Abstract
In stochastic modelling of infectious spread, it is often assumed that infection confers permanent immunity, a susceptible-infective-removed (SIR) model. We show how results concerning long-term (endemic) behaviour may be extended to a susceptible-infective-removed-susceptible (SIRS) model, in which immunity is temporary. Since the full SIRS model with demography is rather intractable, we also consider two simpler models: the susceptible-infective-susceptible (SIS) model with demography, in which there is no immunity; and the SIRS model in a closed population. For each model, we first analyse a deterministic model, then approximate the quasi-stationary distribution (equilibrium distribution conditional upon non-extinction of infection) using a moment closure technique. We look in particular at the effect of the immune period upon infection prevalence and upon time to fade-out of infection. Our main findings are that a shorter average immune period leads to higher infection prevalence in quasi-stationarity, and to longer persistence of infection in the population.
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Clancy, D., Mendy, S.T. The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection. J. Math. Biol. 61, 527–544 (2010). https://doi.org/10.1007/s00285-009-0313-4
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DOI: https://doi.org/10.1007/s00285-009-0313-4
Keywords
- Quasi-stationary distribution
- Stochastic infection model
- Diffusion approximation
- Cumulant equations
- Moment closure