AAM  >> Vol. 3 No. 3 (August 2014)

    Qualitative Analysis of a Stochastic SIR Epidemic Model with Saturated Incidence Rates

  • 全文下载: PDF(356KB) HTML    PP.127-133   DOI: 10.12677/AAM.2014.33019  
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谭 杨:铜仁职业技术学院,铜仁;

传染病模型饱和感染率高斯白噪声稳定分布Epidemic Model Saturated Incidence Rates Gaussian White Noise Stationary Distribution



A stochastically mathematical model of a stochastic SIR epidemic model with saturated incidence rates is proposed and analyzed, setting that all the death rate and incident rate are similarly per-turbed by an independent Gaussian white noise. First the paper shows that the infective population and recovered individuals will tend to zero exponentially almost surely under some additional condition. In addition, a sufficient condition for the stationary distribution around the endemic infection equilibrium state of the corresponding deterministic model is derived and the solution is ergodic.

谭杨, 郭子君. 具有饱和感染率的随机SIR传染病模型的性质分析[J]. 应用数学进展, 2014, 3(3): 127-133. http://dx.doi.org/10.12677/AAM.2014.33019


[1] Hethcote, H.W. and Levin, S.A. (1989) Periodicity in epidemiological models. In: Applied Mathematical Ecology, Springer-Verlag, Berlin.
[2] Hu, X.L. (2007) The existence of periodic solutions for a SIR epidemic model with con-stant birth rate. Pure and Applied Mathematics, 23, 372-376, 380.
[3] Bai, Z. and Zhou, Y. (2011) Existence of two periodic solutions for a non-autonomous SIR epidemic model. Applied Mathematical Modelling, 35, 382-391.
[4] Mao, X.R., Marion, G. and Renshaw, E. (2002) Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes and their Application, 97, 95-110.
[5] Tornatore, E., Buccellato, V. and Shaikhet, L. (2012) Stability of a stochastic SIR system. Physica A, 354, 111-126.
[6] Ji, C.Y., Jiang, D.Q. and Shi, N.Z. (2010) The behavior of an SIR epidemic model with stochastic perturbation. Stochastic Analysis and Applications, 30, 121-131.
[7] Dalal, N., Greenhalgh, D. and Mao, X.R. (2008) A stochastic model for internal HIV dynamics. Mathematical Analysis and Applications, 341, 1084-1101.
[8] Li, S. and Zhang, X. (2013) Qualitative analysis of a stochastic predator-prey system with disease in the predator. International Journal of Biomathematics, 6, 12500681-125006813.
[9] Abta, A., Kaddar, A. and Alaoui, H.T. (2012) Global stability for delay SIR and SEIR ep-idemic models with saturated incidence rates. Electronic Journal of Differential Equations, 23, 1-13.
[10] Liu, Z.J. (2013) Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates. Nonlinear Analysis: Real World Applications, 14, 1286-1299.
[11] Lahrouz, A. and Omari, L. (2013) Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence. Statistics & Probability Letters, 83, 960-968.
[12] Mao, X.R. (1997) Stochastic differential equation and applications. Horwood Publishing Limited, Chichester.