Systematic Approximations to Susceptible-Infectious-Susceptible Dynamics on Networks

Network-based infectious disease models have been highly effective in elucidating the role of contact structure in the spread of infection. As such, pair- and neighbourhood-based approximation models have played a key role in linking findings from network simulations to standard (random-mixing) results. Recently, for SIR-type infections (that produce one epidemic in a closed population) on locally tree-like networks, these approximations have been shown to be exact. However, network models are ideally suited for Sexually Transmitted Infections (STIs) due to the greater level of detail available for sexual contact networks, and these diseases often possess SIS-type dynamics. Here, we consider the accuracy of three systematic approximations that can be applied to arbitrary disease dynamics, including SIS behaviour. We focus in particular on low degree networks, in which the small number of neighbours causes build-up of local correlations between the state of adjacent nodes that are challenging to capture. By examining how and when these approximation models converge to simulation results, we generate insights into the role of network structure in the infection dynamics of SIS-type infections.Author Summary: Networks are now widely used to model infectious diseases, but have posed significant mathematical challenges. Recently analytic results have been obtained for ‘one-off’ network epidemics that follow the SIR paradigm, but these results do not carry over to other scenarios—most significantly to many sexually transmitted infections, where accounting for network structure is vital. Here, we show that it is possible to obtain the large-population dynamics of such diseases on networks through systematic approximations. We focus on a mathematically challenging case of SIS dynamics on networks with low degree.