Global stability of a time-delayed multi-group SIS epidemic model with nonlinear incidence rates and patch structure

Wang, Jinliang¹ ; Muroya, Yoshiaki² ; Kuniya, Toshikazu³

¹ School of Mathematical Science, Heilongjiang University, Harbin 150080, China
² Department of Mathematics, Waseda University 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169- 8555, Japan
³ Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

Journal of nonlinear sciences and its applications, Tome 8 (2015) no. 5, p. 578-599.

Résumé

In this paper, we formulate and study a multi-group SIS epidemic model with time-delays, nonlinear incidence rates and patch structure. Two types of delays are incorporated to concern the time-delay of infection and that for population exchange among different groups. Taking into account both of the effects of crossregion infection and the population exchange, we define the basic reproduction number $\mathcal{R}_0$ by the spectral radius of the next generation matrix and prove that it is a threshold value, which determines the global stability of each equilibrium of the model. That is, it is shown that if $\mathcal{R}_0\leq 1$, the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 > 1$, the system is permanent, an endemic equilibrium exists and it is globally asymptotically stable. These global stability results are achieved by constructing Lyapunov functionals and applying LaSalle's invariance principle to a reduced system. Numerical simulation is performed to support our theoretical results.

Détail

DOI : 10.22436/jnsa.008.05.11

Classification : 34D23, 34K20, 92D30
Keywords: SIS epidemic model, time-delay, nonlinear incidence rate, patch structure.

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