American Institute of Mathematical Sciences

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2014, 11(4): 1003-1025. doi: 10.3934/mbe.2014.11.1003

Stochastic dynamics of SIRS epidemic models with random perturbation

Qingshan Yang 1, and  Xuerong Mao 2,

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China
2. 

Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH

Received  February 2013 Revised  September 2013 Published  March 2014

In this paper, we consider a stochastic SIRS model with parameter perturbation, which is a standard technique in modeling population dynamics. In our model, the disease transmission coefficient and the removal rates are all affected by noise. We show that the stochastic model has a unique positive solution as is essential in any population model. Then we establish conditions for extinction or persistence of the infectious disease. When the infective part is forced to expire, the susceptible part converges weakly to an inverse-gamma distribution with explicit shape and scale parameters. In case of persistence, by new stochastic Lyapunov functions, we show the ergodic property and positive recurrence of the stochastic model. We also derive the an estimate for the mean of the stationary distribution. The analytical results are all verified by computer simulations, including examples based on experiments in laboratory populations of mice.
Keywords: extinction, Ergodic property, positive recurrence..
Mathematics Subject Classification: 37H10, 37H15, 60J6.
Citation: 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
References:
[1]

E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stoch Anal Appl., 26 (2008), 274-297. doi: 10.1080/07362990701857129.

[2]

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

[3]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Second edition. Hafner Press [Macmillan Publishing Co., Inc.] New York, 1975.

[4]

G. K. Basak and R. N. Bhattacharya, Stability in distribution for a class of singular diffusions, Ann Probab., 20 (1992), 312-321. doi: 10.1214/aop/1176989928.

[5]

P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Biol., 63 (2011), 433-457. doi: 10.1007/s00285-010-0376-2.

[6]

S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin, 1993. doi: 10.1007/978-3-642-75301-5.

[7]

G. Chen and T. Li, Stability of stochastic delayed SIR model, Stoch Dynam., 9 (2009), 231-252. doi: 10.1142/S0219493709002658.

[8]

Y. S. Chow, Local convergence of martingales and the law of large numbers, Ann. Math. Statist., 36 (1965), 552-558. doi: 10.1214/aoms/1177700166.

[9]

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.

[10]

R. Z. Hasminskii, Stochastic Stability of Differential Equations, Alphen aan den Rijn, The Netherlands, 1980.

[11]

H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47. doi: 10.1007/BF00276034.

[12]

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations., 217 (2005), 26-53. doi: 10.1016/j.jde.2005.06.017.

[13]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[14]

Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004.

[15]

W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.

[16]

W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[17]

Q. Lu, Stability of SIRS system with random perturbations, Phys. A., 388 (2009), 3677-3686. doi: 10.1016/j.physa.2009.05.036.

[18]

W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time-delay, Appl. Math. Lett., 17 (2004), 1141-1145. doi: 10.1016/j.aml.2003.11.005.

[19]

X. Mao, Stablity of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific & Technical Harlow, UK, 1991.

[20]

X. Mao, Exponentially Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.

[21]

X. Mao, Stochastic Differential Equations and Their Applications, 2nd ed., Horwood Publishing, Chichester, 1997.

[22]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/b98869.

[23]

I. Nasell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6.

[24]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A., 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.

[25]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM. J. Control. Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.

show all references

References:
[1]

E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stoch Anal Appl., 26 (2008), 274-297. doi: 10.1080/07362990701857129.

[2]

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

[3]

N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Second edition. Hafner Press [Macmillan Publishing Co., Inc.] New York, 1975.

[4]

G. K. Basak and R. N. Bhattacharya, Stability in distribution for a class of singular diffusions, Ann Probab., 20 (1992), 312-321. doi: 10.1214/aop/1176989928.

[5]

P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Biol., 63 (2011), 433-457. doi: 10.1007/s00285-010-0376-2.

[6]

S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin, 1993. doi: 10.1007/978-3-642-75301-5.

[7]

G. Chen and T. Li, Stability of stochastic delayed SIR model, Stoch Dynam., 9 (2009), 231-252. doi: 10.1142/S0219493709002658.

[8]

Y. S. Chow, Local convergence of martingales and the law of large numbers, Ann. Math. Statist., 36 (1965), 552-558. doi: 10.1214/aoms/1177700166.

[9]

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.

[10]

R. Z. Hasminskii, Stochastic Stability of Differential Equations, Alphen aan den Rijn, The Netherlands, 1980.

[11]

H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47. doi: 10.1007/BF00276034.

[12]

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations., 217 (2005), 26-53. doi: 10.1016/j.jde.2005.06.017.

[13]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[14]

Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004.

[15]

W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.

[16]

W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[17]

Q. Lu, Stability of SIRS system with random perturbations, Phys. A., 388 (2009), 3677-3686. doi: 10.1016/j.physa.2009.05.036.

[18]

W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time-delay, Appl. Math. Lett., 17 (2004), 1141-1145. doi: 10.1016/j.aml.2003.11.005.

[19]

X. Mao, Stablity of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific & Technical Harlow, UK, 1991.

[20]

X. Mao, Exponentially Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.

[21]

X. Mao, Stochastic Differential Equations and Their Applications, 2nd ed., Horwood Publishing, Chichester, 1997.

[22]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/b98869.

[23]

I. Nasell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19. doi: 10.1016/S0025-5564(02)00098-6.

[24]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A., 354 (2005), 111-126. doi: 10.1016/j.physa.2005.02.057.

[25]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM. J. Control. Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343.

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