Infectious diseases and social distancing under state-dependent probabilities

We analyze the implications of infectious diseases and social distancing in an extended SIS framework to allow for the presence of stochastic shocks with state dependent probabilities. Random shocks give rise to the diffusion of a new strain of the disease which affects both the number of infectives and the average biological characteristics of the pathogen causing the disease. The probability of such shock realizations changes with the level of disease prevalence and we analyze how the properties of the state-dependent probability function affect the long run epidemiological outcome which is characterized by an invariant probability distribution supported on a range of positive prevalence levels. We show that social distancing reduces the size of the support of the steady state distribution decreasing thus the variability of disease prevalence, but in so doing it also shifts the support rightward allowing eventually for more infectives than in an uncontrolled framework. Nevertheless, social distancing is an effective control measure since it concentrates most of the mass of the distribution toward the lower extreme of its support.