Multiple endemic equilibria in an environmentally-transmitted disease with three disease stages

We construct, analyze and interpret a mathematical model for an environmental transmitted disease characterized for the existence of three disease stages: acute, severe and asymptomatic. Besides, we consider that severe and asymptomatic cases may present relapse between them. Transmission dynamics driven by the contact rates only occurs when a parameter $R_{\ast }>1$, as normally occur in directly-transmitted or vector-transmitted diseases, but it will not adequately correspond to a basic reproductive number as it depends on environmental parameters. In this case, the forward transcritical bifurcation that exists for $R_{\ast }<1$, becomes a backward bifurcation, producing multiple steady-states, a hysteresis effect and dependence on initial conditions. A threshold parameter for an epidemic outbreak, independent of $R_{\ast }$ is only the ratio of the external contamination inflow shedding rate to the environmental clearance rate. $R_{\ast }$ describes the strength of the transmission to infectious classes other than the $I$-(acute) type infections. The epidemic outbreak conditions and the structure of $R_{\ast }$ appearing in this model are both responsible for the existence of endemic states.