o.p. as a function of te mosquito per person index for from te second scenario. R MF /R F from scenario II k R MF /R F for i.t.o.p. III wit =.11 R MF /R F for i.t.o.p. I wit =.91 R MF /R F for i.t.o.p. II wit =.19 R MF /R F for i.t.o.p. V wit =.15 R MF /R F for i.t.o.p. IV wit =.48 Tresold Figure 1: Te ratio of R MF /R F for te dengue outbreak at te beginning of te t.o.p. as a function of ratio of te rate of te infected mosquito per uman index k for from te second scenario. R A /R F from scenario II R A /R F for i.t.o.p. II wit =.19 R A /R F for i.t.o.p. I wit =.91 R A /R F for i.t.o.p. III wit =.11 R A /R F for i.t.o.p. IV wit =.48 R A /R F for i.t.o.p. V wit =.15 Tresold Figure 14: Te ratio of R A /R F for te dengue outbreak at te beginning of te t.o.p. as a function of te infected mosquito per uman index ρ for from te second scenario. ρ scenario (Table 4, Figures 7 and 9) and R MF > 1.4 from te second scenario, respectively (Table 6, Figures 7 and 9). From te first scenario, we obtained te te basic reproductive ratio was lowest in te t.o.p. II [1.1,.33) and largest in te t.o.p. III [3.1,9.13) (Table 1, Figures 8 and 1). Furtermore, from tesecondscenario,tebasicreproductiveratiowaslowest in te t.o.p. II [.37,7.93) and largest in te t.o.p. III [5.48,11.78) (Table 11, Figures 8 and 1). We believe tat R MF and R A are a good R estimation at te beginning of te take-off period of infection. Tus, our results can be considered in te control of a dengue fever epidemic, in te dengue endemic regions of Bandung, Indonesia. But te effects of dengue incidence data were obtained from SB Hospital, Bandung, to te value of R estimation, and describe te transmission of dengue fever, because only te dengue incidence data of te same serotype were considered. Some of te results of te basic reproductive ratio estimation for dengue fever in compartmental model were reported in several publications. Favier et al. in [1] estimated a basic reproductive ratio, wic ranged widely from. to 13 for dengue epidemics occurring during te years 1996 to 3 in 9 Brazilian regions. Te upper bound of teir range, in estimating te basic reproductive ratio, is significantly iger tan te upper bound estimate reported ere. Te lower bound of teir range estimate, of basic reproductive

13 Computational and Matematical Metods in Medicine 13 ratio, is relatively smaller tan te lower bound estimate reported ere for some take-off period but te oters are relatively iger tan te lower bound estimate reported ere (Tables 4 and 6). Cowell et al. in [11] estimated te basic reproductive ratio, by using te relationsip between tebasicreproductiveratioandtegrowtrateinteearly pase of dengue fever transmission in Toceman, Mexico. Teir mean estimate of te basic reproductive ratio is 4., wic is significantly iger tan te mean estimate reported ere for every take-off period wic is obtained by R F. Te weekly number of dengue cases in Peru for te period was analyzed in [1] wic obtained tat te basic reproductive ratio ad a median 1.76 ( ), were tese values inside te value of R are reported ere; see Table 7. During te epidemic, occurring in 1 cities of te state of Sao Paulo, Brazil, [4] estimated tat te basic reproductive ratio was in te range [3.6, 1.9]. More recently [14] estimated te reproductive ratio by using te relationsip between te basic reproductive ratio and te force of infection of dengue fever transmission in Salvador from periods 1996 to 1997, and. Tey obtain an estimation of R =.85(.77, 3.7) in te periods 1996 to 1997, and R =.65(.5, 3.3) for. Te newest report of R estimation is in [] for dengue epidemics in dengue outbreak in Easter Island, Cile. Teir mean of te basic reproductive ratio R at 7. is significantly iger tan every value of R estimation reported ere for five initial take-off periods of dengue infection from two scenarios for estimating of.our estimation results, for te basic reproductive ratio by using R F range widely, are from 1.17 to 3.89 for te first scenario and from 1.35 to 3. for te second scenario wic belongs to te range of te first scenario. Our R MF estimation accommodates for te difference of te take-off rate between mosquito and uman. If tere are no differences in te take-off rate ten [1] estimates are exactly te same as ours. Oterwise R F could overestimate or underestimate depending on te value of k. Tables4and6sowtatR A accommodates differences in ρ. R F is exactly te same as oursecondestimatesforacertainpairofρ and.oterwise R F could overestimate or underestimate depending on te value of ρ, and so fort. Terefore, our R MF and R A constructions are more realistic for estimating te value of te basic reproductive ratio, tan te R F. In te first scenario, te control of dengue fever, wic isbasedontestandardvaluer F, is still realistic. Furtermore, better control of dengue fever, wen based on te maximum value of R MF or maximum value of R A,issown in Table 6 and Figures 7 1. Wile for te second scenario, wic is sown by Table 6 and Figures 11 14, it is better to coose one value from te interval as a standard value used to control dengue fever. Terefore, te second construction is better coice, for dengue control strategies in Bandung, Indonesia. Neverteless, te estimation of R as limitations, because it is only done from te fourt day until te sevent day,soteeffectofteincreaseinteincidenceofdenguer is no longer considered. Furtermore, te dengue incidence data used, from SB Hospital, do not distinguis between te latentpopulationandtedengueinfectedumanpopulation, so te estimate of R is limited by te data caracteristics. In reality, nobody can distinguis between te latent population and te dengue infected population. Furtermore te latent population as a sort incubation time period. 6. Conclusion We ave conducted two different constructions of basic reproductive ratio estimations based on incidence data. Tese constructions modify R F model by allowing te take-off rates of uman and mosquito to vary. In te first construction, we assumed tat te take-off rate of an infected mosquito is proportional to te take-off rate of a uman. Tis construction leads to a new form of modified basic reproductive ratio estimation for R MF was a function of te take-off rate of a uman () and te take-off rate of a mosquito (k). A sensitivity analysis indicated tat a small variation of a mosquito take-off rate could affect significant cange in te basic reproductive ratio. In te second construction we assumed tat, during te take-off period, te initial growt of infected umans is sligtly affected by te growt of infected mosquitoes, and vice versa. Tis construction leads to a one-parameter coupling of te initial growt of infected uman and mosquitoes. Te modified basic reproductive ratio estimation for R A resulted from tis construction, and it depends on te mosquitoes per person index ρ. Tisratio increases for small take-off rate and decreases for larger takeoff rate. A numerical analysis of R MF /R F and R A /R F ratios sows variation in a relatively large range of values for R F,inrespectto(, k) and(, ρ) for te first and second scenario for determined,respectively. Conflict of Interests Te autors declare tat tere is no conflict of interests regarding te publication of tis paper. Acknowledgments Tis work was supported by te Hiba Stretegis Nasional, under te DIKTI Indonesia sceme. Te autors would like to tank to anonymous reviewers for teir very elpful suggestions and comments tat elped to improve te paper. References [1] M. N. Burattini, M. Cen, A. Cow et al., Modelling te control strategies against dengue in Singapore, Epidemiology and Infection,vol.136,no.3,pp ,8. 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