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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

Yanan Zhao 1, , Daqing Jiang 2, , Xuerong Mao 3, and  Alison Gray 3,

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.
Keywords: extinction, Lyapunov function., threshold, persistence, SIRS epidemic model.
Mathematics Subject Classification: Primary: 60H10; Secondary: 92D25, 92D3.
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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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

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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

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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

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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

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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

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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

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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

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