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Asymptotic Behavior and Stability of Stochastic SIR Model with Variable Diffusion Rates
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, China
College of Mathematics and Econometrics, Hunan University, Changsha, China
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, China
- 1 Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, China
- 2 College of Mathematics and Econometrics, Hunan University, Changsha, China
- 3 Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, China
Applied Mathematics·Volume 08 (2017)·Pages 1031–1044·Published 7 August 2017·DOI10.4236/am.2017.88080
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Abstract
In this paper, we propose random fluctuation on contact and recovery rates in deterministic SIR model with disease deaths in nonparametric manner and derive a new stochastic SIR model with distributed time delay and general diffusion coefficients. By analysis of the introduced model, we obtain the sufficient conditions for the regularity, existence and uniqueness of a global solution by means of Lyapunov function. Moreover, we also investigate the stochastic asymptotic stability of disease free equilibria and endemic equilibria of this model. Finally, we illustrate our general results by applications.
KeywordsSIR ModelRegularityLyapunov FunctionStochastic Asymptotic Stability
- Allen, E.J., Allen, L.J.S. and Schurz, H. (2005) A Comparison of Persistence-Time Estimation for Discrete and Continuous Stochastic Population Models That Include Demographic and Environmental Variability. Mathematical Biosciences, 196, 14- 38. https://doi.org/10.1016/j.mbs.2005.03.010
- Allen, E.J. (1999) Stochastic Differential Equations and Persistence Time for Two Interacting Populations. Dynamics of Continuous, Discrete and Impulsive Systems, 5, 271-281.
- Arciniega, A. and Allen, E. (2004) Shooting Methods for Numerical Solution of Stochastic Boundary-Value Problems. Stochastic Analysis and Applications, 22, 1295-1314. https://doi.org/10.1081/SAP-200026465
- Arnold, L. (1974) Stochastic Differential Equation. Wiley, New York.
- Bartholomew, D.J. (1982) Stochastic Models for Social Processes. Wiley, New York.
- Bartlett, M.S., Gower, J.C. and Leslie, P.H. (1960) A Comparison of Theoretical and Empirical Results for Some Stochastic Population Models. Biometrika, 47, 1-11 https://doi.org/10.1093/biomet/47.1-2.1
- Beretta, E., Kolmanovskii, V. and Shaikhet, L. (1998) On the General Structure of Epidemic System, Global Asymptotic Stability. Computers & Mathematics with Applications, 12, 677-694. https://doi.org/10.1016/0898-1221(86)90054-4
- Beretta, E. and Takeuchi, Y. (1995) Global Stability of an SIR Epidemic Model with Time Delays. Journal of Mathematical Biology, 33, 250-260 https://doi.org/10.1007/BF00169563
- Carletti, M. (2002) On the Stability Properties of a Stochastic Model for Phage- Bacteria Interaction in Open Marine Environment. Mathematical Biosciences, 175, 117-131. https://doi.org/10.1016/S0025-5564(01)00089-X
- Gard, T.C. (1988) Introduce to Stochastic Differential Equations. Marcel Dekker, Basel.
- Golec, J. and Sathananthan, S. (2003) Stabilitty Analysis of a Stochastic Logistic Model. Mathematical and Computer Modelling, 38, 585-593 https://doi.org/10.1016/S0895-7177(03)90029-X
- Hanson, F.B. and Tuckwell, H.C. (1981) Logistic Growth with Random Density Independent Disasters. Theoretical Population Biology, 19, 1-11. https://doi.org/10.1016/0040-5809(81)90032-0
- Hethcote, H.W. (1976) Qualitative Analyses of Communicable Disease Models. Mathematical Biosciences, 28, 335-356. https://doi.org/10.1016/0025-5564(76)90132-2
- Schurz, H. (2001) Moment Attractivity, Stability and Contractivity Exponents of Stochastic Dynamical Systems. Discrete and Continuous Dynamical Systems— Series A, 7, 487-515. https://doi.org/10.3934/dcds.2001.7.487