Modeling direct and indirect disease transmission using multi-group model

The survival of pathogens outside the host in the environment is an important factor for some diseases transmission. Environment-to-individuals is an indirect mode of transmission, which also plays a role in the dynamics of some disease. In this paper, we proposed a general multi-group epidemic model with nonlinear direct and indirect transmission incidence rates, nonlinear pathogen shedding rates, and common environmental contamination for indirect transmission. Under the certain assumptions, the basic reproduction number of the model is identified. We proved the global stability of the equilibria by using global Lyapunov functions with the specific coefficients and graph-theoretical approach theorem, which are determined by the basic reproduction number. The main result of our model is that we give the specific coefficients of global Lyapunov functions. From the discussion of our model, we conclude that our model contains earlier cholera models, brucellosis models, and other general disease multi-group models as special cases.