# 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:
 [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

show all references

##### 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
 [1] Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 [2] Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119 [3] Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93 [4] Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041 [5] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [6] 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 [7] Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080 [8] Toshikazu Kuniya, Yoshiaki Muroya, Yoichi Enatsu. Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1375-1393. doi: 10.3934/mbe.2014.11.1375 [9] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191 [10] Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169 [11] M. P. Moschen, A. Pugliese. The threshold for persistence of parasites with multiple infections. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1483-1496. doi: 10.3934/cpaa.2008.7.1483 [12] Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057 [13] Qun Liu, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi. Dynamical behavior of a multigroup SIRS epidemic model with standard incidence rates and Markovian switching. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5683-5706. doi: 10.3934/dcds.2019249 [14] Alan E. Lindsay, Michael J. Ward. An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1139-1179. doi: 10.3934/dcdsb.2010.14.1139 [15] Sebastian J. Schreiber. On persistence and extinction for randomly perturbed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 457-463. doi: 10.3934/dcdsb.2007.7.457 [16] Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971 [17] Qingshan Yang, Xuerong Mao. Stochastic dynamics of SIRS epidemic models with random perturbation. Mathematical Biosciences & Engineering, 2014, 11 (4) : 1003-1025. doi: 10.3934/mbe.2014.11.1003 [18] Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033 [19] Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064 [20] Shui-Nee Chow, Yongfeng Li. Model reference control for SIRS models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 675-697. doi: 10.3934/dcds.2009.24.675

2018 Impact Factor: 1.008