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

TL;DR

This paper establishes a precise mathematical relationship between invasion probability and endemic prevalence in Markovian SIS models on networks, revealing that individuals with high endemic prevalence are more likely sources of invasion.

Contribution

It provides a novel theoretical link between invasion probability and endemic prevalence for SIS dynamics on networks, applicable to both directed and undirected cases.

Findings

In undirected networks, invasion probability equals endemic prevalence for each individual.

The total endemic prevalence in the population matches the average invasion probability.

High prevalence individuals are likely sources of successful invasions.

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.