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

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

  • 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[J]. Mathematical Biosciences and Engineering, 2016, 13(4): 723-739. doi: 10.3934/mbe.2016016

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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.


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