Different types of backward bifurcations due to density-dependent treatments

A set of deterministic SIS models with density-dependent treatments are studied to understand the disease dynamics when different treatment strategies are applied. Qualitative analyses are carried out in terms of general treatment functions. It has become customary that a backward bifurcation leads to bistable dynamics. However, this study finds that finds that bistability may not be an option at all; the disease-free equilibrium could be globally stable when there is a backward bifurcation. Furthermore, when a backward bifurcation occurs, the fashion of bistability could be the coexistence of either dual stable equilibria or the disease-free equilibrium and a stable limit cycle. We also extend the formula for mean infection period from density-independent treatments to density-dependent ones. Finally, the modeling results are applied to the transmission of gonorrhea in China, suggesting that these gonorrhea patients may not seek medical treatments in a timely manner.