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.   Google Scholar

[2]

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

[3]

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

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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