Advances in Applied Probability

SIR epidemics with stages of infection

Claude Lefèvre and Massimiliano Giorgio

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In this paper we are concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population when an infective can go through several stages of infection before being removed. The transitions between stages are governed by either a Markov process or a semi-Markov process. An infective of any stage makes contacts amongst the population at the points of a Poisson process. Our main purpose is to derive the distribution of the final epidemic size and severity, as well as an approximation by branching, using simple matrix analytic methods. Some illustrations are given, including a model with treatment discussed by Gani (2006).

Article information

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 768-791.

First available in Project Euclid: 19 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 60K15: Markov renewal processes, semi-Markov processes 92D30: Epidemiology

Epidemic model final size and severity phase-type distribution Markovian or semi-Markovian infection process matrix analytic method


Lefèvre, Claude; Giorgio, Massimiliano. SIR epidemics with stages of infection. Adv. in Appl. Probab. 48 (2016), no. 3, 768--791.

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