Backward bifurcation and local dynamics of epidemic model on adaptive networks with treatment

A Susceptible-Infected-Susceptible (SIS) model with limited treatment capacity on adaptive networks is presented to study the effects of the treatment on epidemic spreading and local stability and bifurcation behavior of the system. We derive the occurring condition of backward bifurcation or forward bifurcation at the disease-free equilibrium in the system. By theoretical analysis and numerical simulations, it is found (i) if a backward bifurcation occurs at the disease-free equilibrium of the system, then bistability exists regardless of the size of the capacity, and (ii) if a forward bifurcation occurs at the disease-free equilibrium of the system, then bistable endemic equilibria exist when the capacity is low. It is also shown that the range of bistability becomes smaller as the capacity increases when the capacity is smaller, otherwise unchanges for larger treatment capacity.