Effect of Mosquito Repellent on the Transmission Model of Chikungunya Fever

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1 Aerican Journal of Applied Sciences 9 (4): , ISSN Science Publications Effect of Mosquito Repellent on te Transission Model of Cikungunya Fever Surapol Naowarat, Prasit Tongjae and I. Ming Tang Departent of Mateatics, Faculty of Science and Tecnology, Surattani Rajabat University, Surat Tani, 84, Tailand Departent of Pysics, Faculty of Science, Maidol University, Bangkok, 4, Tailand Abstract: Proble stateent: In 8 tere was a large outbreak of Cikungunya fever in soutern Tailand. Cikungunya fever is an eerging disease wic tends to affect rubber plantation and fruit orcard workers ore tan oter occupation. Tis study we considers te efficacy of using osquito repellent as a way to prevent and control te spread of Cikungunya fever. Te ateatical odel of te dynaic of tis disease is proposed and analyzed. Approac: A standard dynaical odeling etod was applied for analysis te dynaical odel. Te stability of te odel was deterined by using Rout-Hurwitz criteria. Results: Te conditions for disease free and endeic state are found. To deterine te basic reproductive nuber (R ) wic is te tresold paraeter, if R <, te disease free equilibriu point is locally asyptotically stable. If R >, tere exist te endeic equilibriu state, wic is locally asyptotically stable. Conclusion: It was found tat te use of osquito repellent significantly reduce transission and infection of tis disease wic it ay be an alternative intervention for counities to prevent and control te disease. Key words: Mateatical odel, cikungunya fever, basic reproductive nuber, equilibriu point, stability analysis, osquito repellent INTRODUCTION Cikungunya fever is an artropod-borne viral disease caused by an alpavirus of Togavivirus faily, wic was isolated in 953 in Tanzania (Pioloux et al., 7). Te nae, Cikungunya, originates fro te Monkonde dialec of Tanzania and refers to te patient s contorted posture as a result of severe joint paints (Tilston et al., 9). Te syptos of Cikungunya include sudden onset of fever, cills, eadace, nausea, voiting, joints paints wit and witout swelling, low back pain, an ras. Tese syptos are very siilar to tose of dengue fever; tere is no eorraagic or sock syndroe for (Massad et al., 8). In te ajor outbreak of tis disease in 5 on te island of Reunion, 44, out of a of 775, inabitants reported tat tey ad experienced tese syptos (Moulay et al., ). Cikungunya virus (CHIKV) is transitted to uans by Aedes osquitoes, widespread in soe tropical regions. Ae. albopictus is igly copetent for CHIKV (Poletti et al., ). Tere ave been any explanation tat for te occurrence of large-scale Cikungunya epideics. New strains of virus are particularly infectious to Ae. albopictus, tereby increasing te vector copetence of tis species. However, vector copetence consists of a variety of factors including vector life span, preference for biting uan ore tan non-uan osts, infectiousness wen infected, density and transovarial transission(tsetsarkin et al., 7). Wit te ongoing outbreak in Tailand, a new strain of te CHIKV was isolated in bot osquito serotypes (Tavara et al., 9). Studies ave sown tat Ae. albopictus is ore susceptible to te virus and terefore a ore effective transitter. Te virus was found in bot sexes of tose two species of te osquitoes, indicating a possible role for transovarial transission of virus. Mateatical odels ave becoe te iportant tools for understanding te spread and control of disease. In tis study, we are interested in te role of applying osquito repellent on te dynaical of disease and ow te efficacy of osquito repellent relates to prevent and control te spread of disease. MATERIALS AND METHODS Model forulation: In our odel, we assue tat uan and osquito are constant denoted by N and N, respectively. Corresponding Autor: Surapol Naowarat, Departent of Mateatics, Faculty of Science and Tecnology, Surattani Rajabat University, Surat Tani, 84, Tailand 563

2 A. J. Applied Sci., 9 (4): , R S,I,R N N N S S I I µ µ S,I N A / ( ) N A / ( ) Fig. : Flow cart for te transission of Cikungunya fever Te dynaics of te disease is depicted in te copartent diagra, Fig.. Te uan is divided into te susceptible uan (S ), te infected uan (I ) and te recover uan (R ) copartent. Te osquito is divided into two copartents, te susceptible osquito (S ), te infected osquito (I ), te recovered osquito does not exist, since te osquito infected for over all teir life. Te transission dynaics of te cikungunya fever are described by te following ordinary differential equations Eq. a and b: ds di dr µ µ (a) ( p)n ( ) S N ( ) (r )I (b) N r I µ R (c) ds di A ( ) µ S (d) N µ I (e) N Te Eq. c and d can be canceled since te uan and osquito are constant. i.e., R N S I and S N I. By doing tis we get Since te total uan and osquito s are constant, tus te tie rate of cange of uan ds di dr equal to zero, i.e., + +. It eans tat te birt rate and te deat rate of uan are equal, tat is λ µ. Te total nuber A of Aedes osquito at equilibriu equal to µ. Were: N Te total uan N Te total osquito S, I, R Te nuber of susceptible, infected, recovered uan, respectively S (I ) Te nuber of susceptible (infected) osquito λ, (µ ) Te birt (deat) rate of uan A Te recruitent rate of osquito r (b) Te recovery (biting) rate of uan γ Te transission rate of CHIKV fro infected osquito to uan : γ () N were, γ is te transission rate of CHIKV fro infected uan to osquito : γ N were, µ is te deat rate of osquito. And p is te efficacy of osquito repellent for protecting te osquito in uan. Te reduced odel is depicted as following: ds b β (A / µ ) µ ( p) µ S (3) N tree dependent variables, wic we pick to be S, I, I To analyze te odel by noralizing te Eq. a, b, e and defining new variables Eq. : 564 di b β (A / µ )SI (r )I (4) N

3 A. J. Applied Sci., 9 (4): , di b β ( I )I µ I (5) Analysis of te odel: Equilibriu points: Te odel will be analyzed to investigate te equilibriu points and its stability. Te syste as two possible equilibriu points: te disease free equilibriu point and an endeic equilibriu point. Two equilibriu points are found by setting te RHS of Eq. 3-5 to zero. We obtained Disease Free Equilibriu Point (E ): In te absence of disease in te counity, tat is I, I. We obtained S p, ten E ( p,, ). Endeic Equilibriu Point (E ): In te oter case wen te disease is presented in te counity, I,I, we obtained: ( p) µ M (r )M I (r )( + M ) I M + I I S ( p)(m + I ) M + ( + M )I Wit M A µ and M µ µ N b β (6) (7) (8) Tus, te endeic equilibriu point E (S,I,I ). is Local asyptotical stability: Te local stability of an equilibriu point is deterined fro te Jacobian atrix of te syste of ordinary differential Eq. 3-5 evaluated at eac equilibriu point. Te Jacobian atrix at E is sown as: Were: A r, A µ (r ) M ( p) Fro te caracteristic Eq. 9, we get one of eigenvalue is λ µ <. Next, to deterine te oter eigenvalues fro te caracteristic equation λ + Aλ + A. Te root of tis equation is negative if it is satisfied wit two conditions of Rout-Hurwitz criteria. E is local asyptotically stable if λ + Aλ + A satisfies te following conditions: ) A > ; ) A > To deterine te stability of te equilibriu point (E ), we can see tat A is always positive and A is positive wen: M ( p) < µ (r ) Moreover, we found tat te disease free equilibriu point is locally asyptotically stable wen R o <were: R M ( p) (r ) µ Disease endeic equilibriu point: To deterine te stability of te endeic equilibriu point, E, by finding te eigenvalues of Jacobian atrix at E, as follow: µ M I M S µ I µ J M S M I M S were, S,I,I are given by Eq Te caracteristic equation of Jacobian atrix at E, given by Eq. 3-5, is: µ M ( p) J (r ) M ( p) µ Te eigenvalues of te J are obtained by solving det (J λi). We obtained te caracteristic equations: ( λ )( λ + A λ + A ) (9) λ + B λ + B λ + B 3 3 Were: B r + µ + I + M I, B ( µ + I )(µ + r + M I ) + ( µ + M I )(r ) (b β )(M S ), B (r )( µ + M I )( µ + I ) + 3 (M S + MI ) ( µ + MI )(b β )(MS ), 565

4 Te tree eigenvalues of λ 3 +B λ + B λ+b 3 ave negative real part if tey satisfy te Rout- Hurwitz criteria. Tus, E is local asyptotically stable for R > if λ 3 +B λ + B λ+b 3 satisfies te following conditions: ) B > ; ) B > ; 3 3) B B > B 3 RESULTS A. J. Applied Sci., 9 (4): , Nuerical results: In tis study, we are interested in te transission of Cikungunya fever wit te effect of efficacy of osquito repellent for protecting te osquito. Te paraeters are obtained fro te field study for soe paraeters suc as µ / (65 365) day corresponds to a life expectancy of 65 years in uan, te recovery rate of uan r /7 day, wic is corresponded to te recovery rate of uan is 7 days, te ean life of osquito µ /7 day, correspond to te expectancy of 7 days. Oter paraeters are arbitrarily cosen, suc as te probability tat CHIKV is transitted to uan is β.5 and te probability tat CHIKV is transitted to osquito is β. as sown in Table. (a) Stability of disease free state: Fro te values of paraeters listed in Table, we obtained te eigenvalues and basic reproductive nuber is: λ.596, λ.33737, λ.4, R.469 < 3 (b) Since all of tese eigenvalues are to be negative and te basic reproductive nuber to be less tan one, te equilibriu state will be te disease free state, E (.,,) as sown in Fig. (a) Susceptible uan (S ) and (b) Infected uan (I ), Infected osquito (I ), proportion. Te values of paraeters are in te text. Te solutions converge to te disease free equilibriu state as sown. Stability of endeic state: Next we cange te value of efficacy rate of te use of osquito repellent for protecting in uan to be p.3 and β.7, β.5. Te oter values of paraeters are listed in Table, we obtained te eigenvalues and basic reproductive nuber is: λ.45, λ.743, λ > , R Fig. : Tie series evolution of te copartent s of te case at te disease free state Table : Paraeter values used in nuerical siulations at Disease Free State Paraeters Description Values µ Birt (deat) rate of uan / (65 365) day A Recruitent rate 5 of Aedes osquito β Probability tat.5 CHIKV transit fro Aedes osquito to uan β Probability tat CHIKV. transit fro uan to Aedes osquito r Recovery rate of uan /7 day µ Deat rate of /7 day Aedes osquito N Nuber of uan p Efficacy rate of te use.9 of osquito repellent for protecting in uan

A. J. Applied Sci., 9 (4): 563-569, (a) (b) Fig.

5 A. J. Applied Sci., 9 (4): , (a) (b) Fig. 3: Tie series evolution of te copartents of te case at te endeic state Since all of tese eigenvalues are to be negative and te basic reproductive nuber to be greater tan one, te equilibriu state will be te endeic state, E as sown in Fig. 3. (a) Susceptible uan, (b) Infected uan,(c) infected osquito, proportion. Te values of paraeters are in te text, te solutions oscillate to te endeic equilibriu state. In particular, we are interested in te nuerical results wen te value of efficacy rate of te use of osquito repellent for protecting in uan by te different values of p.3, p.4, p.5, p.6 and p.7 wen te oter paraeters are te sae 567 (c) as using at endeic state. Te nuber of infected uans was decreased as p increased as sown in Fig. 4. DISCUSSION We forulate te transission odel of Cikungunya by considering te effect of osquito repellent to protect osquito biting. Te basic reproductive nuber is R R were: R M ( p) (r ) µ

6 A. J. Applied Sci., 9 (4): , (a) (b) (c) (d) Fig. 4: Tie series of infected uan proportions. Te values of paraeters are te sae as in endeic state wit different value of p, (a) p.3, (b) p.4, (c) p.5, (d) p.6, (e) p.7. Te nuber of infected uans was decreased as p increased 568 (e)

7 A. J. Applied Sci., 9 (4): , R represent te nuber of secondary cases tat arise wen a single infective ost is introduced into a susceptible ost troug patogen transission by te vector (Anderson and May, 99). R is te tresold condition and te stability of te solutions are sown in Fig. and 3. For endeic state te value of R is very ig due to uan as low natural iunity wit tis strain of CHIKV, include te ig obility of uan in te counity and a genetic cange at te position 6 in te gene for te glycoprotein E/E created a utated CHIKV strain wic ad an increased capability for replication in te Ae. albopictus (Tsetsarkin, 9) and tis osquito is aggressive day tie biter (3-46 ties per our), lives a long life (4-8 weeks)and as long fligt radius (4-6 eters) (Pioloux et al., 7). CONCLUSION We can conclusion tat wen te nubers of uan wic used te osquito repellents increasing, it is decreasing te nubers of te susceptible uan to receive te CHIKV. In Tailand, as Cikungunya fever as an epideic. Tese alternative intervention could decreasing te nubers of infected uan by capaign te outdoor workers using wisdo of counity, tat is, a osquito repellent (Citrinella grass product) to protect teselves. Tus, te results of tis study will te appropriate way to control te outbreak of tis disease. ACKNOWLEDGEMENT Surapol Naowarat would like to tank Surattani Rajabat University for financial support according to Code:554A593. REFERENCES Massad, E., S. Ma, M.N. Burattini, Y. Tun, F.A.B. Coutino et al., 8. Te risk of cikungunya fever in a dengue-endeic area. J. Travel Med., 5: DOI:./j x Moulay, D., M.A. Aziz-Alaoui and M. Cadivel,. Te Cikungunya disease: Modeling, vector and transission global dynaics. Mat. Biosci., 9: PMID: 7789 Pioloux, G., B.A. Gauzere, S. Jaueguiberry and M. Strobel, 7. Cikungunya, an epideic arbovirosis. Lancet Infect. Dis., 7: PMID: Poletti, P., G. Messeri, M. Ajelli, R. Vallorani and C. Rizzo et al.,. Transission potential of cikungunya virus and control easures: Te case of Italy. PLoS ONE, 6: e886-e886. DOI:.37/journal.pone.886 Tavara, U., A. Tawatsin, T. Pengsakul, P. Bakdeenuan, S. Canaa et al., 9. Outbreak of cikungunya fever in Tailand and virus detection in field of vector osquitoes, Aedes aegypti (L.) and Aedes albopictus Skuse (Diptera: Culicidae). Souteast Asian J. Trop. Med. Public Healt, 4: PMID: Tilston, N., C. Skelly and P. Weinstein, 9. Pan- European Cikungunya surveillance: designing risk stratified surveillance zones. Int. J. Healt Geograpics, 8: 6-6. DOI:.86/476-7X- 8-6 Tsetsarkin, K.A., 9. Adaptation of Cikungunya virus to Aedes albopictus osquitoes: Te role of utations in te E and E glycoprotein. ProQuest, LLC. Tsetsarkin, K.A., D.L. Vanlandinga, C.E. McGee and S. Higgs, 7. A single utation in cikungunya virus affects vector specificity and epideic potential. PLos Patog 3, e- e. PMID: Anderson, R.M. and R.M. May, 99. Infectious Diseases of Huans: Dynaics and Control. st Edn., Oxford University Press, Oxford, ISBN: 98544X, pp:

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