# American Institute of Mathematical Sciences

June  2015, 20(4): 1277-1295. doi: 10.3934/dcdsb.2015.20.1277

## The threshold of a stochastic SIRS epidemic model in a population with varying size

 1 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024 3 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom

Received  August 2013 Revised  February 2014 Published  February 2015

In this paper, a stochastic susceptible-infected-removed-susceptible (SIRS) epidemic model in a population with varying size is discussed. A new threshold $\tilde{R}_0$ is identified which determines the outcome of the disease. When the noise is small, if $\tilde{R}_0<1$, the infected proportion of the population disappears, so the disease dies out, whereas if $\tilde{R}_0>1$, the infected proportion persists in the mean and we derive that the disease is endemic. Furthermore, when ${R}_0 > 1$ and subject to a condition on some of the model parameters, we show that the solution of the stochastic model oscillates around the endemic equilibrium of the corresponding deterministic system with threshold ${R}_0$, and the intensity of fluctuation is proportional to that of the white noise. On the other hand, when the noise is large, we find that a large noise intensity has the effect of suppressing the epidemic, so that it dies out. These results are illustrated by computer simulations.
Citation: Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277
##### References:
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##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361.  doi: 10.1038/280361a0.  Google Scholar [2] R. M. Anderson and R. M. May, Population biology of infectious diseases II,, Nature, 280 (1979), 455.   Google Scholar [3] R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts,, Philos. Trans. R. Soc. Lond. B, 291 (1981), 451.  doi: 10.1098/rstb.1981.0005.  Google Scholar [4] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913.  doi: 10.1088/0951-7715/18/2/022.  Google Scholar [5] S. Busenberg, K. L. Cooke and M. A. Pozio, Analysis of a model of a vertically transmitted disease,, J. Math. Biol., 17 (1983), 305.  doi: 10.1007/BF00276519.  Google Scholar [6] 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.  Google Scholar [7] S. Busenberg, K. L. Cooke and H. Thieme, Demographic change and persistence of HIV/AIDS in a heterogeneous population,, SIAM J. Appl. Math., 51 (1991), 1030.  doi: 10.1137/0151052.  Google Scholar [8] M. Carletti, K. Burrage and P. M. Burrage, Numerical simulation of stochastic ordinary differential equations in biomathematical modelling,, Math. Comput. Simulation, 64 (2004), 271.  doi: 10.1016/j.matcom.2003.09.022.  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] M. Fan, M. Y. Li and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size,, Math. Biosciences, 170 (2001), 199.  doi: 10.1016/S0025-5564(00)00067-5.  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. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525.  doi: 10.1137/S0036144500378302.  Google Scholar [13] C. Y. Ji, D. Q. Jiang and N. Z. Shi, The behavior of an SIR epidemic model with stochastic perturbation,, Stochastic Anal. Appl., 30 (2012), 755.  doi: 10.1080/07362994.2012.684319.  Google Scholar [14] X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008).  doi: 10.1533/9780857099402.  Google Scholar [15] M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size,, Math. Biosciences, 182 (2003), 1.  doi: 10.1016/S0025-5564(02)00184-0.  Google Scholar [16] R. M. May, R. M. Anderson and A. R. Mclean, Possible demographic consequences of HIV/AIDS epidemics. I. assuming HIV infection always leads to AIDS,, Math. Biosciences, 90 (1988), 475.  doi: 10.1016/0025-5564(88)90079-X.  Google Scholar [17] C. J. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination,, Applied Mathematical Modelling, 34 (2010), 2685.  doi: 10.1016/j.apm.2009.12.005.  Google Scholar [18] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system,, Physica A, 354 (2005), 111.  doi: 10.1016/j.physa.2005.02.057.  Google Scholar [19] C. Vargas-De-Leon, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,, Chaos, 44 (2011), 1106.  doi: 10.1016/j.chaos.2011.09.002.  Google Scholar [20] Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations,, Nonlinear Anal. RWA, 14 (2013), 1434.  doi: 10.1016/j.nonrwa.2012.10.007.  Google Scholar [21] 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.  Google Scholar [22] 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.  Google Scholar
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