SI infection on a dynamic partnership network: characterization of R_0

We model the spread of an $${ SI}$$SI (Susceptible $$\rightarrow $$→Infectious) sexually transmitted infection on a dynamic homosexual network. The network consists of individuals with a dynamically varying number of partners. There is demographic turnover due to individuals entering the population at a constant rate and leaving the population after an exponentially distributed time. Infection is transmitted in partnerships between susceptible and infected individuals. We assume that the state of an individual in this structured population is specified by its disease status and its numbers of susceptible and infected partners. Therefore the state of an individual changes through partnership dynamics and transmission of infection. We assume that an individual has precisely $$n$$n ‘sites’ at which a partner can be bound, all of which behave independently from one another as far as forming and dissolving partnerships are concerned. The population level dynamics of partnerships and disease transmission can be described by a set of $$(n+1)(n+2)$$(n+1)(n+2) differential equations. We characterize the basic reproduction ratio $$R_0$$R0 using the next-generation-matrix method. Using the interpretation of $$R_0$$R0 we show that we can reduce the number of states-at-infection $$n$$n to only considering three states-at-infection. This means that the stability analysis of the disease-free steady state of an $$(n+1)(n+2)$$(n+1)(n+2)-dimensional system is reduced to determining the dominant eigenvalue of a $$3\times 3$$3×3 matrix. We then show that a further reduction to a $$2\times 2$$2×2 matrix is possible where all matrix entries are in explicit form. This implies that an explicit expression for $$R_0$$R0 can be found for every value of $$n$$n.