A Mathematical Model on Chikungunya Disease with Standard Incidence and Disease Induced Death Rate

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1 A Matematical Model on Cikungunya Disease wit Standard ncidence and Disease nduced Deat Rate Meena Mandwariya, Vikram University, ndia, Pradeep Porwal, Vikram University, ndia, Sandeep Tiwari, Vikram University, ndia, Abstract- n tis paper, we extend te work of aowarat, Tongjaem and Tang by introducing te fact tat te model follows te disease related deats. Te model equation were solved and obtained equilibrium points of te system and analyzed for stability. t is sown tat te stability of te equilibrium points can be controlled by te basic reproduction number of te disease. Keywords- Matematical Modeling, Stability, Reproduction number, Equilibrium point, Cikungunya Disease.. TRODUCTO Cikungunya is an alpavirus tat infects umans troug bites from Aedes spp. mosquitoes. Symptoms are similar to tose of dengue fever during te acute pase and include ras and ig fever tat, in a small proportion of cases, can develop into a life-treatening aemorragic fever [7]. Additionally, joint pain tat is frequently associated wit infection can persist for over a year [9, 11] and is responsible for its name wic means tat wic bends in te Makonde language of Soutern Tanzania and ortern Mozambique. n 4, a major epidemic in Lamu, Kenya resulted in 13,5 cases [1]. Tis epidemic sparked a four-year period in wic te virus spread troug numerous islands of te ndian Ocean, ndia and parts of Souteast Asia. Cases were imported to Europe and ort America troug returning travellers, and subsequent autoctonous transmission events occurred due to te wide geograpical distribution of te vectors. Cikungunya is endemic in Asia causing several clinical cases and, sometimes, deats. Te principal vector of te Cikungunya is Aedes albopictus (sometimes called te Asian tiger because it originated from Asia and it is an aggressive mosquito), wic is also a prospective vector for Dengue transmission. Matematical models ave become te important tools for understanding te spread and control of disease. Many researcer give te study on infectious disease wit matematical modeling for te Cikungunya epidemic suc as Cabra and Mittal [3], Dubrulle et al.[4], Dumont et. al. [5], Martin et al.[6], Bacaer [], Moulay et al.[7], Anderson and May [1], Porwal and Badsa [1], Staples et. al.[13], Vazeille et. al. 7, 8 [14,15] and Woodruff et. al. [16]. n tis study, we are interested to study te Cikungunya diseases in te role of applying diseases induced deat on te dynamics of te disease. We present a matematical model and obtained a disease free and endemic equilibrium of te system and analyzed for stability. Te matematical model is based on [8]. 1

2 . THE MATHEMATCAL MODEL Te uman population is divided into te susceptible uman (S), te infected uman () and te recover uman population (R) compartment. Te mosquito population is divided into two compartments, te susceptible mosquito (S o ), te infected mosquito ( o ), te recovered mosquito does not exist. Let and denote te uman and mosquito population size. Te scematic description of model (.1) is in following fig.1. For mosquito: A S For Humans: B(1 p) S R 1 γ Fig.1: Flow cart for te Transmission of Cikungunya fever Te model equations are as follows: Human Population: ds dt d dt dr R dt B(1 p) 1 S 1S (.1) Mosquito Population: ds A S dt d S dt were B(µ) = Te birt (deat) rate of Human Population A = Te recruitment rate of Mosquito Population γ(b) = Te recovery (biting) rate of Human Population = Diseases induced deat rate = Te transmission rate of CHKV from infected mosquito to uman population = Te transmission rate of CHKV from infected uman to mosquito population µ = Te deat rate of Mosquito Population p = Te efficiency of mosquito repellent for protecting te mosquito in uman population

3 Reducing te model using normalizing te equation as: S R S S,, R and Sm, Ten te model (.1) becomes 1S A ds m B(1 p) S dt m 1S A d m ( ) dt dm Sm m (.) dt were R S and S EQULBRUM POTS Diseases free equilibrium points- Let E S,, m be te disease free equilibrium points of te system (.). Te disease free equilibrium points of te model (.) is obtained by equating te time derivative equals to zero, tat is ds d dm. dt dt dt B(1 p) S (Because ) We found tat te disease free equilibrium point B(1 p) E S,, m,, Endemic equilibrium point- Te endemic equilibrium of te model (.) is given by E S,, m By equation (.) S m m m M, were Again, By equation (.) B(1 p) S, MM 1 were M Finally, By equation (.) MM 1 B(1 p) ( ) ( ) MM 1 M 1 Sm 1A 3

4 Tus te disease endemic equilibrium point E1 S,, m were S B(1 p) MM 1 MM B(1 p) ( ) ( ) MM 1 1 m M 1 MM B(1 p) ( ) ( ) M 1 STABLTY OF THE MODEL To study te stability, system (.) can be written as 1S A m F1 B(1 p) S 1S A m F ( ) S m 3 m F Te Jacobian matrix of model (.) is given by 1mA 1SA 1mA 1SA J ( ) Sm Te Jacobian matrix at disease free equilibrium point M1S J ( ) MS 1 Sm Te eigen values of te Jacobian matrix are obtained by solving ( ) a1 a Clearly one Eigen values are negative and oter two Eigen values are given by te roots of te quadratic equation a1a were a1 SmM1S a ( ) t is clear tat all te coefficients of te caracteristics polynomial are positive if SmM1S R 1 ( ) 4

5 Ten by Rout-Hurwitz criterion te disease free equilibrium point E is locally asymptotically stable if R 1. Stability at endemic equilibrium point- Te Jacobian matrix at E1 S,, m M1m M1S 1 1 m ( ) 1 J M MS Sm its eigen value is obtained by solving 3 b1 b b3 were b M 1 1 m SmM1S b ( M1m )( ) ( ) SmM1S b3 ( M1m )( ) Te Eigen values of above caracteristics equation ave negative real part if tey satisfy te Rout-Hurwitz criteria. Tus, E 1 is local asymptotically stable for R >1 and satisfies te following conditions: 1) b 1 > ; ) b 3 > ; 3) b 1 b > b 3. COCLUSO n tis researc work, we consider te dynamics of Cikengunya epidemic model wit disease induced deat rate. Te resulting model equation were solved and analyzed. Te disease free and endemic equilibrium of te system was establised and analyzed for stability. t was found tat te disease free equilibrium to be locally asymptotically stable for R 1 and if R 1te endemic equilibrium exists and is locally asymptotically stable. REFERECES [1] Anderson, R.M. and R.M. May, nfectious Diseases of Humans: Dynamics and Control. 1st Edn., Oxford University Press, Oxford, SB: X, 199, pp: 768. [] Bacaer, Approximation of te basic reproduction number R for vector-borne diseases wit aperiodic vector population. Bull Mat Biol 69, 7, [3] Cabra M, Mittal V, Battacarya D, Rana U, Lal S, Cikungunya fever: a re emerging viral infection. ndian J Med Microbiol 6: 8, 5 1. [4] Dubrulle M, Mousson L, Moutailler S, Vazeille M, Failloux A-B, Cikungunya virus and Aedes mosquitoes: Saliva is infectious as soon as two days after oral infection. PLoS One 4(6), 9. [5] Dumont Y, Ciroleu F, Domerg C, On a temporal model for te Cikungunya disease: modeling, teory and numerics. Mat Biosci 13: 8, [6] Martin E, Moutailler S, Madec Y, Failloux AB, Differential responses of te mosquito Aedes albopictus from te ndian Ocean region to two cikungunya isolates. BMC Ecol 1:8, 1. [7] Moulay, D., M.A. Aziz-Alaoui and M. Cadivel, Te Cikungunya disease: Modeling, vector and transmission global dynamics. Mat. Biosci., 9: 11, PMD: [8] aowarat, S, Tongjaem, P and Tang M., Effect of Mosquito Repellent on te Transmission Model of Cikungunya Fever, American Journal of Applied Sciences 9 (4): 1, [9] Queyriaux B, Simon F, Grandadam M, Micel R, Tolou H, et al., Clinical burden of cikungunya virus infection. Lancet nfectious Diseases 8: 8, 3 [1] Porwal P and Badsa V. H., Dynamical Study of an Sirs Epidemic Model wit Vaccinated Susceptibility, Canadian Journal of Basic and Applied Sciences (4), 14, 9-96 [11] Robillard PY, Boumani B, Gerardin P, Micault A, Fourmaintraux A, et al., Vertical maternal fetal transmission of te cikungunya virus. Ten cases among 84 pregnant women. Presse Medicale 35: 6, [1] Sergon K, juguna C, Kalani R, Ofula V, Onyango C, et al., Seroprevalence of Cikungunya Virus (CHKV) nfection on Lamu sland, Kenya, October 4. Te American Journal of Tropical Medicine and Hygiene 78: 8,

6 [13] Staples JE, Breiman RF, Powers AM, Cikungunya Fever: An Epidemiological Review of a Re-Emerging nfectious Disease. Clinical nfectious Diseases 49: 9, [14] Vazeille M, Moutailler S, Coudrier D, Rousseaux C, Kun H, et al., Two Cikungunya solates from te Outbreak of La Reunion (ndian Ocean) Exibit Different Patterns of nfection in te Mosquito, Aedes albopictus. PLoS OE, 7. [15] Vazeille M, Jeannin C, Martin E, Scaffner F, Failloux AB, Cikungunya: a risk for Mediterranean countries? Acta Trop 15: 8,. [16] Woodruff AW, Bowen ET, Platt GS, Viral infections in travellers from tropical Africa. Br Med J 1: 1978,