# 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 and 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), 361-367. doi: 10.1038/280361a0. [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, The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond. B, 291 (1981), 451-524. doi: 10.1098/rstb.1981.0005. [4] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936. doi: 10.1088/0951-7715/18/2/022. [5] S. Busenberg, K. L. Cooke and M. A. Pozio, Analysis of a model of a vertically transmitted disease, J. Math. Biol., 17 (1983), 305-329. doi: 10.1007/BF00276519. [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), 257-270. doi: 10.1007/BF00178776. [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), 1030-1052. doi: 10.1137/0151052. [8] M. Carletti, K. Burrage and P. M. Burrage, Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Math. Comput. Simulation, 64 (2004), 271-277. doi: 10.1016/j.matcom.2003.09.022. [9] 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. [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), 199-208. doi: 10.1016/S0025-5564(00)00067-5. [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), 876-902. doi: 10.1137/10081856X. [12] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [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), 755-773. doi: 10.1080/07362994.2012.684319. [14] X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402. [15] M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size, Math. Biosciences, 182 (2003), 1-25. doi: 10.1016/S0025-5564(02)00184-0. [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), 475-505. doi: 10.1016/0025-5564(88)90079-X. [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), 2685-2697. doi: 10.1016/j.apm.2009.12.005. [18] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057. [19] C. Vargas-De-Leon, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos, Solitons & Fractals, 44 (2011), 1106-1110. doi: 10.1016/j.chaos.2011.09.002. [20] Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal. RWA, 14 (2013), 1434-1456. doi: 10.1016/j.nonrwa.2012.10.007. [21] 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. [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), 4916-4927. 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: 10.1038/280361a0. [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, The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond. B, 291 (1981), 451-524. doi: 10.1098/rstb.1981.0005. [4] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936. doi: 10.1088/0951-7715/18/2/022. [5] S. Busenberg, K. L. Cooke and M. A. Pozio, Analysis of a model of a vertically transmitted disease, J. Math. Biol., 17 (1983), 305-329. doi: 10.1007/BF00276519. [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), 257-270. doi: 10.1007/BF00178776. [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), 1030-1052. doi: 10.1137/0151052. [8] M. Carletti, K. Burrage and P. M. Burrage, Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Math. Comput. Simulation, 64 (2004), 271-277. doi: 10.1016/j.matcom.2003.09.022. [9] 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. [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), 199-208. doi: 10.1016/S0025-5564(00)00067-5. [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), 876-902. doi: 10.1137/10081856X. [12] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. [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), 755-773. doi: 10.1080/07362994.2012.684319. [14] X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402. [15] M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size, Math. Biosciences, 182 (2003), 1-25. doi: 10.1016/S0025-5564(02)00184-0. [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), 475-505. doi: 10.1016/0025-5564(88)90079-X. [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), 2685-2697. doi: 10.1016/j.apm.2009.12.005. [18] E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057. [19] C. Vargas-De-Leon, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos, Solitons & Fractals, 44 (2011), 1106-1110. doi: 10.1016/j.chaos.2011.09.002. [20] Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal. RWA, 14 (2013), 1434-1456. doi: 10.1016/j.nonrwa.2012.10.007. [21] 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. [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), 4916-4927. doi: 10.1016/j.physa.2013.06.009.
 [1] Adel Settati, Aadil Lahrouz, Mustapha El Jarroudi, Mohamed El Fatini, Kai Wang. On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1985-1997. doi: 10.3934/dcdsb.2020012 [2] Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6131-6154. doi: 10.3934/dcdsb.2021010 [3] Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335 [4] Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447 [5] 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 [6] Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93 [7] Yaru Hu, Jinfeng Wang. Dynamics of an SIRS epidemic model with cross-diffusion. Communications on Pure and Applied Analysis, 2022, 21 (1) : 315-336. doi: 10.3934/cpaa.2021179 [8] Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041 [9] Nguyen Thanh Dieu, Vu Hai Sam, Nguyen Huu Du. Threshold of a stochastic SIQS epidemic model with isolation. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021262 [10] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [11] 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 [12] Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6229-6252. doi: 10.3934/dcdsb.2021016 [13] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [14] Tomás Caraballo, Mohamed El Fatini, Idriss Sekkak, Regragui Taki, Aziz Laaribi. A stochastic threshold for an epidemic model with isolation and a non linear incidence. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2513-2531. doi: 10.3934/cpaa.2020110 [15] 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 [16] 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 [17] Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191 [18] Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169 [19] M. P. Moschen, A. Pugliese. The threshold for persistence of parasites with multiple infections. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1483-1496. doi: 10.3934/cpaa.2008.7.1483 [20] 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 and Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057

2020 Impact Factor: 1.327