We study extensions of the classical SIR model of epidemic spread. First, we consider a single population modified SIR epidemics model in which the contact rate is allowed to be an arbitrary function of the fraction of susceptible and infected individuals. This allows one to model either the reaction of individuals to the information about the spread of the disease or the result of government restriction measures, imposed to limit social interactions and contain contagion. We study the effect of both smooth dependancies of the contact rate for which we prove the existence of a threshold phenomenon that generalizes the well-known dichotomy associated to the reproduction rate parameter in the classical SIR model, and discontinuous feedback terms, which can be studied using tools from sliding mode control. Finally, we consider network SIR models involving different subpopulations that interact on a contact graph and present some preliminary simulations of modified versions of the classic SIR network.
On SIR epidemic models with feedback-controlled interactions and network effects