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 & 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. Google Scholar

[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). doi: 10.1371/journal.pone.0049085. Google Scholar

[3]

L. Arnold, Stochastic Differential Equations: Theory and Applications,, A Wiley-Interscience Publication, (1971). Google Scholar

[4]

J. R. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463. doi: 10.1126/science.197.4302.463. Google Scholar

[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. doi: 10.1007/s11538-009-9480-0. Google Scholar

[6]

K. W. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention,, Discrete and Continuous Dynamical Systems, 11 (2009), 587. doi: 10.3934/dcdsb.2009.11.587. Google Scholar

[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). Google Scholar

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L. Cai and Xuezhi Li, Analysis of a simple vector-host epidemic model with direct transmission,, Discrete Dynamics in Nature and Society, (2010). doi: 10.1155/2010/679613. Google Scholar

[9]

N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use,, J. Math. Anal. Appl., 325 (2007), 36. doi: 10.1016/j.jmaa.2006.01.055. Google Scholar

[10]

L. C. Evans, An Introduction to Stochastic Differential Equations,, University of California, (2013). doi: 10.1090/mbk/082. Google Scholar

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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. doi: 10.1137/10081856X. Google Scholar

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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. Google Scholar

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D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

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M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission,, Applied Mathematical Modelling, 36 (2012), 5214. doi: 10.1016/j.apm.2011.11.087. Google Scholar

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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. Google Scholar

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X. Ling, Modeling and Analysis of Vector-borne Diseases on Complex Networks,, PhD Thesis, (2013). Google Scholar

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S. Mandal, R. R. Sarkar and S. Sinha, Mathematical Models Of Malaria - A Review,, Malar J., (2011). Google Scholar

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X. Mao, Stochastic Differential Equations and Applications,, Woodhead Publishing, (2008). doi: 10.1533/9780857099402. Google Scholar

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M. Martcheva, An Introduction to Mathematical Epidemiology,, Texts in Applied Mathematics, (2015). doi: 10.1007/978-1-4899-7612-3. Google Scholar

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F. E. Mckenzie, Why Model Malaria?,, Parasitology Today, 16 (2000), 511. doi: 10.1016/S0169-4758(00)01789-0. Google Scholar

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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. doi: 10.1016/S0895-7177(00)00169-2. Google Scholar

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K. Okosunl and O. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate,, Appl. Comput. Math., 12 (2013), 20. Google Scholar

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Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure,, Computers and Mathematics with Applications, 56 (2008), 3118. doi: 10.1016/j.camwa.2008.09.002. Google Scholar

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K. Rafail, Stochastic Stability of Differential Equations,, Springer-Verlag Berlin Heidelberg, (2012). doi: 10.1007/978-3-642-23280-0. Google Scholar

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R. Rosenberg and C. B. Beard, Vector-borne infections,, CDCEID journal, 17 (2011). doi: 10.3201/eid1705.110310. Google Scholar

[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. doi: 10.1016/j.apm.2012.03.029. Google Scholar

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Smallpox: Disease, Prevention, and Intervention., The CDC and the World Health Organization,, History and Epidemiology of Global Smallpox Eradication From the training course, (): 16. Google Scholar

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M. O. Souza, Multiscale analysis for a vector-borne epidemic model,, J. Math. Biol., 68 (2014), 1269. doi: 10.1007/s00285-013-0666-6. Google Scholar

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D. R. Stirzaker, A perturbation method for the stochastic recurrent epidemic,, EpidemicIMA J Appl Math., 15 (1975), 135. doi: 10.1093/imamat/15.2.135-a. Google Scholar

[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. Google Scholar

[35]

J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment,, PNAS, 100 (2003), 9067. doi: 10.1073/pnas.1436273100. Google Scholar

[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. doi: 10.12733/jics20101856. Google Scholar

[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. doi: 10.1016/j.jmaa.2007.12.058. Google Scholar

[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. doi: 10.1007/978-3-540-78911-6_14. Google Scholar

[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. doi: 10.1007/s11424-010-8436-7. Google Scholar

[40]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems,, Control Optim, 46 (2007), 1155. doi: 10.1137/060649343. Google Scholar

[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, (3204). doi: 10.1155/2014/320460. Google Scholar

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. Google Scholar

[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). doi: 10.1371/journal.pone.0049085. Google Scholar

[3]

L. Arnold, Stochastic Differential Equations: Theory and Applications,, A Wiley-Interscience Publication, (1971). Google Scholar

[4]

J. R. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463. doi: 10.1126/science.197.4302.463. Google Scholar

[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. doi: 10.1007/s11538-009-9480-0. Google Scholar

[6]

K. W. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention,, Discrete and Continuous Dynamical Systems, 11 (2009), 587. doi: 10.3934/dcdsb.2009.11.587. Google Scholar

[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). Google Scholar

[8]

L. Cai and Xuezhi Li, Analysis of a simple vector-host epidemic model with direct transmission,, Discrete Dynamics in Nature and Society, (2010). doi: 10.1155/2010/679613. Google Scholar

[9]

N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use,, J. Math. Anal. Appl., 325 (2007), 36. doi: 10.1016/j.jmaa.2006.01.055. Google Scholar

[10]

L. C. Evans, An Introduction to Stochastic Differential Equations,, University of California, (2013). doi: 10.1090/mbk/082. Google Scholar

[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. doi: 10.1137/10081856X. Google Scholar

[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. Google Scholar

[13]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

[14]

, href=, (). Google Scholar

[15]

, , (). Google Scholar

[16]

, , (). Google Scholar

[17]

, , (). Google Scholar

[18]

M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission,, Applied Mathematical Modelling, 36 (2012), 5214. doi: 10.1016/j.apm.2011.11.087. Google Scholar

[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. Google Scholar

[20]

X. Ling, Modeling and Analysis of Vector-borne Diseases on Complex Networks,, PhD Thesis, (2013). Google Scholar

[21]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical Models Of Malaria - A Review,, Malar J., (2011). Google Scholar

[22]

X. Mao, Stochastic Differential Equations and Applications,, Woodhead Publishing, (2008). doi: 10.1533/9780857099402. Google Scholar

[23]

M. Martcheva, An Introduction to Mathematical Epidemiology,, Texts in Applied Mathematics, (2015). doi: 10.1007/978-1-4899-7612-3. Google Scholar

[24]

F. E. Mckenzie, Why Model Malaria?,, Parasitology Today, 16 (2000), 511. doi: 10.1016/S0169-4758(00)01789-0. Google Scholar

[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. doi: 10.1016/S0895-7177(00)00169-2. Google Scholar

[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. Google Scholar

[27]

Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure,, Computers and Mathematics with Applications, 56 (2008), 3118. doi: 10.1016/j.camwa.2008.09.002. Google Scholar

[28]

K. Rafail, Stochastic Stability of Differential Equations,, Springer-Verlag Berlin Heidelberg, (2012). doi: 10.1007/978-3-642-23280-0. Google Scholar

[29]

R. Rosenberg and C. B. Beard, Vector-borne infections,, CDCEID journal, 17 (2011). doi: 10.3201/eid1705.110310. Google Scholar

[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. doi: 10.1016/j.apm.2012.03.029. Google Scholar

[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. Google Scholar

[32]

M. O. Souza, Multiscale analysis for a vector-borne epidemic model,, J. Math. Biol., 68 (2014), 1269. doi: 10.1007/s00285-013-0666-6. Google Scholar

[33]

D. R. Stirzaker, A perturbation method for the stochastic recurrent epidemic,, EpidemicIMA J Appl Math., 15 (1975), 135. doi: 10.1093/imamat/15.2.135-a. Google Scholar

[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. Google Scholar

[35]

J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment,, PNAS, 100 (2003), 9067. doi: 10.1073/pnas.1436273100. Google Scholar

[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. doi: 10.12733/jics20101856. Google Scholar

[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. doi: 10.1016/j.jmaa.2007.12.058. Google Scholar

[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. doi: 10.1007/978-3-540-78911-6_14. Google Scholar

[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. doi: 10.1007/s11424-010-8436-7. Google Scholar

[40]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems,, Control Optim, 46 (2007), 1155. doi: 10.1137/060649343. Google Scholar

[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, (3204). doi: 10.1155/2014/320460. Google Scholar

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