A dengue transmission model in Thailand considering sequential infections with all four serotypes

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1 Originl Article A dengue trnsmission model in Thilnd considering sequentil infections with ll four serotypes Eriko Chikki, Hirofumi Ishikw Deprtment of Humn Ecology, Grdute School of Environmentl Science, Okym University, , Okym, Jpn Abstrct Bckground: Dengue fever/dengue hemorrhgic fever is prevlent in Thilnd, where ll serotypes re found nd the dominnt serotype hs chnged irregulrly. Although lmost ll primry infections present with slight symptoms or re symptomtic, little is known bout the infectiousness of dengue fever. Methodology: A mthemticl model of the trnsmission for dengue virus ws constructed covering the possibility of sequentil infections with ll four different serotypes. The model ws combined with the sesonl popultion dynmics of Aedes egypti, the principl vectors of dengue virus in Thilnd. The contributions of inpprent cses in the trnsmission to mosquito vectors nd ntibody-dependent enhncement were incorported into the model. Moreover, the hypothesis of n unnturl infection route ws exmined, where person cquires immunity by infection during cross-immunity period, through model simultions. Results: A comprtive study on the trnsmission probbilities of inpprent cses to mosquito vectors showed tht the prevlence of dengue infection could be immeditely stmped out fter severe outbrek if inpprent cses hd no infectiousness. The simultion under n unnturl infection route ssumption resulted of yerly chnges in the dominnt serotype nd shrp, irregulr vritions in outbreks. Conclusion: The supposition tht inpprent cses hd no infectiousness ws not in ccord with the ctul sitution in Thilnd. Furthermore, the simultion result supported the unnturl infection route s hving n influence on epidemics of dengue. Key words: dengue, inpprent infection, Thilnd, ntibody-dependent-enhncement, mthemticl model J Infect Dev Ctries 2009; 3(9): Received 10 Mrch Accepted 30 August 2009 Copyright 2009 Chikki nd Ishikw. This is n open-ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Introduction Dengue infection (dengue fever (DF), dengue hemorrhgic fever (DHF) nd dengue shock syndrome (DSS)), which is cused by dengue virus (DENV), is n endemic disese with low mortlity rte in southestern Asin countries. Becuse most infections in this re occur in Thilnd, DF/DHF is serious disese in this country [1,2]. In Thilnd, serotypes 1-4 of DENV re widespred [3]. Most cses re infected with dengue in childhood, so significnt proportion of children hve obtined immunity to ll serotypes [4]. In Thilnd, dengue epidemics hve occurred every yer in the lst 40 yers. The incidence ws recorded s 325 per 100,000 when n outbrek of DF occurred in 1987 [3]. Queen Sirikit Ntionl Institute of Child Helth (QSNICH) in Bngkok dignosed 15,376 dengue infection ptients during It ws observed tht ll four serotypes were found, nd tht the dominnt serotype, which ccounted for the most ptients mong the four serotypes, chnged irregulrly [3]. Frequently, the sttistics on dengue ptients re gthered in reports from regionl helth mngement orgniztions. In Thilnd, cliniclly dignosed cses of DF, DHF nd DSS re regulrly reported to the Provincil Helth Offices [5,6]. The totl number of infected persons remins unknown without ctive detection surveillnce becuse slight or symptomtic persons rrely visit hospitl for tretment [7]. In this rticle, mthemticl model of the trnsmission for dengue virus ws constructed covering the possibility of sequentil infections with ll four different serotypes to investigte how inpprent cses ffect the prevlence of dengue infection, to estimte the incidence of DF/DHF including inpprent cses, nd to study the succession of the dominnt serotype. Aedes egypti ws recognized s the principle vector of DENV in Thilnd [8-10]. The model of dengue trnsmission ws combined with the sesonl popultion dynmics of A. egypti. There re mny susceptible-infected-recovered (SIR)-trnsmission models of vrious infectious

2 Figure 1. Distribution of confirmed dengue cses by serology in QSNICH, Bngkok during [3]. () Distinction between the primry infections (ornge), re-infections (yellow) nd unknown (sky-blue). (b) Distributions of isolted serotypes in ptients with primry infections nd re-infections. Ech serotype is denoted by different color (Den-1, blue; Den-2, red; Den-3, green; nd Den-4, purple) diseses, including DF. Derouich [11] studied the mintennce in dengue epidemic with two serotypes by bsic SIR model. Brtley [12] developed two-serotype dengue trnsmission model in considertion of the sesonl popultion dynmics of A. egypti. Wering [13] constructed fourserotype model for DF in Thilnd considering the influence of ntibody-dependent enhncement (ADE). The present model incorportes sesonl vritions in the popultion of A. egypti, inpprent cses tht influence prevlence, nd the influence of ADE. The epidemiologicl prmeters were estimted from investigtions in Kmpheng Phet Province [1,14,15] to improve the model relisticlly. Comprtive studies on trnsmission probbilities of inpprent cses to mosquito vectors showed tht the prevlence of dengue infection could be immeditely stmped out fter severe outbrek if inpprent cses hd no infectiousness, which ws not in ccord with the ctul sitution in Thilnd. Moreover, n exmintion ws mde of n "unnturl" infection route of dengue trnsmission [16] tht Ngo nd Koelle [17] dopted in their trnsmission model. The ssumption of n unnturl infection route mens tht person during cross-immunity period cn be symptomticlly infected with nother serotype thn the serotypes they hve cquired immunity to. The simultion showed the unnturl infection route ssumption occurred with cler chnge of the dominnt serotype nd with shrp nd irregulr vrition in outbreks, while the dominnt serotype chnged in stedy regulr rottion without the unnturl infection route ssumption. Mterils nd methods Avilble dt The report on dengue ptients under 18 yers old t QSNICH, Bngkok, during contined the detection of the serotype of DENV by blood serum nlysis of ptients in cute or convlescent phses, the differentil dignosis of primry DF nd DHF/DSS, nd the distinction between primry infection nd re-infection. The report stted tht 87% of ptients were clssified s re-infection, nd the distribution of serotypes in ptients in primry nd reinfection cses [3] (Figure1). Active detection surveillnce of dengue infection ws performed for bout 2,000 students in 12 primry schools in Kmpheng Phet province in northern Thilnd during [1,14,15]. In the study, ll students bsent due to illness were dignosed s DF or not. In ddition, students in good helth were checked for their height nd weight, nd blood smple for dengue serology ws tken three times during ech surveillnce period (June 1-August 15, August 16-November 15) in the dengue seson. It ws reported tht the incidence of pprent nd inpprent cses were estimted s 4.3%, 3.6% (1998), 3.2%, 3.3% (1999), nd 1.4%, 0.8% (2000), respectively (Figure 2). The trnsition of infection stge (in humns) Persons who re bitten by n infected mosquito develop DF fter the incubtion period (1/ ) of 2-12 dys [16], 5.7 (DEN-1), 6.0 (DEN-4) dys [18]. However, most persons infected for the first time re symptomtic, while most re-infected persons develop pprent symptoms [14]; therefore, infected 712

3 Figure 2. The distribution of dengue cses in Kmpheng Phet during mong pprent nd inpprent cses [1,14,15]. Purple nd ornge show pprent nd inpprent cses, respectively. The number of pprent nd inpprent cses. The rtios of pprent nd inpprent cses in primry infection nd re-infection. sttes were distinguished between pprent nd inpprent infections. In pprent cses, symptoms lst for two to seven dys (the infectious period, ) [19,20]. Recently, it ws reported tht three cses of simultneous infections with multiple serotypes were detected in Thilnd [21]; however, the model ws limited to single serotype infection t time. The model lso ssumed tht immunity to re-infection with previously experienced serotype holds lifelong, which ws consistent with experimentl dt [22]. A temporry cross-immunity with other serotypes occurs fter recovery nd lsts for two to nine months [16,22,23]. ADE, which is negtive immune rection, occurs fter the temporl cross-immunity period [13,24]. ADE my increse the risk of progression to DHF nd the trnsmission probbilities of DENV from vector to humn nd lso from humn to vector. H(i) stnds for the trnsmission probbility from vector infected with i-serotype DENV to humn. To reflect the influence of ADE in the model, the trnsmission probbility for re-infection with i-serotype is represented s the product of H(i) by the enhncement multiple (i). An increse of DHF-development risk by ADE is represented s boost in the rtio of pprent symptoms in ptients in the model becuse of the lck of distinction between DHF nd DF. V(i) stnds for the trnsmission probbility from person infected with i-serotype DENV to vector. There ws little informtion bout the infectiousness of inpprent cses. It ws reported tht DHF ptients showed pek viremi levels 100-1,000 times higher thn DF ptients, which suggests the reltion of infectiousness to the severity of symptoms [25]. As the mount of DENV prticles in n inpprent cse is ssumed to be less thn tht in n pprent cse, the infectiousness of inpprent cses should be lower thn tht of pprent cses. So, the trnsmission probbility from n inpprent cse to vector is represented s the product of V(i) nd reduction weight (y ( )). Estimtes of trnsmission probbility According to the report of QSNICH [3], DEN-3 ws the most frequent isolte (49%), followed by DEN-1 (44%), DEN-2 (5%), DEN-4 (2%) in primry infections, nd DEN-2 ws the most frequent isolte (37%) nd followed by DEN-3 (29%), DEN-1 (22%), DEN-4 (12%) in re-infection cses. The trnsmission rte for ech serotype ( H(i) ) in primry infections ws rrnged by size in order of DEN-3, DEN-1, DEN-2, nd DEN-4, which were fixed by the product of the multiplier nd, when the trnsmission rte of DEN- 4 ( ) ws chosen s the stndrd. The trnsmission rte for ech serotype in re-infection cses ws given by the product of the enhncement multiple ( (i) ) nd H(i), then the order by size s follows: DEN-2, DEN- 3, DEN-1, nd DEN-4. For re-infection, the vlue of (3) H(3) ws lrger thn tht of the other serotypes in contrst to the order bove. Becuse most people re infected with DEN-3 in primry infections, significnt number of persons hve obtined immunity to DEN-3 nd few persons cn be infected with DEN-3 in re-infection cses. The influence of ADE in third nd fourth infections ws still unknown, but the risk for DHF with the third nd 713

4 Tble 1. Trnsmission probbility from vector nd enhncement multiples by serotype. Serotype(i) Trnsmission probbility from Enhncement multiple i) vector infected by i- serotype H(i) fourth infection ws not different from tht with the second infection in QSNICH [26]. The influence of ADE in third nd fourth infections ws ssumed to be equl to the second infection. Then, the multipliers in H(i) nd (i) were chosen s stisfying the observed orders of isolted serotypes in both primry infection nd re-infection cses [3] (Tble 1). According to the surveillnce in Kmpheng Phet province [1,15], there were 154 students with symptoms nd 177 students with no symptoms of bout 2,000 primry school students during the investigtion period over three yers. The incidence per yer ws ssumed to be 110/2,000 in the schoolchildren ge ctegory nd the incidences of pprent nd inpprent cses were ssumed to be equl. Then, we set nd x 1 = 0.1, x 2 = x 3 = x 4 = 0.5 (Tble 2). The trnsmission rtes for ll serotypes from humn to mosquito were ssumed to be equl nd were set t V(1) = V(2) = V(3) = V(4) = 0.3 (Tble 2). Popultion dynmics of mosquito vectors The principl vector of DENV in Thilnd is A. egypti [9,27]. In Thilnd, tempertures re mintined t level suitble for breeding of A. egypti throughout the yer, which is the reson DF previls in ll sesons [28]. The popultion of A. egypti vries by seson. A survey of A. egypti in Chchoengso province in Thilnd, which is locted 100 km est of Bngkok nd hs similr climte to Bngkok, showed tht the number of lrve of A. egypti in the wet seson ws more thn tht in the hot nd cool sesons, nd tht in the hot seson ws greter thn in the cool seson [9,10]. The life expectncy of A. egypti (1/ V ) ws estimted s 14 dys [29]. A model of the popultion dynmics of A. egypti ws constructed. The wet seson is divided into two prts by the level of rinfll; therefore, the model hd four sesons: Februry-April s the hot seson, My-July s the wet seson 1, August- October s wet seson 2, nd November-Jnury s the cool seson. The emergence rte of A. egypti ( V ) ws estimted from the bove survey [9,10] (Tble 3). Mthemticl model Popultion dynmics by ge In Thilnd, cses of dengue infection re frequently infected in childhood. The susceptibility nd the seriousness of dengue infection seem to vry by ge [30]. Therefore, the totl popultion ws clssified into five ge ctegories from school ge, which were represented by Nh ( = 1,, 5). The verge lifespns of men nd women in Thilnd in 2006 were 69 nd 75 yers, respectively [31]. The model ssumed tht the verge lifespn ws 70 yers nd the deth rte ( H ) ws 1/70 (yer -1 ) nd tht the popultion of every ctegory nd lso the totl popultion held stble (Nh /dt = 0), which led to the determintion of the birth rte being equl to the deth rte ( H ) nd the trnsfer rtes from Nh to Nh +1 (Tble 4). Model scheme without n unnturl infection route ssumption The effect of n unnturl infection route ssumption ws investigted on the dengue epidemic, where person during the cross-immunity period cn be infected, which ws dopted in the model by Ngo nd Koelle [17]. When the unnturl infection route ssumption ws not dopted, the totl popultion ws divided into five epidemiologicl clsses tht ech contined five ge ctegories: susceptible clss tht could be infected by ny of the serotypes except for the serotypes tht it hd lredy cquired immunity for; n exposed clss tht hd been bitten by n infected mosquito but hd not yet reched infectivity; n pprently infected clss tht hd infectivity with symptoms; n inpprently infected clss tht hd infectivity without symptoms; nd cross-immunity clss tht hd immunity to ll four serotypes just fter recovery. The bove epidemiologicl clsses were superscripted to Sh, Eh, Ih, Ih, Ch with ge ctegories nd the serotypes current infection nd those tht hd lredy obtined immunity were subscripted. The model scheme of the primry infection of dengue with the epidemiologicl clsses nd the trnsfers mong them re shown in Figure 3. As person my be infected four times with the four serotypes, the model could llow up to four 714

5 Tble 2. Estimted prmeter vlues in simultion. Symbol Description Estimted vlue H Humn birth rte =deth rte 1/70 yer -1 H(i) Trnsmission probbility from vector infected with i-serotype dengue virus to humn See Tble 1 H(4) 0.35 Enhncement multiple for the See Tble 1 i -serotype (i) x m x 1 Rtio of pprent symptom 0.1 x 2,x 3,x 4 ptients in the m-th infection 0.5 1/ H Incubtion period in humns 5 dys 1/ H Infectious period in humns 6 dys 1/ H Cross-immunity period in 3 months humns V Mosquito emergence rte See Tble 3 V Mosquito deth rte 1/14 dy -1 Trnsmission probbility 0.3 from host infected i -serotype dengue virus to vector V(i) 1/ V Incubtion period in mosquitoes 10 dys b Humn biting rte 1/2 dy -1 Figure 3. Model scheme of dengue in primry infections in humns. 715

6 Tble 3. Estimted emergence rte of mosquitoes. Seson Emergence rte V (dy -1 ) Februry-April (hot seson) 1/15 My-July (wet seson 1) 1/13.5 August-October (wet seson 2) 1/12.4 November-Jnury (cool seson) 1/15.5 Tble 4. Age ctegories nd the trnsfer rtes mong ge ctegories. Symbol Age ctegory (yers) Trnsfer rte f (dy -1 ) Popultion Nh Nh Nh Nh Nh Totl infections with different serotypes, where the model scheme of the second, third, nd fourth infections ws similr to the primry one. Model scheme with n unnturl infection route ssumption When n unnturl infection route ssumption ws dopted, the new exposed clss (Eh ) ws dded to the previous model. If person is infected with serotype other thn the serotypes tht he or she hd cquired immunity to during the cross-immunity period, the ptient moves to the newly exposed clss Eh nd therefter to the inpprently infected clss (Ih ) becuse he or she is probbly symptomtic or shows slight symptoms due to the effect of crossimmunity (Figure 4). Model scheme in mosquitoes The totl mosquito popultion (Nv) ws divided into three epidemiologicl clsses: susceptible clss, n exposed clss, nd n infected clss, which were represented by Sv, Ev nd Iv, respectively. An infected mosquito hs infectiousness for its lifetime. The model ws limited to single serotype infection (Figure 5). Pepin et l. [32] reported tht competition between serotypes could ffect virus titers in multiple infections in mosquitoes, which suggested tht the trnsmission probbility ws decresed. Results The progress of dengue epidemic ws simulted using trnsmission model. The simultions postulted tht the popultion of mosquitoes, which vry sesonlly, followed emergence rtes in the four sesons, comprising 300,000 on every Jnury 1st (Figure 6) nd tht the humn popultion ws set s bout 26,500 to give second ge ctegory (7-12 yers) of 2,000 in ccordnce with the number of schoolchildren in Kmpheng Phet province (Tble 4). The results of simultions for 10 yers were obtined fter burn-in period of 10 yers by introducing four persons who were infected with the different serotypes into ll negtive popultions. Comprison mong situtions with nd without n unnturl infection route ssumption The numbers of pprent nd inpprent cses in the totl popultion nd in the schoolchildren ge ctegory (7-12 yers) were observed under situtions with nd without n unnturl infection route ssumption. The reduction weight of the trnsmission probbility in the inpprently infected clss (y) ws ssumed to be reduced to 50% of tht in the pprently infected clss (y = 0.5) becuse n inpprent cse hd lower infectiousness thn n pprent cse. The numbers of pprent nd inpprent cses chnged sesonlly ccording to the dynmics of mosquito vectors. The dominnt serotype chnged in stedy rottion without the unnturl infection route ssumption, while it chnged irregulrly with n unnturl infection route ssumption (Figure 7). Comprison of the number of pprent cses in two ge group ctegories, 716

7 Figure 4. Model scheme with the unnturl infection route ssumption in humns. Figure 5. Model scheme of dengue in vectors. Figure 6. Sesonl profile of the A. egypti popultion. schoolchildren nd older, mong situtions with nd without n unnturl infection route ssumption re shown in Figure 8. Comprison mong reduction weights of trnsmission probbility in the inpprently infected clss For the sitution tht inpprent cses were ssumed to hve no infectiousness (y = 0.0), mjor epidemic broke out once, but it took long time until the next epidemic occurred without the unnturl infection route ssumption, while DF ws eliminted fter mjor outbrek becuse lmost ll persons gined immunity to ll four serotypes rpidly with the unnturl infection route ssumption (Figure 9). Sensitivity for the incidence to vritions in the trnsmission probbility prmeter vlues The sensitivity for the incidences of both pprent nd inpprent dengue cses to vritions in the vlues of the trnsmission probbilities from vector to humn ( ), from humn to vector ( V ), nd the reduction weight of the trnsmission probbility in inpprent cses (y) were exmined through simultions for 10 yers. Figure 10 shows the reltive incidences for these prmeter vlues with rnge of V = 0.3, y = 0.5) in the model under situtions without nd with the unnturl infection route ssumption. Mild fluctutions in the incidences of both pprent nd inpprent cses were found when the vlue of the trnsmission probbilities holds the bove-dopted vlue or over. On the other hnd, the incidence decreses when the vlue tends towrds zero. Discussion The report on DF by QSNICH indicted tht only 11% of ptients were primry infections nd tht other 87% were re-infection cses [3]. The surveillnce of DF in Kmpheng Phet lso showed 717

8 Figure 7. The trnsition of inpprent nd pprent cses for 4-serotypes in the sitution of y = 0.5 without (-d) nd with (e-h) the unnturl infection route ssumption. (, b, e, f), (c, d, g, h) show the second ge ctegory (7-12 yers) nd the sum of ll ge ctegories, respectively. (, c, e, g) nd (b, d, f, h) show pprent nd inpprent cses. Ech serotype is denoted by different color (Den-1, blue; Den-2, red; Den-3, green; nd Den-4, purple). 718

9 Figure 8 Comprison of the number of pprent cses mong two ge group ctegories: school children nd older without () nd with (b) the unnturl infection route ssumption. Blue nd red show school children nd 50- yers ge group ctegories, respectively. Figure 9 The trnsition of incidences for the sitution where inpprent cses hve no infectiousness without (light blue) nd with (ornge) the unnturl infection route ssumption. Four persons ech who were infected with different serotype were introduced into fully negtive popultion t the beginning. Figure 10 Reltive incidences of pprent (solid line) nd inpprent (broken line) dengue cses for vrious vlues of prmeters compred with those for the dopted vlues in the present study (dimond mrk) without () nd with (b) the unnturl infection route ssumption. Blue, red, nd green show the situtions for vritions in the trnsmission probbilities from vector to humn ( ), from humn to vector ( V ), nd the reduction weight of the trnsmission probbility in inpprent cses (y), respectively. 719

10 tht 3.9% of ptients were primry infections nd the other 96.1% were re-infection cses [14,15]. These reports suggested tht dengue infections in primry infections produced slight symptoms or were symptomtic, while tht in re-infection generted more cute symptoms [3,33]. It ws considered tht ADE could influence the incresed risks of progress to symptomtic stte nd incresed trnsmission probbility for re-infection. In the model, ADE ws ssumed to hve n influence t ny time of reinfection, while Brtley et l. [12] nd Wering et l. [13] ssumed tht ADE worked briefly fter recovery. In Thilnd, the temperture is suitble for A. egypti breeding throughout the yer. Becuse the popultion of A. egypti ws recognized to hve sesonl fluctutions [9,10], the model dopted the estimted emergence rtes by seson ccording to rinfll, to reflect the trnsmission cpcity of vectors. In res such s Thilnd where DF is endemic, most cses re infected with DF in childhood, so significnt proportion of children hve been infected nd obtined immunity to ll serotypes [4]. The present model introduced the clssifiction of n ge ctegory structure, so tht it ws possible to exmine the incidence of DF by ge. The simultion showed tht 3.9% of ptients were primry infections nd the other 96.1% were re-infection cses [14,15]. These reports suggested tht dengue infections in primry infections produced slight symptoms or were symptomtic, while tht in re-infection generted more cute symptoms [3,33]. It ws considered tht ADE could influence the incresed risks of progress to symptomtic stte nd incresed trnsmission probbility for re-infection. In the model, ADE ws ssumed to hve n influence t ny time of reinfection, while Brtley et l. [12] nd Wering et l. [13] ssumed tht ADE worked briefly fter recovery. In Thilnd, the temperture is suitble for A. egypti breeding throughout the yer. Becuse the popultion of A. egypti ws recognized to hve sesonl fluctutions [9,10], the model dopted the estimted emergence rtes by seson ccording to rinfll, to reflect the trnsmission cpcity of vectors.in res such s Thilnd where DF is endemic, most cses re infected with DF in childhood, so significnt proportion of children hve been infected nd obtined immunity to ll serotypes [4]. The present model introduced the clssifiction of n ge ctegory structure, so tht it ws possible to exmine the incidence of DF by ge. The simultion showed tht the number of infected persons decresed in the older ctegories, nd tht the incidence in the fifth ge ctegory (50 yers nd over) ws extremely smll compred with the children s ge ctegory (7-12 yers) (Figure 8). The effect of n unnturl infection route ssumption on the dengue epidemic ws investigted, which ws dopted in the model by Ngo nd Koelle [17]. The simultion showed tht when the unnturl infection route ssumption ws dded, cler chnge in dominnt serotype occurred on n irregulr bsis long with shrp nd irregulr vritions in outbreks, while the dominnt serotype chnged in nerly regulr rottion without the unnturl infection route ssumption. Thilnd hs mintined dengue epidemic in recent yers nd big outbrek hs occurred irregulrly every few yers. It is difficult to predict when severe outbreks will occur, but they were supposed to be cused by meteorologicl chnges [34] or by the introduction of infected persons from outside the re [35]. The simultion result supported the unnturl infection route s hving n influence on epidemics of dengue. It is cler tht there re mny inpprent cses in DF [14,15], but the role of inpprent cses in dengue trnsmission is uncertin. A trnsmission model for DENV, including the inpprently infected clss, ws constructed to investigte how the inpprent infection ffects the prevlence. There ws little informtion bout the infectiousness of inpprent cses, lthough inpprent cses could ccount for bout hlf of dengue infections [14,15]. When it ws ssumed tht inpprent cses hd no infectiousness, mjor outbrek occurred once, nd fterwrds the epidemic diminished immeditely (Figure 9), which ws not in ccord with the ctul sitution in Thilnd. The simultion results under n unnturl infection route ssumption lso greed with the ctul sitution in Thilnd. Becuse its construction ws bsed on the dengue epidemic in Thilnd, this dengue trnsmission model is suited for simulting the trnsition of dengue epidemic in the circumstnces tht DF/DHF previls in ll sesons nd tht ll four serotypes of DENV re spred. It cn be helpful for nlyzing vrious hypotheses on the trnsmission process. Further creful reserch is necessry before the plnning of preventive mesures ginst DF, including vector control, becuse indequte vector control my strengthen the seriousness of future dengue epidemics. 720

11 Acknowledgements This work ws supported in prt by Grnt-in-Aid from the Ministry of Helth, Lbour nd Welfre of Jpn for Reserch for Emerging nd Re-emerging Infectious Diseses (Grnt no. H20-Sinkou-ippn-015) nd Grnt-in-Aid from the Jpn Society for the Promotion of Science ( ). References 1. Anderson KB, Chunsuttiwt S, Nislk A, Mmmen MP, Librty DH, Rothmn AL, Green S, Vughn DW, Ennis FA, Endy TP (2007) Burden of symptomtic dengue infection in children t primry school in Thilnd: prospective study. Lncet 369: Mckenzie JS, Gubler DJ, Petersen LR (2004) Emerging flviviruses: the spred nd resurgence of Jpnese encephlitis, West Nile nd dengue viruses. Nture Med 10: S Nislk A, Endy TP, Nimmnnity S, Klynrooj S, Thisykorn U, Scott RM, Burke DS, Hoke CH, Innis BL, Vughn DW (2003) Serotype-specific dengue virus circultion nd dengue disese in Bngkok, Thilnd from 1973 to Am J Trop Med Hyg 68: Gubler DJ (1998) Dengue nd dengue hemorrhgic fever. 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12 Ecol 38: Kittigul L, Pitkrnjnkul P, Sujirrt D, Siripnichgon K (2007) The differences of clinicl mnifesttions nd lbortory findings in children nd dults with dengue virus infection. J Clin Virol 39: World Helth Orgniztion (2008) World Helth Sttistics WHO, Genev. 32. Pepin KM, Lmbeth K, Hnley KA (2008) Asymmetric competitive suppression between strins of dengue virus. BMC Microbiol 8: Murgue B, Roche C, Chungue E, Depris X (2000) Prospective study of the durtion nd mgnitude of viremi in children hospitlised during the dengue-2 outbrek in French Polynesi. J Med Virol 60: Czelles B, Chvez M, McMichel AJ, Hles S (2005) Nonsttionry influence of El Niño on the synchronous dengue epidemics in Thilnd. PLoS Med 2:e Gubler DJ (1987) Dengue nd dengue hemorrhgic fever in the Americs. P R Helth Sci J 6: Corresponding uthor Dr. Hirofumi Ishikw Deprtment of Humn Ecology Grdute School of Environmentl Science Okym University, Okym, Jpn Tel , Fx E-mil: ishikw@ems.okym-u.c.jp Conflict of Interest: No conflict of interest is declred Appendix The model is governed by the following differentil equtions. The runs over the ge ctegories 1-5; the symbol stnds for the trnsfer rte f from Nh to Nh +1 for =1, 2, 3, 4 but formlly 0 = 5 =0, 1, is 1 for =1 or 0 for otherwise. Humns Primry infection with i serotype dsh (0,0,0,0)/dt = 1, H Nh+ -1 Sh -1 (0,0,0,0)-( + H(i) + H )Sh (0,0,0,0) (A1) deh (i,0,0,0)/dt = -1 Eh -1 (i,0,0,0)+ -1 H(i) Sh - 1 (0,0,0,0) H(i) Sh (0,0,0,0) -( + H + H )Eh (i,0,0,0) (A2) dih (i,0,0,0)/dt = -1 Ih -1 (i,0,0,0)+ -1 (1-x 1 ) H Eh -1 (i,0,0,0)+(1- )(1- x 1 ) H Eh (i,0,0,0) -( + H + H )Ih (i,0,0,0) (A3) dih (i,0,0,0) /dt = -1-1 Ih (i,0,0,0) + -1 x 1 H Eh -1 (i,0,0,0)+(1- )x 1 H Eh (i,0,0,0) -( + H + H )Ih (i,0,0,0) (A4) dch (i,0,0,0)/dt = -1 Ch -1 (i,0,0,0)+ -1 H (Ih -1 (i,0,0,0)+ih -1 (i,0,0,0)) +(1- ) H (Ih (i,0,0,0)+ih (i,0,0,0))- ( + H + H )Ch (i,0,0,0) (A5) H(i) = H(i) Iv (i) b/nh (A6) Re-infection with k serotype, without n unnturl infection route ssumption As n exmple of the third k serotype infection fter the first i serotype nd the second j serotype infections, where the serotypes i, j, k re different from ech other. dsh (i,j,0,0)/dt = -1 Sh -1 (i,j,0,0)+ -1 H Ch -1 (i,j,0,0)+(1- ) H Ch (i,j,0,0) -( + H(k) + H )Sh (i,j,0,0) (A7) deh (i,j,k,0)/dt= -1 Eh - 1 (i,j,k,0) + -1 H(k) Sh -1 (i,j,0,0)+(1- ) H(k) Sh (i,j,0,0) -( + H + H )Eh (i,j,k,0) (A8) dih (i,j,k,0)/dt = -1 Ih -1 (i,j,k,0)+ -1 (1-x 3 ) H Eh -1 (i,j,k,0)+(1- )(1- x 3 ) H Eh (i,j,k,0) -( + H + H )Ih (i,j,k,0) (A9) dih (i,j,k,0) /dt = -1-1 Ih (i,j,k,0) + -1 x 3 H Eh -1 (i,j,k,0)+(1- )x 3 H Eh (i,j,k,0) -( + H + H )Ih (i,j,k,0) (A10) dch (i,j,k,0)/dt = -1 Ch -1 (i,j,k,0)+ -1 H (Ih -1 (i,j,k,0)+ih -1 (i,j,k,0)) +(1- ) H (Ih (i,j,k,0)+ih (i,j,k,0) )- ( + H + H )Ch (i,j,k,0) (A11) H(k) = (k) H(k) Iv (k) b/nh (A12) Re-infection with k-serotype, with n unnturl infection route ssumption Applying the unnturl infection route ssumption, equtions (A5), (A7) nd (A10) re replced s (A5 ), (A7 ) nd (A10 ) nd the new eqution (A13) ws dded. dch (i,0,0,0)/dt = -1 Ch -1 (i,0,0,0) + -1 H (Ih -1 (i,0,0,0)+ih -1 (i,0,0,0)) +(1- ) H (Ih (i,0,0,0) +Ih (i,0,0,0) )- ( + H(k) H + H )Ch (i,0,0,0) (A5 ) dsh (i,j,0,0)/dt = -1 Sh -1 (i,j,0,0)+ -1 (1- k3 H(k) Ch -1 (i,j,0,0))+(1- )(1- k H(k) Ch (i,j,0,0)) -( + H(k) + H )Sh (i,j,0,0) (A7 ) deh (i,j,k,0) /dt = -1 Eh -1 (i,j,k,0)+ -1 H(k) Ch -1 (i,j,0,0)+(1- ) H(k) Ch (i,j,0,0) -( + H + H )Eh (i,j,k,0) (A13) dih (i,j,k,0)/dt = -1 Ih -1 (i,j,k,0)+ -1 H (x 3 Eh -1 (i,j,k,0)+eh -1 (i,j,k,0)) +(1- ) H (x 3 Eh (i,j,k,0)+eh (i,j,k,0))-( + H + H )Ih (i,j,k,0) (A10 ) Here, k mens the summtion of ll k-serotypes except i, j. Vector infection with i -serotype dsv/dt = V Nv-( V(i) + V )Sv dev (i) /dt = V(i) Sv-( V + V )Ev (i) div (i) /dt = V Ev (i) - V Iv (i) V(i) = V(i) (totl(ih (i))+y totl(ih (i)))b/nh (A14) (A15) (A16) (A17) Here, totl(ih (i)) or totl(ih (i)) runs over ll ge-ctegories ( = 1,, 5) nd ll possible combintions of j, k, l in (i,0,0,0), (j,i,0,0), (j,k,i,0), (j,k,l,i) for the epidemiologicl clsses Ih or Ih. 722