Journal of Mathematical Biology

, Volume 75, Issue 3, pp 521–541

# Hybrid Markov chain models of S–I–R disease dynamics

• Nicolas P. Rebuli
• N. G. Bean
• J. V. Ross
Article

## Abstract

Deterministic epidemic models are attractive due to their compact nature, allowing substantial complexity with computational efficiency. This partly explains their dominance in epidemic modelling. However, the small numbers of infectious individuals at early and late stages of an epidemic, in combination with the stochastic nature of transmission and recovery events, are critically important to understanding disease dynamics. This motivates the use of a stochastic model, with continuous-time Markov chains being a popular choice. Unfortunately, even the simplest Markovian S–I–R model—the so-called general stochastic epidemic—has a state space of order $$N^2$$, where N is the number of individuals in the population, and hence computational limits are quickly reached. Here we introduce a hybrid Markov chain epidemic model, which maintains the stochastic and discrete dynamics of the Markov chain in regions of the state space where they are of most importance, and uses an approximate model—namely a deterministic or a diffusion model—in the remainder of the state space. We discuss the evaluation, efficiency and accuracy of this hybrid model when approximating the distribution of the duration of the epidemic and the distribution of the final size of the epidemic. We demonstrate that the computational complexity is $${\mathcal {O}}(N)$$ and that under suitable conditions our approximations are highly accurate.

## Keywords

Markov population processes Epidemiology Fluid approximation Diffusion approximation

## Mathematics Subject Classification

60J27 60J28 92D30 60J22 60J60

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