一类具有随机扰动的非自治SIRI流行病模型的动力学行为

Abstract

Abstract:

In this paper, we study the dynamics of a stochastic nonautonomous Susceptible-Infective-Removed-Infective (SIRI) epidemic model. By employing the Lyapunov function method, we show that there exists at least one nontrivial positive T-periodic solution of the system. Moreover, sufficient conditions for extinction of the disease are established. Some numerical simulations are carried out to illustrate the theoretical results.

Key words: Stochastic SIRI epidemic model, Periodic solutions, Extinction, Lyapunov function

CLC Number:

O211.6

Zhongwei Cao,Xiangdan Wen,Wei Feng,Li Zu. Dynamics of a Nonautonomous SIRI Epidemic Model with Random Perturbations[J].Acta mathematica scientia,Series A, 2020, 40(1): 221-233.

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References 24

1	Kermack W O , McKendrick A G . Contributions to the mathematical theory of epidemics-I. Bull Math Biol, 1991, 53 (1/2): 33- 55
2	Ma Z N, Zhou Y C, Wu J H. Modeling and Dynamics of Infectious Diseases. Beijing:Higher Education Press, 2009
3	Zhang J , Jin Z , Sun G Q , et al. Analysis of rabies in China:transmission dynamics and control. PloS One, 2011, 6 (7): e20891 doi: 10.1371/journal.pone.0020891
4	Buonomo B , D'Onofrio A , Lacitignola D . Global stability of an SIR epidemic model with information dependent vaccination. Math Biosci, 2008, 216 (1): 9- 16
5	Korobeinikov A , Wake G C . Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl Math Lett, 2002, 15 (8): 955- 960 doi: 10.1016/S0893-9659(02)00069-1
6	Korobeinikov A . Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission. Bull Math Biol, 2006, 30, 615- 626
7	Blower S . Modelling the genital herpes epidemic. Herpes, 2004, 3, 138A- 146A
8	Wildy P, Field H J, Nash A A. Classical herpes latency revisited//Mahy B W J, Minson A C, Darby G K, et al. Virus Persistence Symposium. Cambridge:Cambridge University Press, 1982, 33:133-168
9	Tudor D . A deterministic model for herpes infections in human and animal populations. SIAM Rev, 1990, 32 (1): 136- 139 doi: 10.1137/1032003
10	Vargas-De-León C . On the global stability of infectious disease models with relapse. Abstraction and Application, 2013, 9, 50- 61
11	Gray A , Greenhalgh D , Hu L , et al. A stochastic differential equation SIS epidemic model. SIAM J Appl Math, 2011, 71 (3): 876- 902 doi: 10.1137/10081856X
12	Lin Y G , Jiang D Q , Wang S . Stationary distribution of a stochastic SIS epidemic model with vaccination. Physica A, 2014, 394, 187- 197 doi: 10.1016/j.physa.2013.10.006
13	Lahrouz A , Omari L , Kiouach D . Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model. Nonlinear Anal Model Control, 2011, 16 (1): 59- 76
14	Lahrouz A , Settati A . Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation. Appl Math Comput, 2014, 233, 10- 19
15	Bai Z G , Zhou Y C . Existence of two periodic solutions for a non-autonomous SIR epidemic model. Appl Math Model, 2011, 35 (1): 382- 391 doi: 10.1016/j.apm.2010.07.002
16	Bai Z G , Zhou Y C , Zhang T L . Existence of multiple periodic solutions for an SIR model with seasonality. Nonlinear Anal TMA, 2011, 74 (11): 3548- 3555 doi: 10.1016/j.na.2011.03.008
17	Kuniya T . Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients. Appl Math Lett, 2014, 27, 15- 20 doi: 10.1016/j.aml.2013.08.008
18	Li T , Li Y G , Hethcote H W . Periodic traveling waves in SIRS endemic models. Math Comput Model, 2009, 49 (1/2): 393- 401
19	Lin Y G , Jiang D Q , Liu T . Nontrivial periodic solution of a stochastic epidemic model with seasonal variation. Appl Math Lett, 2015, 45, 103- 107 doi: 10.1016/j.aml.2015.01.021
20	Ji C Y , Jiang D Q . The threshold of a non-autonomous SIRS epidemic model with stochastic perturbations. Math Meth Appl Sci, 2016, 40 (5): 1773- 1782
21	Liu Q , Jiang D Q , Shi N Z , et al. Nontrivial periodic solution of a stochastic non-autonomous SISV epidemic model. Physica A, 2016, 462, 837- 845 doi: 10.1016/j.physa.2016.06.041
22	Mao X R. Stochastic Differential Equations and Their Applications. Chichester:Horwood, 1997
23	Khasminskii R. Stochastic Stability of Differential Equations. Berlin:Springer, 2011
24	Higham D J . An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev, 2001, 43, 525- 546 doi: 10.1137/S0036144500378302

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Dynamics of a Nonautonomous SIRI Epidemic Model with Random Perturbations

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