September  2013, 18(7): 1873-1887. doi: 10.3934/dcdsb.2013.18.1873

Long-time behaviour of a perturbed SIR model by white noise

1. 

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

Received  August 2012 Revised  March 2013 Published  May 2013

In this paper, we consider a stochastic SIR model with perturbed disease transmission coefficient. We present sufficient conditions for the disease to extinct exponentially. In the case of persistence, we analyze long-time behaviour of densities of the distributions of the solution. We will prove that the densities of the solution can converge in $L^1$ to an invariant density under appropriate conditions. Also we find the support of the invariant density. Specially, when the intensity of white noise is relatively small, we find a new threshold for an epidemic to occur.
Citation: Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873
References:
[1]

S. Aida, S. Kusuoka and D. Strook, On the support of Wiener functionals,, in, (1993), 3.

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford University Press, (1992).

[3]

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

[4]

G. B. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II),, Probab. Theory Relat. Fields, 90 (1991), 377. doi: 10.1007/BF01193751.

[5]

D. R. Bell, "The Malliavin Calculus,", Dover publications, (2006).

[6]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107. doi: 10.1016/S0362-546X(01)00528-4.

[7]

N. M. Ferguson, D. J. Nokes and R. M. Anderson, Dynamical complexity in age-structured models of the transmission of measles virus,, Math. BioSci., 138 (1996), 101. doi: 10.1016/S0025-5564(96)00127-7.

[8]

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.

[9]

B. T. Grenfell, B. M. Bolker and A. Kleczkowski, Seasonality and extinction in chaotic metapopulations,, Proc. Roy. Soc. Lond. B, 259 (1995), 97. doi: 10.1098/rspb.1995.0015.

[10]

H. B. Guo, M. Y. Li and Z. S. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Can. Appl. Math. Q., 14 (2006), 259.

[11]

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

[12]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation,, Physica A, 390 (2011), 1747.

[13]

C. Y. Ji, D. Q. Jiang, Q. S. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation,, Automatica, 48 (2012), 121. doi: 10.1016/j.automatica.2011.09.044.

[14]

M. J. Keeling and B. T. Grenfell, Disease extinction and community size: modeling the persistence of measles,, Science, 275 (1997), 65. doi: 10.1126/science.275.5296.65.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics (part I),, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700.

[16]

M. Y. Li and Z. S. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003.

[17]

R. M. May and R. M. Anderson, Population biology of infectious diseases, part II,, Nature, 280 (1979), 455. doi: 10.1038/280455a0.

[18]

X. Z. Meng and L. S. Chen, The dynamics of a new SIR epidemic model concerning pulse vaccination strategy,, Appl. Math. Comput., 197 (2008), 528. doi: 10.1016/j.amc.2007.07.083.

[19]

D. Mollison, V. Isham and B. Grenfell, Epidemics: Models and data,, J. Roy. Stat. Soc. A, 157 (1994), 115. doi: 10.2307/2983509.

[20]

K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems,, J. Math. Anal. Appl., 215 (1997), 56. doi: 10.1006/jmaa.1997.5609.

[21]

P. Rohani, D. J. D. Earn and B. T. Grenfell, Opposite patterns of synchrony: In sympatric disease metapopulations,, Science, 286 (1999), 968. doi: 10.1126/science.286.5441.968.

[22]

M. Roy and R. D. Holt, Effects of predation on host-pathogen dynamics in SIR models,, Theor. Popul. Biol., 73 (2008), 319. doi: 10.1016/j.tpb.2007.12.008.

[23]

R. Rudnicki, Long-time behaviour of a stochastic prey-predator model,, Stochastic Process. Appl., 108 (2003), 93. doi: 10.1016/S0304-4149(03)00090-5.

[24]

R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems,, Math. Biosci., 206 (2007), 108. doi: 10.1016/j.mbs.2006.03.006.

[25]

D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle,, in, (1972), 333.

[26]

J. M. Tchuenche, A. Nwagwo and R. Levins, Global behaviour of an SIR epidemic model with time delay,, Math. Methods Appl. Sci., 30 (2007), 733. doi: 10.1002/mma.810.

[27]

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.

[28]

F. P. Zhang, Z. Z. Li and F. Zhang, Global stability of an SIR epidemic model with constant infectious period,, Appl. Math. Comput., 199 (2008), 285. doi: 10.1016/j.amc.2007.09.053.

show all references

References:
[1]

S. Aida, S. Kusuoka and D. Strook, On the support of Wiener functionals,, in, (1993), 3.

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford University Press, (1992).

[3]

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

[4]

G. B. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II),, Probab. Theory Relat. Fields, 90 (1991), 377. doi: 10.1007/BF01193751.

[5]

D. R. Bell, "The Malliavin Calculus,", Dover publications, (2006).

[6]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Anal., 47 (2001), 4107. doi: 10.1016/S0362-546X(01)00528-4.

[7]

N. M. Ferguson, D. J. Nokes and R. M. Anderson, Dynamical complexity in age-structured models of the transmission of measles virus,, Math. BioSci., 138 (1996), 101. doi: 10.1016/S0025-5564(96)00127-7.

[8]

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.

[9]

B. T. Grenfell, B. M. Bolker and A. Kleczkowski, Seasonality and extinction in chaotic metapopulations,, Proc. Roy. Soc. Lond. B, 259 (1995), 97. doi: 10.1098/rspb.1995.0015.

[10]

H. B. Guo, M. Y. Li and Z. S. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Can. Appl. Math. Q., 14 (2006), 259.

[11]

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

[12]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, Multigroup SIR epidemic model with stochastic perturbation,, Physica A, 390 (2011), 1747.

[13]

C. Y. Ji, D. Q. Jiang, Q. S. Yang and N. Z. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation,, Automatica, 48 (2012), 121. doi: 10.1016/j.automatica.2011.09.044.

[14]

M. J. Keeling and B. T. Grenfell, Disease extinction and community size: modeling the persistence of measles,, Science, 275 (1997), 65. doi: 10.1126/science.275.5296.65.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics (part I),, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700.

[16]

M. Y. Li and Z. S. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003.

[17]

R. M. May and R. M. Anderson, Population biology of infectious diseases, part II,, Nature, 280 (1979), 455. doi: 10.1038/280455a0.

[18]

X. Z. Meng and L. S. Chen, The dynamics of a new SIR epidemic model concerning pulse vaccination strategy,, Appl. Math. Comput., 197 (2008), 528. doi: 10.1016/j.amc.2007.07.083.

[19]

D. Mollison, V. Isham and B. Grenfell, Epidemics: Models and data,, J. Roy. Stat. Soc. A, 157 (1994), 115. doi: 10.2307/2983509.

[20]

K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems,, J. Math. Anal. Appl., 215 (1997), 56. doi: 10.1006/jmaa.1997.5609.

[21]

P. Rohani, D. J. D. Earn and B. T. Grenfell, Opposite patterns of synchrony: In sympatric disease metapopulations,, Science, 286 (1999), 968. doi: 10.1126/science.286.5441.968.

[22]

M. Roy and R. D. Holt, Effects of predation on host-pathogen dynamics in SIR models,, Theor. Popul. Biol., 73 (2008), 319. doi: 10.1016/j.tpb.2007.12.008.

[23]

R. Rudnicki, Long-time behaviour of a stochastic prey-predator model,, Stochastic Process. Appl., 108 (2003), 93. doi: 10.1016/S0304-4149(03)00090-5.

[24]

R. Rudnicki and K. Pichór, Influence of stochastic perturbation on prey-predator systems,, Math. Biosci., 206 (2007), 108. doi: 10.1016/j.mbs.2006.03.006.

[25]

D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle,, in, (1972), 333.

[26]

J. M. Tchuenche, A. Nwagwo and R. Levins, Global behaviour of an SIR epidemic model with time delay,, Math. Methods Appl. Sci., 30 (2007), 733. doi: 10.1002/mma.810.

[27]

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.

[28]

F. P. Zhang, Z. Z. Li and F. Zhang, Global stability of an SIR epidemic model with constant infectious period,, Appl. Math. Comput., 199 (2008), 285. doi: 10.1016/j.amc.2007.09.053.

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