An Exact Relationship Between Invasion Probability and Endemic Prevalence for Markovian SIS Dynamics on Networks

Abstract

Understanding models which represent the invasion of network-based systems by infectious agents can give important insights into many real-world situations, including the prevention and control of infectious diseases and computer viruses. Here we consider Markovian susceptible-infectious-susceptible (SIS) dynamics on finite strongly connected networks, applicable to several sexually transmitted diseases and computer viruses. In this context, a theoretical definition of endemic prevalence is easily obtained via the quasi-stationary distribution (QSD). By representing the model as a percolation process and utilising the property of duality, we also provide a theoretical definition of invasion probability. We then show that, for undirected networks, the probability of invasion from any given individual is equal to the (probabilistic) endemic prevalence, following successful invasion, at the individual (we also provide a relationship for the directed case). The total (fractional) endemic prevalence in the population is thus equal to the average invasion probability (across all individuals). Consequently, for such systems, the regions or individuals already supporting a high level of infection are likely to be the source of a successful invasion by another infectious agent. This could be used to inform targeted interventions when there is a threat from an emerging infectious disease.

Understanding the influence of all nodes in a network

“…In over 20,000 simulated continuous-time SIS spreading processes, no processes which went extinct reached more than 20 nodes, while processes which did not go extinct reached the majority of the network. It has been argued that such bifurcation in outcomes is predicted by theory 38 . Given that the distribution of the number of infected nodes is characterized by two well separated modes, the mean is best seen as an indirect estimate of the likelihood of the higher mode.…”

Persistence time of SIS infections in heterogeneous populations and networks

“…(ii) For the SIS model on a finite network, in which each individual u makes contact with each other individual v at rate β uv , it is known that, provided infectious periods are exponentially distributed, the decay parameter of the process is unchanged under transposition of the matrix of infection rates {β uv }. This follows from the property of 'network duality', see [28,18,16]. In our context, this implies that the mean time to extinction from quasi-stationarity, τ , is identical if we interchange the roles of λ, µ.…”

Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations

“…As Clancy (2018) points out, see also Wilkinson and Sharkey (2013), it follows from the network duality results of Harris (1976) and Holley and Liggett (1975) that provided infectious periods are exponentially distributed, the value of τ is unchanged if we interchange the roles of λ, μ. Theorem 1(ii) then follows immediately from Theorem 1(i).…”