# American Institute of Mathematical Sciences

September  2016, 21(7): 2109-2127. doi: 10.3934/dcdsb.2016039

## Analysis of stochastic vector-host epidemic model with direct transmission

 1 Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849 2 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, United States

Received  December 2015 Revised  February 2016 Published  August 2016

In this paper, we consider the stochastic vector-host epidemic model with direct transmission. First, we study the existence of a positive global solution and stochastic boundedness of the system of stochastic differential equations which describes the model. Then we introduce the basic reproductive number $\mathcal{R}^s_0$ in the stochastic model, which reflects the deterministic counterpart, and investigate the dynamics of the stochastic epidemic model when $\mathcal{R}^s_0 <1$ and $\mathcal{R}^s_0 >1$. In particular, we show that random effects may lead to extinction in the stochastic case while the deterministic model predicts persistence. Additionally, we provide conditions for the extinction of the infection and stochastic stability of the solution. Finally, numerical simulations are presented to illustrate some of the theoretical results.
Citation: Yanzhao Cao, Dawit Denu. Analysis of stochastic vector-host epidemic model with direct transmission. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2109-2127. doi: 10.3934/dcdsb.2016039
##### References:
 [1] M. Aguiar, N. Stollenwerk and B. W. Kooi, Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study,Epidemiology Insights,, ISBN: 978-953-51-0565-7, (): 978. [2] M. Andraud, N. Hens, C. Marais and P. Beutels, Dynamic epidemiological models for dengue transmission: A systematic review of structural approaches, PLoS One, 7 (2012), e49085. doi: 10.1371/journal.pone.0049085. [3] L. Arnold, Stochastic Differential Equations: Theory and Applications, A Wiley-Interscience Publication, 1971. [4] J. R. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. doi: 10.1126/science.197.4302.463. [5] K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of west nile virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0. [6] K. W. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 587-611. doi: 10.3934/dcdsb.2009.11.587. [7] Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic dynamics of an sirs epidemic model with ratio-dependent incidence rate, Abstract and Applied Analysis, 2013 (2013), Article ID 172631, 11pp. [8] L. Cai and Xuezhi Li, Analysis of a simple vector-host epidemic model with direct transmission, Discrete Dynamics in Nature and Society, (2010), Article ID 679613, 12pp. doi: 10.1155/2010/679613. [9] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53. doi: 10.1016/j.jmaa.2006.01.055. [10] L. C. Evans, An Introduction to Stochastic Differential Equations, University of California, Berkeley, CA, 2013. doi: 10.1090/mbk/082. [11] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation sis epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X. [12] D. O, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, Math. Biol., 35 (1990), 503-522. [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [14] , href=, (). [15] [16] [17] [18] M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission, Applied Mathematical Modelling, 36 (2012), 5214-5228. doi: 10.1016/j.apm.2011.11.087. [19] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992. doi: 10.1007/978-3-662-12616-5. [20] X. Ling, Modeling and Analysis of Vector-borne Diseases on Complex Networks, PhD Thesis, Kansas State University, 2013. [21] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical Models Of Malaria - A Review, Malar J., 10, 2011, 202. [22] X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, second edition, January 13, 2008. doi: 10.1533/9780857099402. [23] M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61. Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3. [24] F. E. Mckenzie, Why Model Malaria?, Parasitology Today, 16 (2000), 511-516. doi: 10.1016/S0169-4758(00)01789-0. [25] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2. [26] K. Okosunl and O. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math., 12 (2013), 20-32. [27] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Computers and Mathematics with Applications, 56 (2008), 3118-3129. doi: 10.1016/j.camwa.2008.09.002. [28] K. Rafail, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [29] R. Rosenberg and C. B. Beard, Vector-borne infections, CDCEID journal, 17 (2011), 2pp. doi: 10.3201/eid1705.110310. [30] M. Samsuzzoha, M. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915. doi: 10.1016/j.apm.2012.03.029. [31] Smallpox: Disease, Prevention, and Intervention., The CDC and the World Health Organization,, History and Epidemiology of Global Smallpox Eradication From the training course, (): 16. [32] M. O. Souza, Multiscale analysis for a vector-borne epidemic model, J. Math. Biol., 68 (2014), 1269-1293. doi: 10.1007/s00285-013-0666-6. [33] D. R. Stirzaker, A perturbation method for the stochastic recurrent epidemic, EpidemicIMA J Appl Math., 15 (1975), 135-160. doi: 10.1093/imamat/15.2.135-a. [34] N. Stollenwerk, M. Aguiar, S. Ballesteros, J. Boto, B. Kooi and L. Mateus, Dynamic Noise, Chaos and Parameter Estimation in Population Biology, Interface Focus, 2012. doi: 10.1098/rsfs.2011.0103. [35] J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment, PNAS, 100 ( 2003), 9067-9072. doi: 10.1073/pnas.1436273100. [36] Q. Wei, Z. Xiong and F. Wang, Dynamic of a Stochastic SIR Model Under Regime Switching, Journal of Information & Computational Science, 10 (2013), 2727-2734. doi: 10.12733/jics20101856. [37] H. M. Wei, X. Z. Li and M. Martcheva, An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342 (2008), 895-908. doi: 10.1016/j.jmaa.2007.12.058. [38] M. J. Wonham and M. A. Lewis, A Comparative Analysis of Models for West Nile Virus, Mathematical Epidemiology Lecture Notes in Mathematics, 1945 (2008), 365-390. doi: 10.1007/978-3-540-78911-6_14. [39] H. Yang, H. Wei and X. Li, Global stability of an epidemic model for vector-borne disease, J Syst Sci Complex Journal, 23 (2010), 279-292. doi: 10.1007/s11424-010-8436-7. [40] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Control Optim, 46 (2007), 1155-1179. doi: 10.1137/060649343. [41] L. Zu, D. Jiang and D. O'Regan, Stochastic Permanence, Stationary Distribution and Extinction of a Single-Species Nonlinear Diffusion System withRandom Perturbation, Abstract and Applied Analysis, Article ID 320460, 2014. doi: 10.1155/2014/320460.

show all references

##### References:
 [1] M. Aguiar, N. Stollenwerk and B. W. Kooi, Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study,Epidemiology Insights,, ISBN: 978-953-51-0565-7, (): 978. [2] M. Andraud, N. Hens, C. Marais and P. Beutels, Dynamic epidemiological models for dengue transmission: A systematic review of structural approaches, PLoS One, 7 (2012), e49085. doi: 10.1371/journal.pone.0049085. [3] L. Arnold, Stochastic Differential Equations: Theory and Applications, A Wiley-Interscience Publication, 1971. [4] J. R. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. doi: 10.1126/science.197.4302.463. [5] K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of west nile virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0. [6] K. W. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 587-611. doi: 10.3934/dcdsb.2009.11.587. [7] Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic dynamics of an sirs epidemic model with ratio-dependent incidence rate, Abstract and Applied Analysis, 2013 (2013), Article ID 172631, 11pp. [8] L. Cai and Xuezhi Li, Analysis of a simple vector-host epidemic model with direct transmission, Discrete Dynamics in Nature and Society, (2010), Article ID 679613, 12pp. doi: 10.1155/2010/679613. [9] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53. doi: 10.1016/j.jmaa.2006.01.055. [10] L. C. Evans, An Introduction to Stochastic Differential Equations, University of California, Berkeley, CA, 2013. doi: 10.1090/mbk/082. [11] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation sis epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902. doi: 10.1137/10081856X. [12] D. O, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, Math. Biol., 35 (1990), 503-522. [13] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [14] , href=, (). [15] [16] [17] [18] M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission, Applied Mathematical Modelling, 36 (2012), 5214-5228. doi: 10.1016/j.apm.2011.11.087. [19] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992. doi: 10.1007/978-3-662-12616-5. [20] X. Ling, Modeling and Analysis of Vector-borne Diseases on Complex Networks, PhD Thesis, Kansas State University, 2013. [21] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical Models Of Malaria - A Review, Malar J., 10, 2011, 202. [22] X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, second edition, January 13, 2008. doi: 10.1533/9780857099402. [23] M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61. Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3. [24] F. E. Mckenzie, Why Model Malaria?, Parasitology Today, 16 (2000), 511-516. doi: 10.1016/S0169-4758(00)01789-0. [25] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2. [26] K. Okosunl and O. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math., 12 (2013), 20-32. [27] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Computers and Mathematics with Applications, 56 (2008), 3118-3129. doi: 10.1016/j.camwa.2008.09.002. [28] K. Rafail, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [29] R. Rosenberg and C. B. Beard, Vector-borne infections, CDCEID journal, 17 (2011), 2pp. doi: 10.3201/eid1705.110310. [30] M. Samsuzzoha, M. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915. doi: 10.1016/j.apm.2012.03.029. [31] Smallpox: Disease, Prevention, and Intervention., The CDC and the World Health Organization,, History and Epidemiology of Global Smallpox Eradication From the training course, (): 16. [32] M. O. Souza, Multiscale analysis for a vector-borne epidemic model, J. Math. Biol., 68 (2014), 1269-1293. doi: 10.1007/s00285-013-0666-6. [33] D. R. Stirzaker, A perturbation method for the stochastic recurrent epidemic, EpidemicIMA J Appl Math., 15 (1975), 135-160. doi: 10.1093/imamat/15.2.135-a. [34] N. Stollenwerk, M. Aguiar, S. Ballesteros, J. Boto, B. Kooi and L. Mateus, Dynamic Noise, Chaos and Parameter Estimation in Population Biology, Interface Focus, 2012. doi: 10.1098/rsfs.2011.0103. [35] J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment, PNAS, 100 ( 2003), 9067-9072. doi: 10.1073/pnas.1436273100. [36] Q. Wei, Z. Xiong and F. Wang, Dynamic of a Stochastic SIR Model Under Regime Switching, Journal of Information & Computational Science, 10 (2013), 2727-2734. doi: 10.12733/jics20101856. [37] H. M. Wei, X. Z. Li and M. Martcheva, An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342 (2008), 895-908. doi: 10.1016/j.jmaa.2007.12.058. [38] M. J. Wonham and M. A. Lewis, A Comparative Analysis of Models for West Nile Virus, Mathematical Epidemiology Lecture Notes in Mathematics, 1945 (2008), 365-390. doi: 10.1007/978-3-540-78911-6_14. [39] H. Yang, H. Wei and X. Li, Global stability of an epidemic model for vector-borne disease, J Syst Sci Complex Journal, 23 (2010), 279-292. doi: 10.1007/s11424-010-8436-7. [40] C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Control Optim, 46 (2007), 1155-1179. doi: 10.1137/060649343. [41] L. Zu, D. Jiang and D. O'Regan, Stochastic Permanence, Stationary Distribution and Extinction of a Single-Species Nonlinear Diffusion System withRandom Perturbation, Abstract and Applied Analysis, Article ID 320460, 2014. doi: 10.1155/2014/320460.
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