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 and 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 "Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotic" (eds. K. D. Elworthy, N. Ikeda), Longman Scient. Tech., (1993), 3-34.

[2]

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

[3]

R. M. Anderson and R. M. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367. 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), 377-402. doi: 10.1007/BF01193751.

[5]

D. R. Bell, "The Malliavin Calculus," Dover publications, New York, 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), 4107-4115. 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), 101-130. 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), 876-902. 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), 97-103. 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), 259-284.

[11]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. 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), 1747-1762.

[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), 121-131. 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), 65-67. 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), 700-721.

[16]

M. Y. Li and Z. S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20. 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), 455-461. 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), 528-597. 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), 115-149. 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), 56-74. 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), 968-971. 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), 319-331. 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), 93-107. 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), 108-119. 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 "Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. III," University of California Press, Berkeley, (1972), 333-359.

[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), 733-749. doi: 10.1002/mma.810.

[27]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126. 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), 285-291. 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 "Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotic" (eds. K. D. Elworthy, N. Ikeda), Longman Scient. Tech., (1993), 3-34.

[2]

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

[3]

R. M. Anderson and R. M. May, Population biology of infectious diseases, part I, Nature, 280 (1979), 361-367. 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), 377-402. doi: 10.1007/BF01193751.

[5]

D. R. Bell, "The Malliavin Calculus," Dover publications, New York, 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), 4107-4115. 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), 101-130. 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), 876-902. 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), 97-103. 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), 259-284.

[11]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. 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), 1747-1762.

[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), 121-131. 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), 65-67. 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), 700-721.

[16]

M. Y. Li and Z. S. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20. 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), 455-461. 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), 528-597. 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), 115-149. 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), 56-74. 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), 968-971. 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), 319-331. 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), 93-107. 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), 108-119. 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 "Proc. Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. III," University of California Press, Berkeley, (1972), 333-359.

[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), 733-749. doi: 10.1002/mma.810.

[27]

E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Physica A, 354 (2005), 111-126. 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), 285-291. doi: 10.1016/j.amc.2007.09.053.

[1]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146

[2]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735

[3]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203

[4]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic and Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569

[5]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469

[6]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6253-6265. doi: 10.3934/dcdsb.2021017

[7]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[8]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[9]

Wenbin Yang, Yujing Gao, Xiaojuan Wang. Diffusion modeling of tumor-CD4$ ^+ $-cytokine interactions with treatments: asymptotic behavior and stationary patterns. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1285-1300. doi: 10.3934/dcdsb.2021090

[10]

Françoise Pène. Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 567-587. doi: 10.3934/dcds.2009.24.567

[11]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

[12]

Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051

[13]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[14]

Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems and Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427

[15]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147

[16]

Guangliang Zhao, Fuke Wu, George Yin. Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems. Mathematical Control and Related Fields, 2015, 5 (2) : 359-376. doi: 10.3934/mcrf.2015.5.359

[17]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

[18]

Weiyi Zhang, Ling Zhou. Global asymptotic stability of constant equilibrium in a nonlocal diffusion competition model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022062

[19]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105

[20]

Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure and Applied Analysis, 2021, 20 (3) : 975-994. doi: 10.3934/cpaa.2021002

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (235)
  • HTML views (0)
  • Cited by (35)

Other articles
by authors

[Back to Top]