Global Dynamic Behavior of a Multigroup Cholera Model with Indirect Transmission

For a multigroup cholera model with indirect transmission, the infection for a susceptible person is almost invariably transmitted by drinking contaminated water in which pathogens, V. cholerae, are present. The basic reproduction number is identified and global dynamics are completely determined by . It shows that is a globally threshold parameter in the sense that if it is less than one, the disease-free equilibrium is globally asymptotically stable; whereas if it is larger than one, there is a unique endemic equilibrium which is global asymptotically stable. For the proof of global stability with the disease-free equilibrium, we use the comparison principle; and for the endemic equilibrium we use the classical method of Lyapunov function and the graph-theoretic approach. 1. Introduction Cholera, a waterborne gastroenteric infection, remains a significant threat to public health in the developing world. Outbreaks of cholera occur cyclically, usually twice per year in endemic areas, and the intensity of these outbreaks varies over longer periods [1]. Hence, in the last few decades, enormous attention is being paid to the cholera disease and several mathematical dynamic models have been developed to study the transmission of cholera [1–7]. In these papers, they consider the population is uniformly mixed, but many factors can lead to heterogeneity in a host population. So in this paper we divide different population into different groups, which can be divided geographically into communities, cities, and countries, to incorporate differential infectivity of multiple strains of the disease agent. In the case of cholera, the transmission usually occurs through ingestion of contaminated water or feces rather than through casual human-human contact [1]. Therefore, direct contact of healthy person with an infected person is not a risk for contracting infection, whereas a healthy person may contract infection by drinking contaminated water in which pathogens, V. cholerae, are present [2]. The members of this bacterial genus (V. cholerae) naturally colonize in lakes, rivers, and estuaries. Therefore we consider that cholera transmits to other individuals via bacteria in the aquatic environment and formulates a multi-group epidemic model for cholera. Let be the total population which is divided into four epidemiological compartments, susceptible compartment , infectious compartment , recovered compartment , and vaccinated compartment . Let be the density of V. cholerae in the aquatic environment. As a consequence of the increase in the density of virulent V.