Multiple Endemic Equilibria in an Environmentally-Transmitted Disease With Three Disease Stages

Abstract

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 where severe and asymptomatic cases may present relapse between them. Transmission dynamics driven by the contact rates (as normally occur in directly-transmitted or vector-transmitted diseases) only occurs when a parameterR*>1. In this case, the forward transcritical bifurcation that exists forR*<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 ofR*is only the ratio of the external contamination inflow shedding rate to the environmental clearance rate.R*describes the strength of the transmission to infectious classes other than theI-(acute) type infections. The epidemic outbreak conditions and the structure ofR*appearing in this model are both resposible for the existence of endemic states.