Characterizing outbreak vulnerability in a stochastic SIS model with an external disease reservoir

In this article, we take a mathematical approach to the study of population-level disease spread, performing a quantitative and qualitative investigation of an SISκ model which is a susceptible-infectious-susceptible (SIS) model with exposure to an external disease reservoir. The external reservoir is non-dynamic, and exposure from the external reservoir is assumed to be proportional to the size of the susceptible population. The full stochastic system is modelled using a master equation formalism. A constant population size assumption allows us to solve for the stationary probability distribution, which is then used to investigate the predicted disease prevalence under a variety of conditions. By using this approach, we quantify outbreak vulnerability by performing the sensitivity analysis of disease prevalence to changing population characteristics. In addition, the shape of the probability density function is used to understand where, in parameter space, there is a transition from disease free, to disease present, and to a disease endemic system state. Finally, we use Kullback-Leibler divergence to compare our semi-analytical results for the SISκ model with more complex susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) models.