The basic reproduction ratio R₀ for a sexually transmitted disease in pair formation model with two types of pairs

We study a model for pair formation and separation with two types of pairs which differ in average duration. A fraction f{hook} of all newly formed pairs have a long duration (denoted by "steady"), the remaining fraction 1 - f{hook} have a short duration ("casual"). This distinction is motivated by data about the survival times of partnerships in a sociological survey. In this population we consider a sexually transmitted disease, which can have different transmission rates in steady and in casual partnerships. We investigate under which conditions an epidemic can occur after introduction of the disease into a population where the process of pair formation and separation is at equilibrium. If there is no recovery we can compute an explicit expression for the basic reproduction ratio R₀; if we take recovery into account we can derive a condition for the stability of the disease-free equilibrium which is equivalent to R₀ < 1. We discuss how R₀ depends on various model parameters.