Cholera dynamics with Bacteriophage infection: A mathematical study

Mathematical modeling of waterborne diseases, such as cholera, including a biological control using Bacteriophage viruses in the aquatic reservoirs is of great relevance in epidemiology. In this paper, our aim is twofold: at first, to understand the cholera dynamics in the region around a water body; secondly, to understand how the spread of Bacteriophage infection in the cholera bacterium V. cholerae controls the disease in the human population. For this purpose, we modify the model proposed by Codeço, for the spread of cholera infection in human population and the one proposed by Beretta and Kuang, for the spread of Bacteriophage infection in the bacteria population [1, 2]. We first discuss the feasibility and local asymptotic stability of all the possible equilibria of the proposed model. Further, in the numerical investigation, we have found that the parameter ϕ, called the phage adsorption rate, plays an important role. There is a critical value, ϕc, at which the model possess Hopf-bifurcation. For lower values than ϕc, the equilibrium E* is unstable and periodic solutions are observed, while above ϕc, the equilibrium E* is locally asymptotically stable, and further shown to be also globally asymptotically stable. We investigate the effect of the various parameters on the dynamics of the infected humans by means of numerical simulations.