On the effect of population heterogeneity on dynamics of epidemic diseases

Abstract

The paper investigates a class of SIS models of the evolution of an infectious disease in a heterogeneous population. The heterogeneity reflects individual differences in the susceptibility or in the contact rates and leads to a distributed parameter system, requiring therefore, distributed initial data, which are often not available. It is shown that there exists a corresponding homogeneous (ODE) population model that gives the same aggregated results as the distributed one, at least in the expansion phase of the disease. However, this ODE model involves a nonlinear "prevalence-to-incidence" function which is not constructively defined. Based on several established properties of this function, a simple class of approximating function is proposed, depending on three free parameters that could be estimated from scarce data. How the behaviour of a population depends on the level of heterogeneity (all other parameters kept equal) - this is the second issue studied in the paper. It turns out that both for the short run and for the long run behaviour there exist threshold values, such that more heterogeneity is advantageous for the population if and only if the initial (weighted) prevalence is above the threshold.