Mathematical Modeling for Dengue Transmission with the Effect of Season

Transcription

1 World Academy of cience, Engineering and ecnology 5 Matematical Modeling for Dengue ransmission wit te Effect of eason R. Kongnuy., and P. Pongsumpun Abstract Matematical models can be used to describe te transmission of disease. Dengue disease is te most significant mosquito-borne iral disease of uman. t now a leading cause of cildood deats and ospitalizations in many countries. Variations in enironmental conditions, especially seasonal climatic parameters, effect to te transmission of dengue iruses te dengue iruses and teir principal mosquito ector, Aedes aegypti. A transmission model for dengue disease is discussed in tis paper. We assume tat te uman and ector populations are constant. We sowed tat te local stability is completely determined by te tresold parameter, B. f B is less tan one, te disease free equilibrium state is stable. f B is more tan one, a unique endemic equilibrium state exists and is stable. e numerical results are sown for te different alues of te transmission probability from ector to uman populations. preenting dengue and DHF is to combat te mosquito ectors []. Enironments, climatic ariables suc as temperature, umidity, season and precipitation significantly influence mosquito deelopment. emperature affects to te deelopment of te mosquitoes, as well as dengue iral deelopment. Dengue infection is endemic in ailand and many oter tropical countries. Fig. sows te incidence rate of dengue disease per,, reported to te Diision of Epidemiology, Ministry of Public Healt classified by mont, ailand during te period As we see, dengue cases generally peak in June, July and August. Keywords Dengue tresold parameters. disease, matematical model, season, D. NRODUCON ENGUE d Feer (DF), Dengue Haemorragic Feer (DHF) and Dengue ock yndrome (D) are increasingly important public ealt problems in te tropical and subtropical areas. Dengue as been recognized in oer countries and.5 billion people lie in areas were dengue is endemic []. Dengue disease caused by four distinct serotypes irus known as DEN-, DEN-, DEN-3 and DEN- 4. t is transmitted to te uman by biting of te infected female Aedes mosquitoes as te primary mosquito ector. nfection by any single type apparently produces permanent immunity to it, but only temporary cross immunity to te oters. e mosquitoes neer recoer from te infection since teir infectie period ends wit teir deat []. nfection wit one of tese iruses caracteristically results in feer, eadace and ras. e clinical spectrum can ary, oweer, from asymptomatic to more seere infections wit bleeding and sock. e manifestations of DHF include emorrage and sock, wic is te result of a sudden loss of intraascular olume consequent to ascular leakage. As no accine presently exists, te only metod of controlling or P. Pongsumpun is wit te Department of Matematics, Faculty of cience, King Mongkut s nstitute of ecnology Ladkrabang, Calongkrung road, Ladkrabang, Bangkok 5, ailand (corresponding autor to proide pone: ext. 696; fax: ext. 84 ; kppuntan@ kmitl.ac.t). R. Kongnuy is wit te Department of Matematics, Faculty of cience, King Mongkut s nstitute of ecnology Ladkrabang, Calongkrung road, Ladkrabang, Bangkok 5, ailand, ( s9685@kmitl. ac.t). Fig. e incidence rate of dengue patients according to te disease seerity mont-by-mont between 999 and 8 [3]. Matematical modeling of infectious disease as a long istory. e starting point is generally taken to be a paper by Daniel Bernoulli [4] on te preention of smallpox by inoculation. o control te dengue effectiely, we sould understand te dynamics of te disease transmission and take into account all of te releant details, suc as te dynamics of te Estea and Vargas [5] deeloped a model for te dengue disease transmission and included te dynamics of te Aedes aegypti mosquitoes into a standard R (susceptible-infectierecoer) epidemic model of a single population. eir model sows tat tere is a tresold number wic is a function of Aedes equilibrium population size and of te Aedes recruitment rate, aboe wic te disease will be endemic and below wic te disease will anis. n tis paper, we deelop a matematical model as an interesting tool for te understanding of te dengue transmission for te difference season. Our interest ere is to derie and analyze te model taking into account te 69

2 World Academy of cience, Engineering and ecnology 5 seasonality dengue compartment in te transmission model. n te next section, we formulate te model. n section 3, we analyze te model. e equilibrium states of tis system and teir stability are obtained. Finally, we gie te numerical results and conclusion.. HE MAHEMACAL MODEL n our model, we assume tat te uman and ector population ae constant size. Let N and N be te uman and ector population sizes. e matematical model for tis transmission is based on te transmission diagram in Fig.. Fig. ransmission diagram of dengue disease wit season. All tables and figures you insert in your document are only to elp you gauge te size of your paper, for te conenience of te referees, and to make it easy for you to distribute preprints. e uman population is subdiided into te susceptible, te infectious uman in ig epidemic season (in June, July and August), te infectious uman in low epidemic season (eery monts except June, July and August),and te recoered R. e ector population is subdiided into te susceptible and te infectious. e dynamical equations for uman population are d b bl N ( ) () dt N m N m d b ( r) () dt N m dl bl ( r)l (3) dt N m dr r( l ) R (4) dt and te dynamical equations for ector population are as follows d b bl A ( l) (5) dt N m N m d b bl ( l ) (6) dt N m N m e equation for R and in (4)-(5) can be eliminated since at time t, we ae l R N and N. o simplify te matematical analysis of tis study, we normalize te equations ()-(6) by defining new ariables N, N, l l N, R R N, N (A ) and N (A ). e total uman and ector populations are constant, tus rates of cange for total uman and ector populations equal to zero. is gies birt and deat rates are equal for te uman population, te total ector population equals to A.We obtain te equations as follows: d b b ( (A ) (A )) dt N m N m l d b dt N m (A ) ( r) d b dt N m l l (A ) ( r)l d b bl ( N ln )( ) () dt N m N m were N is te total uman population, N is te total ector population, is te number of susceptible uman population, is te number of infectious uman in ig epidemic season, l is te number of infectious uman in low epidemic season, R is te number of recoer uman population, is te number of susceptible ector population, is te number of infectious ector population, is te deat rate in te uman population, is te birt rate in te uman population, is te deat rate in te ector population, is te transmission probability from ector to uman (in ig epidemic season), l is te transmission probability from ector to uman (in low epidemic season), is te transmission probability from uman (in ig epidemic season) to ector, l is te transmission probability from uman (in low epidemic season) to ector, r is te recoer rate in te uman population b is te biting rate of ector. (7) (8) (9) 693

3 World Academy of cience, Engineering and ecnology 5 e our model, we define te transmission probability from ector to uman (in ig epidemic season) and te transmission probability from uman (in ig epidemic season) to ector are more tan.5 but not more tan.5, ) and te transmission probability from ector to uman (in low epidemic season) and te transmission probability from uman (in low epidemic season) to ector are more tan or equal zero but not more tan.5, l.5 ) wit te new two conditions ( l l R and.. ANALY OF HE MODEL A. Equilibrium Point n tis section we will find te equilibrium points of equations (7)-() in te region of, wit (,,, ),,,. l l Direct calculation sows tat equations (7)-() as two possible equilibrium points: te disease free equilibrium point and a unique endemic equilibrium point. wo equilibrium points are found by setting te rigt and side of (7)-() equal to zero. is gies ) te disease free equilibrium point E (,,,) and ) te endemic disease equilibrium point E (,, l, ) were () M () M( M) l (3) M( M) M3 M (4) M3 MM wit bn bln bn,, 3, (N m) (N m) (N m) bln b N (A ) 4,, (N m) (N m) b ln (A ) r, M, M, (N m) M B. Local Asymptotical tability e local stability of an equilibrium point is determined from te Jacobian matrix of te rigt and side of te aboe set of differential equations ealuated at te equilibrium point. C. Local Asymptotical tability For te equations defined by (7)-(), te Jacobian matrix ealuated at E is te 4 4matrix gien by M (5) M 3 4 e eigenalues are obtained by soling te matrix equation, det(j 4 ). o ealuate te determinant, we obtained te following caracteristic equation ( )( M )( aa ) (6) were a M, (7) M 3 4 a ( ) (8) Looking at te caracteristic equation, (6), we see tat two of te eigenalues are and M. Bot of tem are negatie. Next, we will ceck te sign of oter eigenalues. From aa, te two conditions of Rout-Hurwitz criteria [6] for local asymptotical stability in second order caracteristic polynomial equation are i) a, ii) a. After we ceck te stability of te equilibrium point, we can see ais always positie and a is positie wen 3 4. Moreoer, we found tat te disease free M equilibrium point is locally stable for 3 4 B. M D. Disease Endemic Equilibrium point e stability of te endemic disease equilibrium point, E, like tat of E, is determined by looking at te eigenalues of te Jacobian ealuated at E. e Jacobian for tis equilibrium point is M M were 3( ) 4( ) 3 4l,, (9) l and are gien by equation ()-(4).e caracteristic equation for te Jacobian matrix ealuated at te second equilibrium state, gien by (7)-(), is 694

4 World Academy of cience, Engineering and ecnology 5 ( M )( b b b ) () 3 t can be seen tat one eigenalue is M. Next we found te oter eigenalues by soling te equation 3 b bb () wit b ( ((M ) ) M )/, () 3 b ( (M ( ( ) M 3 ( ( M ) M ( ) ))) /, (3) 3 3 b ( (M ( M ) M ( ) )/ (4) 3 3 By using Rout-Hurwitz criteria [6], eac equilibrium point is locally stable if te following conditions are satisfied, i) b ii) b iii) bb b We can see b is always positie. For te second and te tird conditions we sown tese conditions by using te following figures Fig. 4 Numerical solutions of (7)-(), demonstrate te times series of, l and respectiely, for B 4.463, ' B.66 wit /(36565) day -, (/4) day -, A,, N,, b / 3,r / 3, m,.7, l.4,.7, l.4. e fractions of populations oscillate to te endemic state (.3557,.65,.35,.664). Fig. 3 e parameter space for te endemic equilibrium point wic satisfies te Rout-Hurwitz criteria wit te alue of parameters: respectiely, for wit /(36565) day -, (/4) day -, A,, N,,b/3,r /3,m, l.4,.7, l.4 and. We found tat E is locally stable for B and E is locally stable for B. E. Numerical imulation n tis paper, we are interested in te transmission of dengue disease wit te effect of season. e alues of te parameter used in tis study are as follows: /(36565) per day corresponds to a life expectancy of 65 years in uman. e mean life of mosquito is 4 days; (/4) per day. e recoery rate equals to /3 per day. We assume tat no alternatie ost. e conditions of parameters are, l l,.5, and l, l.5. e oter parameters are arbitrarily cosen. We presented te numerical solutions of (7)-() for te endemic equilibrium state on te following figures. Fig. 5 Numerical solutions of (7)-(), demonstrate te times series of, l and respectiely, for B 8.496, ' B.945 wit /(36565) day -, (/4) day -, A,, N,,b /3,r / 3,m,.7,.4,.7,.4. e fractions of populations l l 695

5 World Academy of cience, Engineering and ecnology 5 oscillate to te endemic state (.779,.798,.456,.37486). V. DCUON AND CONCLUON We obtain te tresold parameter for te model as 3 4 B M (5) b NA bllna (N m) ( r) (N m) ( r) Analysis of tis model reeals te existence of two equilibrium points. One is te disease free equilibrium and it is locally asymptotically stable if and only if B. e anoter one equilibrium point is te endemic equilibrium point. is equilibrium point will be a locally asymptotically stable endemic point if and only if B. 3 4 Let B is te tresold parameter. e M quantity B B is called te basic reproductie number of te disease, since it represents te aerage number of secondary cases tat one case can produce if introduced into a susceptible population. We consider te time series of uman and ector populations wen te transmission probability from ector to uman (in ig epidemic season) and te transmission probability from uman to ector (in low epidemic season) are difference. We sow in Fig. 6. 6a) 6b) Fig. 6 Numerical solutions of (7)-() demonstrate te equilibrium solution of, l and respectiely, for B (6a) wit /(36565) day -, (/4) day -, A,, N,,b /3,r / 3,m,.5, l.4,.7, l.4. (6b) wit /(36565) day -, (/4) day -, A,, N,, b / 3,r / 3,m,.7,.5,.7,.4. l l e basic reproductie number of te disease for Fig. 4 and Fig. 5 equal to.66 and.945, respectiely. Periods of te oscillations as te simulations approac te endemic equilibrium point are estimated by means of te solutions of te linearized system, obtain 6.7 years for Fig. 4 and 3.99 years for Fig. 5. Moreoer, we compare te transmission of dengue disease for te different transmission probability from ector to uman (in ig epidemic season)and te transmission probability from ector to uman (in low epidemic season). We can see from te alue of B, if te transmission of dengue disease for te different transmission probability from ector to uman (in ig epidemic season) is iger, tis means tat te infectious uman proportion in ig epidemic and infectious ector proportion are ig. For te transmission probability from ector to uman (in low epidemic season) is ig, te infectious uman proportion in low endemic interal is ig too. ACKNOWLEDGMEN e autors would like to tank Prof.Dr.-Ming ang at Maidol Uniersity, ailand. REFERENCE [] World Healt Organization, Dengue Haemorragic Feer: Diagnosis, reatment, Preention and Control. Genea, 997. [] D. J. Gubler, Dengue, CRC: Boca Raton, 986, pp. 3. [3] Diision of Epidemiology, Annual Epidemiological ureillance Report, Ministry of Public Healt Royal ai Goernment, [4] D. Bernoulli, Essai d une nonelle analyse de la mortalite causee par la petite erole, et des adantages de l inocubation pour la preenter, Acad. R. ci, pp. -95, 76. [5] L. Estea, and C. Vargas, Analysis of a dengue disease transmission model, Mat. Bioci, ol. 5, pp. 3-5, 998. [6] M. Robert, tability and Complexity in Model Ecosystems. Princeton Uniersity Press, New Jersey, 997. [7] R. M. Anderson, and R. M. May, nfectious Diseases of Humans, Dynamics and control. Oxford U. Press, Oxford,