Backward bifurcation in simple SIS model
A simple epidemic model of SIS-type is investigated, where the treatment of infected persons is proportional to the infection rate only up to a given value, and is held constant for higher infection rates. The author presents a qualitative analysis in two steps: First, when the disease-induced death rate is zero, and, secondly, when this rate is positive. In the first case, Thieme's theory of asymptotically autonomous systems is used to reduce the (S,I)-system to a single nonsmooth differential equation which easily admits the investigation of the equilibria and their stability as a function of an essential parameter. The second case may not be reduced to a one-dimensional problem. Nevertheless, a rather complete analysis is carried out in the (S,I)-plane. It yields the existence of multiple endemic equilibria, their position and, in particular, their backward bifurcation for certain parameter values. Moreover, results are obtained concerning Hopf bifurcation and the nonexistence of periodic orbits.
scientific article; zbMATH DE number 5632898