In this paper, we conduct a careful global stability analysis for a generalized cholera epidemiological model originally proposed in [J. Wang and S. Liao, A generalized cholera model and epidemic/endemic analysis, J. Biol. Dyn. 6 (2012), pp. 568-589]. Cholera is a water-and food-borne infectious disease whose dynamics are complicated by the multiple interactions between the human host, the pathogen, and the environment. Using the geometric approach, we rigorously prove the endemic global stability for the cholera model in three-dimensional (when the pathogen component is a scalar) and four-dimensional (when the pathogen component is a vector) systems. This work unifies the study of global dynamics for several existing deterministic cholera models. The analytical predictions are verified by numerical simulation results.

We study an SEIRS epidemic model with an isolation and nonlinear incidence rate function. We have obtained a threshold value [Formula: see text] and shown that there is only a disease-free equilibrium point, when [Formula: see text] and an endemic equilibrium point if [Formula: see text]. We have shown that both disease-free and endemic equilibrium point are globally stable.

Abstract. An SIQ epidemic model with isolation and nonlinear incidence rate is studied. We have obtained a threshold value R and shown that there is only a disease free equilibrium point when 1 R < , and there is also an endemic equilibrium point if 1 R > . With the help of Liapunov function, we have shown that disease free-and endemic equilibrium point is globally stable.

In this paper, the positive steady states of the epidemic model with non-monotonic incidence rate are considered. Firstly, it is proved that the unique positive constant steady state is stable for the ODE system and the PDE system. Secondly, a priori estimate of positive steady states is given, and the non-existence of non-constant positive steady states is established by using Poincare inequality and Young inequality. Finally, the existence and bifurcation of non-constant positive steady states are studied by using the degree theory and the global bifurcation theorem.

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