GLOBAL BEHAVIORS FOR A CLASS OF MULTI-GROUP SIRS EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE

In this paper, we study a class of multi-group SIRS epidemic models with nonlinear incidence rate which have cross patch infection between different groups. The basic reproduction number ℛ0is calculated. By using the method of Lyapunov functions, LaSalle’s invariance principle, the theory of the non...

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Bibliographic Details
Published inTaiwanese journal of mathematics Vol. 19; no. 5; pp. 1509 - 1532
Main Authors Tang, Qian, Teng, Zhidong, Jiang, Haijun
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.10.2015
Subjects
Disease models
Epidemics
Liapunov functions
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Summary:In this paper, we study a class of multi-group SIRS epidemic models with nonlinear incidence rate which have cross patch infection between different groups. The basic reproduction number ℛ0is calculated. By using the method of Lyapunov functions, LaSalle’s invariance principle, the theory of the nonnegative matrices and the theory of the persistence of dynamical systems, it is proved that if ℛ0≤ 1 then the disease-free equilibrium is globally asymptotically stable, and if ℛ0> 1 then the disease in the model is uniform persistent. Furthermore, when ℛ0> 1, by constructing new Lyapunov functions we establish the sufficient conditions of the global asymptotic stability for the endemic equilibrium. 2010Mathematics Subject Classification: 34D23, 92D30. Key words and phrases: Multi-group SIRS epidemic model, Nonlinear incidence rate, Basic reproduction number, Extinction and permanence, Global asymptotic stability, Lyapunov function.
ISSN:1027-5487
2224-6851
DOI:10.11650/tjm.19.2015.4205