# 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 & 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. 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. [3] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases,, Berlin, (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., (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. doi: 10.1007/BF01193751. [6] D. R. Bell, The Malliavin Calculus,, Dover publications, (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. 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. 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. doi: 10.1137/10081856X. [10] R. Z. Hasminskii, Stochastic Stability of Differential Equations,, Sijthoff & Noordhoff, (1980). [11] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. [12] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. 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. 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. [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. 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. 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. 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. 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. 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. 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. doi: 10.1006/jmaa.1997.5609. [23] R. Rudnicki, Long-time behaviour of a stochastic prey-predator model,, Stochastic Process. Appl., 108 (2003), 93. 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. 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, (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, (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. 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. 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. 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. 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. 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. [3] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases,, Berlin, (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., (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. doi: 10.1007/BF01193751. [6] D. R. Bell, The Malliavin Calculus,, Dover publications, (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. 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. 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. doi: 10.1137/10081856X. [10] R. Z. Hasminskii, Stochastic Stability of Differential Equations,, Sijthoff & Noordhoff, (1980). [11] H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. [12] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. 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. 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. [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. 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. 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. 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. 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. 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. 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. doi: 10.1006/jmaa.1997.5609. [23] R. Rudnicki, Long-time behaviour of a stochastic prey-predator model,, Stochastic Process. Appl., 108 (2003), 93. 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. 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, (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, (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. 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. 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. 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. doi: 10.1016/j.physa.2013.06.009.
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