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 \(\mathcal {R}_0\) ${\mathcal{R}}_0$ is derived. There is only the disease-free equilibrium \(E_0\) $E_0$ if \(\mathcal {R}_0\le 1\) ${\mathcal{R}}_0\le 1$ , while, besides \(E_0\) $E_0$ , there is also a unique endemic equilibrium \(E^*\) $E^{\ast }$ if \(\mathcal {R}_0>1\) ${\mathcal{R}}_0>1$ . When \(\mathcal {R}_0<1\) ${\mathcal{R}}_0<1$ , \(E_0\) $E_0$ is globally asymptotically stable by using the technique of linearization and the fluctuation lemma. When \(\mathcal {R}_0>1\) ${\mathcal{R}}_0>1$ , \(E^*\) $E^{\ast }$ 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 \(\mathcal {R}_0\) ${\mathcal{R}}_0$ and the infection level at \(E^*\) $E^{\ast }$ to determine the significant parameters affecting disease outbreak and severity, respectively. Influences of the vaccination rate \(\phi \) $\varphi$ and the waning rate of vaccine \(\eta \) $\eta$ on the dynamical behaviors of the model are also discussed.