# American Institute of Mathematical Sciences

2016, 13(4): 723-739. doi: 10.3934/mbe.2016016

## Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate

 1 Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China 2 Mathematics and Science College, Shanghai Normal University, Shanghai 200234

Received  October 2015 Revised  February 2016 Published  May 2016

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold $R_0$ which completely governs the disease dynamics: when $R_0<1$, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when $R_0>1$, the disease is permanent. It is interesting that the threshold value $R_0$ bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.
Citation: Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016
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