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

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  • 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.
    Mathematics Subject Classification: Primary: 60H10; Secondary: 92D25, 92D30.


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  • [1]

    R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature, 280 (1979), 361-367.doi: 10.1038/280361a0.


    R. M. Anderson and R. M. May, Population biology of infectious diseases II, Nature, 280 (1979), 455-461.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.doi: 10.1137/S0036144500378302.


    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.


    X. Mao, Stochastic Differential Equations and Applications, $2^{nd}$ edition, Horwood, Chichester, UK, 2008.doi: 10.1533/9780857099402.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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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