Last modified: 2014-03-31

#### Abstract

How do concurrent partnerships influence the spread of sexually transmitted infections like HIV? We model the spread of an SI (Susceptible -> Infectious) infection on a dynamic homosexual partnership network. The network incorporates demographic turnover due to individuals entering and leaving the sexually active population. Individuals in this network have a dynamically varying number of simultaneous partnerships, with at most n at a time. Transmission may occur in partnerships between infectious and susceptible partners. The state of an individual is specified by its disease status and its number of susceptible and infectious partners. Using a mean field at distance one assumption we can describe the partnership and disease transmission dynamics on the population level by a set of (n+1)(n+2) ordinary differential equations. This model allows for an explicit expression of the basic reproduction number R_0. This is done by first characterizing R_0 using the next-generation-matrix method. Using the interpretation of R_0 we reduce the number of states-at-infection from n to three. Therefore, R_0 can be characterized as the dominant eigenvalue of a 3x3 matrix (that can be even further reduced to a 2x2 matrix). Finally, all entries of this matrix can be calculated explicitly. This means that, for all n, the stability analysis around the disease-free steady state of an (n+1)(n+2)-dimensional system is reduced to the evaluation of one explicit expression.