Dangerous connections: the spread of infectious diseases on dynamic networks

Abstract

Are concurrent partnerships (multiple partnerships at a time) driving HIV epidemics in sub-Saharan Africa? Opinions differ! While simulation studies show the potential impact concurrency can have, empirical evidence is inconclusive. In my PhD research I used mathematical models to understand how network structures, such as concurrency, impact the spread of sexually transmitted infections, such as HIV. The sexual network is dynamic as it changes over time due to individuals entering and leaving the population and partnerships breaking and forming. The main part of my PhD project consisted of the formulation and analysis of mathematical models for the spread of infectious diseases on dynamic networks. We developed a class of that is general and flexible and amenable to mathematical analysis. In the tradition of physiologically structured population models, the model formulation starts at the individual-level. Influences from the ‘outside world’ on an individual are captured by environmental variables. These variables are population-level quantities. A key characteristic is that individuals can be decomposed into a number of conditionally independent components: each individual has a fixed number of ‘binding sites’. In this way, we can distinguish three different levels: binding sites, individuals, and the population. By focusing on the binding-site level and relating the three levels to each other we are able to write down a system of renewal equations for the environmental variables that determine the dynamics on all three levels. This system of renewal equations can be used to characterize population-level epidemiological quantities such as R_0 and the endemic equilibrium. The endemic disease prevalence is characterized implicitly by a fixed-point problem of the system of renewal equations. We also easily derived an explicit expression for the basic reproduction number R_0. This is one of the most important quantities in infectious disease. It is a threshold parameter for an epidemic to take off. In the second part of my dissertation I used the models of the first part to gain qualitative insights in the relation between concurrency and disease dynamics. We isolated specific network characteristics to study their impact on disease dynamics by fixing some key network quantities such as sexually active lifespan, mean partnership duration, mean lifetime number of partners. We showed that concurrency could drive an HIV epidemic by moving the epidemic threshold value R_0 from below to above one. Using the same methodology we also investigated gender asymmetry in concurrent partnerships. We found that more gender asymmetry in this respect is associated with lower levels of disease prevalence in the population. This is especially the case with polygyny (where men may have concurrent partnerships but women are monogamous). The balance in this PhD project has been on the side of model formulation and analysis. This has lead to a lot of enjoyable mathematics, subtle modelling issues and R_0-calculations and -interpretations. More importantly, it has lead the development of a class of models for the spread of infetious diseases on dynamic networks that enrich the mathematical modelling toolbox for understanding infectious disease dynamics.