Abstract
\par We formulated a generic S-I-R-S
(Susceptible-Infected-Recovered-Susceptible) epidemic model of cholera
that incorporate three key features: an Allee Effect on bacteria
dynamic, the loss of immunity of recovered individuals and infection
force to cholera regulated by the contact and logistic dose-reponse of
bacteria. These assumptions are built into a simple model which yields
surprisingly rich dynamics. Having three different disease-free
equilibriums $\mathsf{Q_{0}}$,
$\mathsf{Q_{\rho}}$ and
$\mathsf{Q_{\theta}}$, the dynamic
of the model is essentially characterized by a threshold quantity
$\mathcal R^{0}_{0}$ which represents the basic
reproduction number of the disease-free equilibrium
$\mathsf{Q_{0}}$. The model supports the
possibility of bi-stability, backward bifurcation and forward
bifurcation. The sensibility analysis of the model and theoretical
results supported by numerical simulations suggest that an efficient
control strategy would be to increase the value of
$\theta$ (Allee threshold bacterial population) which
is equivalent to increasing unfavorable conditions for bacteria growth.
These conditions are generally: regular environmental consolidation
measures, compliance with hygiene rules and unfavorable climatic
factors.