Advances in Applied Mathematics
Vol.3 No.03(2014), Article ID:13995,6 pages
DOI:10.12677/AAM.2014.33019

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

Yang Tan1, Zijun Guo2

1Tongren Polytechnic College, Tongren

2Institute of Applied Mathematics, South-China Agricultural University, Guangzhou

Email: 553129685@qq.com

Copyright © 2014 by authors and Hans Publishers Inc.

Received: May 15th, 2014; revised: Jun. 18th, 2014; accepted: Jun. 25th, 2014

ABSTRACT

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 perturbed 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.

Keywords:Epidemic Model, Saturated Incidence Rates, Gaussian White Noise, Stationary Distribution

1铜仁职业技术学院，铜仁

2华南农业大学应用数学研究所，广州

Email: 553129685@qq.com

1. 引言

(1)

(i.e.)研究了如下随机系统的渐近性质：

(2)

, ,

, 得到如下模型：

(3)

.

(4)

.

2. 随机模型(3)的解的存在、唯一性

3. 随机系统(3)关于点的渐近性质

1)

2)

，则由引理有

.

.

.

, (5)

，则

.

(6)

4. 随机系统(3)关于点的分布稳定性

, , ,.

(7)

(8)

(9)

，且结合基本不等式有

(10)

，当

(11)

1. [1]   Hethcote, H.W. and Levin, S.A. (1989) Periodicity in epidemiological models. In: Applied Mathematical Ecology, Springer-Verlag, Berlin.

2. [2]   Hu, X.L. (2007) The existence of periodic solutions for a SIR epidemic model with constant birth rate. Pure and Applied Mathematics, 23, 372-376, 380.

3. [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. [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. [5]   Tornatore, E., Buccellato, V. and Shaikhet, L. (2012) Stability of a stochastic SIR system. Physica A, 354, 111-126.

6. [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. [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. [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. [9]   Abta, A., Kaddar, A. and Alaoui, H.T. (2012) Global stability for delay SIR and SEIR epidemic models with saturated incidence rates. Electronic Journal of Differential Equations, 23, 1-13.

10. [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. [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. [12]   Mao, X.R. (1997) Stochastic differential equation and applications. Horwood Publishing Limited, Chichester.