Dynamic behaviors of a cholera model with nonlinear incidences, multiple transmission pathways, and imperfect vaccine

In this article, we propose a cholera model to study the effects of multiple transmission pathways, imperfect vaccine, nonlinear incidences, and differential infectivity of vibrios. The expression of the basic reproductive number ℛ_0 is derived. There is only the disease-free equilibrium E_0 if ℛ_0≤ 1 , while, besides E_0 , there is also a unique endemic equilibrium E^* if ℛ_0>1 . When ℛ_0<1 , E_0 is globally asymptotically stable by using the technique of linearization and the fluctuation lemma. When ℛ_0>1 , E^* is globally asymptotically stable by the Lyapunov direct method. These theoretical results are supported with numerical simulations for the case with Beddington-DeAngelis incidences. We further perform the sensitivity analyses of ℛ_0 and the infection level at E^* to determine the significant parameters affecting disease outbreak and severity, respectively. Influences of the vaccination rate ϕ and the waning rate of vaccine η on the dynamical behaviors of the model are also discussed.