Abstract
We model the spread of an { 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 ‘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) differential equations. We characterize the basic reproduction ratio R_0 using the next-generation-matrix method. Using the interpretation of R_0 we show that we can reduce the number of states-at-infection 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)-dimensional system is reduced to determining the dominant eigenvalue of a 3times 3 matrix. We then show that a further reduction to a 2times 2 matrix is possible where all matrix entries are in explicit form. This implies that an explicit expression for R_0 can be found for every value of n.
Highlights
The role that concurrent partnerships might play in the spread of HIV in sub-Saharan Africa is the subject of an ongoing debate
While simulation studies have shown the large impact that concurrency potentially has on the epidemic growth rate and the endemic prevalence of HIV (Kretzschmar and Morris 1996; Morris and Kretzschmar 1997, 2000; Eaton et al 2011; Goodreau 2011), the empirical evidence for such a relationship is inconclusive (Lurie and Rosenthal 2010; Reniers and Watkins 2010; Tanser et al 2011; Kenyon and Colebunders 2012)
We show that R0 defined can be interpreted as the basic reproduction ratio for individuals, since individuals can be considered to be collections of n binding sites
Summary
The role that concurrent partnerships might play in the spread of HIV in sub-Saharan Africa is the subject of an ongoing debate. Simulation studies prevail, and general theory is mainly focused on static networks (Diekmann et al 1998; Ball and Neal 2008; House and Keeling 2011; Lindquist et al 2011; Miller et al 2012; Miller and Volz 2013) This motivated us to develop and analyse a mathematical model for the spread of an SI (Susceptible–Infectious) infection along a dynamic network. In a previous paper (Leung et al 2012) a model for a dynamic sexual network of a homosexual population is presented that incorporates demographic turnover and allows for individuals to have multiple partners at the same time, with the number of partners varying over time.
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