Narrow Results

In this paper, a nonlinear deterministic model is proposed with a saturated treatment function. The expression of the basic reproduction number for the proposed model was obtained. The global dynamics of the proposed model was studied using the basic reproduction number and theory of dynamical systems. It is observed that proposed model exhibits backward bifurcation as multiple endemic equilibrium points exist when $R_0<1$. The existence of backward bifurcation implies that making $R_0<1$ is not enough for disease eradication. This, in turn, makes it difficult to control the spread of cholera in the community. We also obtain a unique endemic equilibria when $R_0>1$. The global stability of unique endemic equilibria is performed using the geometric approach. An extensive numerical study is performed to support our analytical results. Finally, we investigate two major cholera outbreaks, Zimbabwe (2008–09) and Haiti (2010), with the help of the present study.

This paper describes a cholera disease transmission model in the human population through the consumption of zooplankton as food by humans. Here the plankton population is classified into two subpopulations such as phytoplankton and zooplankton. Also, human population is classified into two subpopulations such as susceptible human and infected human. The proposed system reflects the impacts of using time delay in the cholera disease transmission. Different possible equilibrium points of our proposed system have been determined. Here local and global stabilities of our proposed system have been analyzed. The existence of Hopf bifurcation has been studied at the interior equilibrium point. The normal form method and center manifold theorem have been used to test the nature of Hopf bifurcation. It is observed that the interior equilibrium is locally asymptotically stable when the time delay in disease transmission term is large, while the change of stability of positive equilibrium will cause a bifurcating periodic solution at the time delay $\tau$ to be at less than its critical value. Finally, some numerical simulation results have been presented for the better understanding of our proposed system.