# American Institute of Mathematical Sciences

September  2016, 21(7): 2363-2378. doi: 10.3934/dcdsb.2016051

## Stationary distribution of stochastic SIRS epidemic model with standard incidence

 1 College of Mathematic, Jilin University, Changchun 130012, Jilin, China, China 2 College of Mathematics, Beihua University, Jilin 132013, Jilin, China 3 College of Science, China University of Petroleum(East China), Qingdao 266580, Shandong, China 4 Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26, Richmond Street, Glasgow G1 1XH

Received  December 2014 Revised  September 2015 Published  August 2016

We study stochastic versions of a deterministic SIRS(Susceptible, Infective, Recovered, Susceptible) epidemic model with standard incidence. We study the existence of a stationary distribution of stochastic system by the theory of integral Markov semigroup. We prove the distribution densities of the solutions can converge to an invariant density in $L^1$. This shows the system is ergodic. The presented results are demonstrated by numerical simulations.
Citation: Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051
##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature, 280 (1979), 361-367. doi: 10.1007/978-3-642-68635-1. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases II, Nature, 280 (1979), 455-461. [3] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases, Berlin, Heidelberg. New York: Springer-Verlag, 1982. doi: 10.1007/978-3-642-68635-1. [4] R. M. Anderson and R. M. May, Infectious Diseases of Human: Dynamics and Control, Oxford: Oxford University Press. 1991. [5] G.B. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II), Probab. Theory Relat. Fields, 90 (1991), 377-402. doi: 10.1007/BF01193751. [6] D. R. Bell, The Malliavin Calculus, Dover publications, New York, 2006. [7] S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28 (1990), 257-270. doi: 10.1007/BF00178776. [8] 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. [9] 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. [10] R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, Netherlands, 1980. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [12] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [13] C. Ji, D. Jiang, Q. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044. [14] W. O. Kermack and A. G. McKendrick, Contribution to mathematical theory of epidemics, P. Roy. Soc. Lond. A Math., 115 (1927), 700-721. [15] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76. doi: 10.1515/rose-2016-0005. [16] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics & Probability Letters, 83 (2013), 960-968. doi: 10.1016/j.spl.2012.12.021. [17] M. Li, J. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. [18] Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873. [19] H. Liu, Q. Yang and D. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825. doi: 10.1016/j.automatica.2012.02.010. [20] Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Analysis: Real World Applications, 14 (2013), 1286-1299. doi: 10.1016/j.nonrwa.2012.09.016. [21] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University, 1973. [22] K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 215 (1997), 56-74. doi: 10.1006/jmaa.1997.5609. [23] R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl., 108 (2003), 93-107. doi: 10.1016/S0304-4149(03)00090-5. [24] R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 206 (2007), 108-119. doi: 10.1016/j.mbs.2006.03.006. [25] D. W. Stroock and S. R. S. Varadhan, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. III, University of California Press, Berkeley, 1972. [26] S. Aida, S. Kusuoka and D. Strook, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1972. [27] Q. Yang, D. Jiang and N. Shi, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 388 (2012), 248-271. doi: 10.1016/j.jmaa.2011.11.072. [28] Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Applied Mathematical Letter, 34 (2014), 90-93. doi: 10.1016/j.aml.2013.11.002. [29] Y. Zhao, D. Jiang, X. Mao and A. Gray, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete Continuous Dynam. Systems - B, 20 (2015), 1277-1295. doi: 10.3934/dcdsb.2015.20.1277. [30] Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Physica A, 392 (2013), 4916-4927. doi: 10.1016/j.physa.2013.06.009.

show all references

##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature, 280 (1979), 361-367. doi: 10.1007/978-3-642-68635-1. [2] R. M. Anderson and R. M. May, Population biology of infectious diseases II, Nature, 280 (1979), 455-461. [3] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases, Berlin, Heidelberg. New York: Springer-Verlag, 1982. doi: 10.1007/978-3-642-68635-1. [4] R. M. Anderson and R. M. May, Infectious Diseases of Human: Dynamics and Control, Oxford: Oxford University Press. 1991. [5] G.B. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II), Probab. Theory Relat. Fields, 90 (1991), 377-402. doi: 10.1007/BF01193751. [6] D. R. Bell, The Malliavin Calculus, Dover publications, New York, 2006. [7] S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28 (1990), 257-270. doi: 10.1007/BF00178776. [8] 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. [9] 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. [10] R. Z. Hasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, Netherlands, 1980. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [12] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [13] C. Ji, D. Jiang, Q. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044. [14] W. O. Kermack and A. G. McKendrick, Contribution to mathematical theory of epidemics, P. Roy. Soc. Lond. A Math., 115 (1927), 700-721. [15] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76. doi: 10.1515/rose-2016-0005. [16] A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statistics & Probability Letters, 83 (2013), 960-968. doi: 10.1016/j.spl.2012.12.021. [17] M. Li, J. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. [18] Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 1873-1887. doi: 10.3934/dcdsb.2013.18.1873. [19] H. Liu, Q. Yang and D. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825. doi: 10.1016/j.automatica.2012.02.010. [20] Z. Liu, Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Analysis: Real World Applications, 14 (2013), 1286-1299. doi: 10.1016/j.nonrwa.2012.09.016. [21] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University, 1973. [22] K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 215 (1997), 56-74. doi: 10.1006/jmaa.1997.5609. [23] R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochastic Process. Appl., 108 (2003), 93-107. doi: 10.1016/S0304-4149(03)00090-5. [24] R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 206 (2007), 108-119. doi: 10.1016/j.mbs.2006.03.006. [25] D. W. Stroock and S. R. S. Varadhan, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. III, University of California Press, Berkeley, 1972. [26] S. Aida, S. Kusuoka and D. Strook, On the Support of Diffusion Processes with Applications to the Strong Maximum Principle, Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1972. [27] Q. Yang, D. Jiang and N. Shi, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 388 (2012), 248-271. doi: 10.1016/j.jmaa.2011.11.072. [28] Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Applied Mathematical Letter, 34 (2014), 90-93. doi: 10.1016/j.aml.2013.11.002. [29] Y. Zhao, D. Jiang, X. Mao and A. Gray, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete Continuous Dynam. Systems - B, 20 (2015), 1277-1295. doi: 10.3934/dcdsb.2015.20.1277. [30] Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Physica A, 392 (2013), 4916-4927. doi: 10.1016/j.physa.2013.06.009.
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