Markovian Susceptible-Infectious-Susceptible (SIS) Dynamics on Finite Networks: Endemic Prevalence and Invasion Probability

Last modified: 2014-03-31

#### Abstract

The susceptible-infectious-susceptible (SIS) model, in which individuals flip back and forth between just two states, can be interpreted as a simple representation for the spread of endemic disease. In order to capture the inherent randomness and heterogeneity of the `real-world' phenomenon, this model can also be constructed as a stochastic process which takes place on a finite transmission network. In this context, it is possible to fruitfully represent the dynamics as a kind of percolation process. This representation naturally reveals an important property of the model known as `duality'.

In the case of strongly connected networks, the system possesses a unique `quasi-stationary distribution' (QSD) which can capture the statistics of the endemic situation. By considering the mechanism by which the QSD has practical relevance, and by applying the idea of duality, an equally relevant quantifier for the probability of `invasion' emerges. Moreover, an exact individual-level relationship between endemic prevalence and invasion probability can be proved.

#### Keywords

susceptible-infectious-susceptible; stochastic; probability; networks; invasion; prevalence; quasi-stationary distribution; contact process