A stochastic SIS epidemic model with heterogeneous contacts

• An exact Markovian SIS epidemic model with heterogeneities is formulated and studied.
• Efficient computational methods are proposed for the quantification of the epidemic.
• The length of an outbreak, the number of infectives and related measures are studied.
• The results are applied for SIS epidemics in small populations (families and ICUs).

A stochastic model for the spread of an SIS epidemic among a population consisting of NN individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when NN is small. The model is formulated as a finite time-homogeneous continuous-time Markov chain XX. Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steady-state measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace–Stieltjes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of XX, which provides a good approximation of the process XX at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R0R0 is defined here as a random variable, rather than an expected value.