9 J. Mat. Fund. Sci., Vol. 46, o. 3, 4, 9-3 A Dynamical Model for Transmission of West ile Virus in Cicken-Mosquito Interaction Jafaruddin,, Juni Wijayanti Puspita 3, uning uraini & Edy Soewono Department of Matematics, Institut Teknologi Bandung Jalan Ganesa, Bandung 43, Indonesia Department of Matematics, Uniersitas usa Cendana Jalan Adisucipto Kampus, Penfui, Kupang 8, usa Tenggara Timur 3 Department of Matematics, Uniersitas Tadulako, Kampus Bumi Tadulako, Jalan Soekarno-Hatta Km 9, Mauti Kolore, Palu Sulawesi Tenga 948 Email: jafaruddin.amid@yaoo.com Abstract. Te West ile irus (WV) is transmitted troug te bites of infected mosquitoes. Te spread of WV in cicken populations is quite unique. Altoug cickens can contract te irus troug a mosquito bite, tey immediately build immunity to te irus and do not sow pysical symptoms of illness and ence cickens are only temporary carriers of te irus. Recently, experimental results ae sown tat mosquitoes do not cange fecundity beaior, yet results indicate tat resistance to infection is associated wit a fitness cost in terms of mosquito surial. We constructed a ost-ector type transmission model for WV in mosquito-cicken populations. Te basic reproductie ratio, R, was obtained. From sensitiity analysis of R it was sown tat under certain conditions tis ratio decrease wit an increase of te lifetime of mosquito infection. Keywords: basic reproductie ratio; dynamical model; mosquito-cicken interaction; stability analysis; WV transmission. Introduction Te spread of te West ile Virus (WV) as been reported in Africa, Asia, and Europe for decades, and recently it as been found in ort America. It was first discoered in Uganda in 937 and in Egypt and India in 95. Causing millions of deats in birds and a few tousand in umans, WV as ad a tremendous impact on public ealt policies [,]. A report from te US Center for Disease Control and Preention indicates tat tere were 6 uman cases of encepalitis, including seen deats in 999 [3]. WV is most commonly transmitted to umans by mosquitoes. Until now tere as been no medication to treat WV infection, nor are tere accines to preent it. Unfortunately, most people infected wit WV will ae no symptoms. About in 5 people wo are infected will deelop a feer wit oter Receied September 4 t, 3, st Reision April 4 t, 4, nd Reision October 4 t, 4, Accepted for publication October 6 t, 4. Copyrigt 4 Publised by ITB Journal Publiser, ISS: 337-576, DOI:.564/j.mat.fund.sci.4.46.3.7
A Dynamical for Trans. of WV in Cic.-Mosq. Interaction 9 symptoms. Less tan % of infected people deelop a serious, sometimes fatal, neurologic illness [,3]. WV can infect many species of liing beings, suc as umans, orses, and also birds, especially cickens [4]. Te symptoms of WV in cickens are not easy to see, but according to Sanne, et al. in [], most accurate information about WV actiity can be analyzed from infection in cickens. Tere is an incubation period before cickens can transmit WV to mosquitoes, i.e. 3-5 days, wile te incubation period for mosquitoes is - days. Matematical models ae been deeloped by many different autors to understand te interaction of WV in animals or umans [5]. A compartment model to describe te interaction between mosquitoes and birds is discussed in Wonam, et al. [6,7]. Te basic reproductie ratio of te birds as endemic indicator was sown analytically, witout including demograpic factors. Howeer, Bowman, et al. in [8] found tat te basic reproductie ratio is not sufficient to caracterize te dynamics of WV, tey also depend on te initial number of all state ariables. A differential equations model wit non-delayautonomous approac was constructed by Lord and Day [9-] in order to understand te impact of bird mortality in WV transmission dynamics. Special interaction between female mosquitoes and birds was introduced by Cruz- Paceco [5]. All of te matematical models from te autors aboe ae represented te spread of WV between mosquitoes and birds. But according to [], different from oter birds, cickens ae teir incubation period after tey are infected by WV. Terefore, in tis paper we introduce our matematical model in order to understand te spread of WV between cickens and mosquitoes. Te immune effect in cicken populations is included in te model to gie a better understanding of te problem. Tis paper is constructed as follows. A ost-ector model of WV transmission is gien in Section. Te basic reproductie ratio and te numerical existence of an endemic equilibrium are discussed in Section 3. A dynamical analysis of te WV transmission is performed in Section 4, to gie a better understanding of te situation in te field. Te conclusion is gien in Section 5. Host-Vector Model WV Transmission We ae constructed a SEIR-SEI ost-ector model for interaction between cicken and mosquito populations. Te ost population (cickens) consists of susceptible ( S ), exposed ( E ), infected ( I ), temporary immune and less
9 Jafaruddin, et al. infectious ( I i ), and recoered ( R ) compartments, wile te ector population (mosquitoes) is diided into tree compartments: susceptible ( S ), exposed ( E ), and infected ( I ). Te transmission sceme is gien in Figure. Here, we assume tat te total ost population S E I Ii R and te total ector population S E I are constant. We gie different natural deat rates of mosquitoes between susceptible and infected mosquitoes []. According to Figure and te parameter descriptions in Table, te dynamical equations of WV transmission are gien by System (). Figure Te sceme of ost-ector WV transmission. WV transmission model in Cicken-Mosquitoes interaction ds b S I ds bs I I dt dt i S A S de b S I de bs I I dt dt i E E di di E I E I dt dt dii I Ii dt dr Ii R dt ()
A Dynamical for Trans. of WV in Cic.-Mosq. Interaction 93 Table Definition and Parameter Values. Parameters in te models Definition Value(s) Dimension A Recruitment rate of cickens - day - A Recruitment rate of mosquitoes - day - b Biting rate (aerage number of bites per day) b day - Transmission rate from day - mosquitoes to cickens Transmission rate from cickens day - to mosquitoes Transmission rate from immune day - cickens to mosquitoes atural deat rate of cickens.4 day - atural deat rate of mosquitoes..5 day - Deat rate of infected..4 day - mosquitoes Transition rate from exposed day - cickens to infected cickens Transition rate from infected.43 day - cickens to immune cickens Transition rate from exposed day - mosquitoes to infected mosquitoes Recoery rate for cickens.5. day - Population size of cickens - - A = Population size of mosquitoes A - - In te next section, we conduct a dynamical analysis to study te dynamics and identify te critical parameters of WV transmission. 3 Basic Reproductie Ratio Te disease-free equilibrium of te system is gien by A A E S, E, I, Ii, R, S, E, I. Because of
94 Jafaruddin, et al. ds de di dii dr is constant, it means tat. Terefore, dt dt dt dt dt A from Eq. () we obtain. Meanwile, te assumption tat is ds de di constant, one obtains and te lifetime of infected dt dt dt mosquitoes greater tan te lifetime of susceptible mosquitoes A A ( ) obtains. Here, we coose. We obtain te next generation matrix (see [],[] ), gien as follows: GM b b b b b Te eigen alues of GM are zero of multiplicity 3 and b b b. Terefore te basic reproductie ratio R is gien in te form b. R (3) As sown in Eq. (3), te basic reproductie ratio of System () depends on all parameters of System (), suc as te ratio between mosquito and cicken population,, te biting rate, and oter transition parameters. We obtain tat E is locally asymptotically stable if and only if R by using te diagonal and determinant metod (see [3]) from te Jacobian matrix of System (). Here, te alue of R represents tat te new WV infections produces less tan one new infected indiidual during te infection period, terefore WV infection cannot spread.
A Dynamical for Trans. of WV in Cic.-Mosq. Interaction 95 Figure Existence and stability of endemic equilibrium wit total population size and ital parameter data set 4,, b, 7,, 8,..4,.8,., 4. From numerical simulation we found tat R.43 and endemic equilibrium E = S *, E *.8, I *.9, I *.7, R 93.3, S 939, E 6.9, I i * * * * 54.. We obtained R, it means tat te WV can inade and spread into te cicken and mosquito population. ote tat it is not possible to express te endemic equilibrium analytically. Te numerical results indicating te stability of te endemic equilibrium are sown in Figure. 4 Sensitiity Analysis We ae used te Maple software to sow leel sets of R for ariation parameters suc as te lifetime of infected mosquitoes, te lifetime of susceptible mosquitoes, te recoery rate of te cickens, te transition rate from infected cickens to immune cickens, and te effect of different lifetimes of mosquitoes to te cange te alue of R. Te first leel sets of R in Figure 3(a) sow relations between te transition rate from infected cickens to immune cickens and te deat rate of infected mosquitoes for.. Te second leel sets of R in Figure 3(b) sow relations between te mortality rate of infected mosquitoes and te recoery rate of te cickens for.43. Bot figures were obtained from ypotetical data A 5,
96 Jafaruddin, et al. A, b, and te alues of te transition parameters from Sanne, et al. [], namely.4,.,.5,.48,.8 and.5.,., Te tird and te fourt leel sets of R in Figure 4 are relations between te transition rate from exposed cickens to infected cickens and te transition rate from exposed mosquitoes to infected mosquitoes. Te leel sets of R for.49.39 and te leel sets of R for.39.49 are sown in Figure 4(a) and Figure 4(b), respectiely. Bot figure use ypotetical data A, A, b and te alues of te transition parameters from Sanne, et al. [], namely.4,.,.,.43,.48,.39,.8, and.. (a) (b) Figure 3 Leel sets of R wit respect to transition rate from infected cickens to immune cickens (a), and to recoery rate of cickens to deat rate of infected mosquitoes (b). Figure 3 sows tat R decreased faster wit te increase of (see Figure 3(a)) as well as wit te increase of (see Figure 3(b)). Suitable, from te experimental data in Sanne, et al. [] sown tat te aerage lifetime of
A Dynamical for Trans. of WV in Cic.-Mosq. Interaction 97 mosquitoes increases due to infection ( ). Also, we conclude tat te basic reproductie ratio may decrease, as sown in Figure 3. (a) (b) Figure 4 Relation between transition rate from exposed cickens to infected cickens, and transition rate from exposed mosquitoes to infected mosquitoes. ext we studied te effect of te ratio between mosquitoes and cickens to te cange te alue of R. Here, we ae used parameter alues.,.8, b, wile te oters parameter alues from Table. Substituting tese alues in Eq. (3), we obtained te alue of R (4).64. Figure 5 Magnitude of R against ratio
98 Jafaruddin, et al. From Figure 4, R is proportional to te ratio between mosquito and cicken population sizes. From Figure 5, we ae R for.4 in Eq. (5). It sows tat te WV infection cannot spread. In oter words, te WV will die out if te number of mosquitoes is smaller tan.4 times te number of cickens. Furtermore,.4 sow te WV is endemic if te number of mosquitoes is more tan.4 times te number of cickens. Te simulation of te dynamics of I, I i, and I is sown in Figure 6. Figure 6 Te dynamics of I, I i, and I population wit te following parameter data set: 4,, b, 7,, 8,.4,.,.8,., 4. (a).49.39, and R.43. (b).39.49, and R.3. Figure 6 sows tat te effect of te increased lifetime of WV infected mosquitoes accelerates te increase of te number of WV infections at te beginning of te epidemic growt and extends te outbreak period. 5 Conclusion We ae obtained a model for WV transmission for cicken-mosquito interaction, taking into account two stages of infection before te recoered cickens and te effect of te prolongation of te lifetime of te infected cickens. Te numerical results indicate tat te increase of lifetime of infected mosquitoes may reduce te basic reproductie ratio.
A Dynamical for Trans. of WV in Cic.-Mosq. Interaction 99 Acknowledgments Te autors would like to tank ITB Decentralization Researc Grant. Te autors tank te anonymous reiewers for teir ery elpful suggestions and comments tat elped to improe te manuscript. We would also like to tank te editors for teir suggestions regarding te discussion. References [] Compton, J., Feldman D., Orizaga, S. & Stein, M., A Model for West ile Virus Transmission VIGRE Summer Program-7, PD Dissertation. Matematics Department, Texas A & M Uniersity, Texas, 7. [] Senne, D.A., Pedersen, J.C., Hutto, D.L., Taylor, W.D., Scmitt B.J. & Panigray, B., Patogenicity of West ile Virus in Cickens, Aian Desease, 44, pp. 64-649,. [3] Centers for Disease Control (CDC), West ile Virus. Diision of Vector- Borne Infectious Diseases, ttp://www.cdc.go/ncidod/dbid/westnile/, ( Marc 4). [4] Tabler, T., Wells, J. & Zai, W. Farmers, Cickens, and West ile Virus, Mississippi State Uniersity, Publication 735B, 3. [5] Cruz-Paceco, G., Estea, L., Montaño-Hirose, J.A. & Vargas C., Modelling te Dynamics of West ile irus, Bull. Mat. Biol., 67(6), p. 57-63, 5. [6] Langeian, S.A., Bunning, M., Dais, B. & Komar,., Experimental Infection of Cickens as Candidate for West ile Virus, Imerging Infectious Diseases, 7(4), pp. 76-79,. [7] Wonam, M.J., De-Camino-Beck, T. & Lewis, M., An Epidemiological Model for West ile Virus: Inasion Analysis and Control Applications, Proc. Roy. Soc. London, Series B 538, pp. 5-57, 4. [8] Bowman, C., Gumel, A.B., Wu, J., Van-den Driessce, P. & Zu, H., A Matematical Model for Assessing Control Strategies Against West ile Virus, Bull. Mat. Biol., 67 (5), pp. 7-6, 5 [9] Lord, C. & Day, J.F., Simulation Studies of St. Louis Encepalitis Virus in Sout Florida, Vector Borne Zoonotic Diseases, (4), pp. 99-36,. [] Lord, C. & Day, J.F., Simulation Studies of St. Louis Encepalitis and West ile Viruses: te Impact of Bird Mortality, Vector Borne Zoonotic Diseases, (4), pp. 37-33,. [] Diekmann, O. & Heesterbeek, J.A.P., Matematical Epidemiologi of Infectious Diseases: Model Building, Analysis and Interpretation, Editor in Cief Simon Lein, Princeton Uniersity, USA, Wiley Series in Matematical and Computational Biology,.
3 Jafaruddin, et al. [] Van den Driessce, P. & Watmoug, J., Reproduction umbers and Subtresold Endemic Equilibria for Compartmental Models of Disease Transmission, Mat. Bioscience, 8, pp. 9-48,. [3] Horn, R.A. & Jonson, C.R., Topics in Matrix Analysis, Corrected Reprint of Te 99 Original, Cambridge Uniersity Press, Cambridge, 994.