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

Shouying  Huang; Jifa  Jiang; Shouying  Huang; Jifa  Jiang

doi:10.3934/mbe.2016016

Mathematical Biosciences and Engineering

2016, Volume 13, Issue 4: 723-739. doi: 10.3934/mbe.2016016

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

Shouying Huang ¹,
Jifa Jiang ²

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

Received: 01 October 2015 Accepted: 29 June 2018 Published: 01 May 2016
MSC : Primary: 92D30, 34D23; Secondary: 05C82.

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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.
- Heterogeneous network,
- nonlinear incidence,
- epidemic spreading,
- equilibrium,
- global stability.
Citation: Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate[J]. Mathematical Biosciences and Engineering, 2016, 13(4): 723-739. doi: 10.3934/mbe.2016016

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Abstract

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.

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