Modelling heterogeneities in individual frailties in epidemic models

A central question in epidemiology is how heterogeneity, in its different manifestations, influences the dynamics and intensity of infections. The individual differences have been named frailty in the demography literature in the context of susceptibility to death. In epidemic theory, heterogeneity in the host population has not be treated systematically before, except for variations in the ability to transmit the disease. In this paper, we propose an epidemic model, taking into account the phenomenon of frailty in the various transition rates, and analyse its effect on the observable rates. We develop expressions for the steady state distribution of the force of infection, number of infected and various observed epidemic parameters. In particular, we show how apparently age-dependent parameters may arise from heterogeneity in situations where no explicit age-dependence is built into the model. Using standard linearization methods, the basic reproduction number is determined. Finally, we show how the force of infection can be determined from an integral equation and apply the methods of functional analysis to prove existence and uniqueness of the endemic equilibrium.