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

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 R0 which completely governs the disease dynamics: when R0 < 1, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when R0 > 1, the disease is permanent. It is interesting that the threshold value R0 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.

Analysis of the SAITS alcoholism model on scale-free networks with demographic and nonlinear infectivity

“…It is rarely observed in the real world. Individuals often contact only a small, clustered, subpopulation in reality, so many people study epidemic or alcoholism models on complex networks recently [9,10,15,18,24,34,38,42]. Pastor-Satorras and Vespiganai [24] studied a dynamical model for the spreading of infectious diseases on scale-free networks, and found the absence of an epidemic threshold and its associated critical behaviour.…”

Epidemic dynamics on complex networks with general infection rate and immune strategies

“…Therefore, the node with more edges has a higher possibility of being infected. Recently, starting with the seminal works by Pastor-Satorras and Vespignani [20,21], there has been a burst of activity on investigating the effects of the network topology on the behavior of epidemic spreading (see [1], [3]- [8], [11], [13]- [16], [18], [19], [23], [25]- [29], [31], [32]).…”

“…In mathematical epidemiology, as we have seen in the literatures [6,8,7,11,12,13,25,27,31,32], one of the most important problem is to study the global behavior of epidemic spreading. For the spreading of infections on heterogeneous networks, Pastor-Satorras and Vespignani [20,21] proposed and studied a wellknown SIS epidemic model, which has inspired a great number of related works.…”

“…Both [25] and [32] show that the endemic equilibrium is globally attractive by using a monotone iterative technique. Recently, Huang and Jiang [8] proved the global asymptotical stability of the endemic equilibria of the model in [25] and the model with the constraint condition λ(k) = λk in [32]. It is noticed that the problem about the global asymptotical stability of the endemic equilibrium of the model without the condition λ(k) = λk in [25] is worth studying further.…”

“…Zhang and Sun [31] presented an SIS epidemic model with a generalized feedback mechanism on weighted networks and obtained the global asymptotical stability of the disease-free equilibrium and the local asymptotical stability of the endemic equilibrium. Huang and Jiang [8] developed and analyzed a new SIS epidemic model with a general nonlinear incidence rate on complex heterogeneous networks. They proved the permanence of the disease and the global asymptotical stability of disease-free equilibrium in detail.…”