Incorporating demographic stochasticity into multi-strain epidemic models: application to influenza A

Pavlo Minayev,Neil Ferguson

doi:10.1098/rsif.2008.0467

Pavlo Minayev, Neil Ferguson

Open Access

https://doi.org/10.1098/rsif.2008.0467

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Abstract

We develop mathematical models of the transmission and evolution of multi-strain pathogens that incorporate strain extinction and the stochastic generation of new strains via mutation. The dynamics resulting from these models is then examined with the applied aim of understanding the mechanisms underpinning the evolution and dynamics of rapidly mutating pathogens, such as human influenza viruses. Our approach, while analytically relatively simple, gives results that are qualitatively similar to those obtained from much more complex individually based simulation models. We examine strain dynamics as a function of cross-immunity and key transmission parameters, and show that introducing strain extinction and modelling mutation as a stochastic process significantly changes the model dynamics, leading to lower strain diversity, reduced infection prevalence and shorter strain lifetimes. Finally, we incorporate transient strain-transcending immunity in the model and demonstrate that it reduces strain diversity further, giving patterns of sequential strain replacement similar to that seen in human influenza A viruses.

Highlights

Modelling of the transmission of multi-strain pathogens to date has either used compartmental deterministic frameworks (Andreasen et al 1997; Gupta et al 1998; Gog & Grenfell 2002) or large-scale individually based simulations ( Ferguson et al 2003; Koelle et al 2006)
This paper has introduced a new class of multi-strain disease models that retain the relatively parsimonious compartmental structure of previous models (Andreasen et al 1997; Gupta et al 1998; Minayev & Ferguson 2008) but accounts for finite population sizes by explicitly modelling strain extinction, and captures the stochasticity of the mutation process
The dynamics of these ‘semi-stochastic’ models are strikingly different from those of the entirely deterministic model, with large amplitude limit cycles and chaotic behaviour typically being replaced by oscillatory dynamics with a limited number of dominant strains circulating at any one time

Summary

INTRODUCTION

Modelling of the transmission of multi-strain pathogens to date has either used compartmental deterministic frameworks (Andreasen et al 1997; Gupta et al 1998; Gog & Grenfell 2002) or large-scale individually based simulations ( Ferguson et al 2003; Koelle et al 2006) While the former retain sufficient simplicity to allow (quasi-) analytical insight into system dynamics, they cannot satisfactorily capture the intrinsically stochastic nature of strain creation (via mutation or recombination) and extinction (Minayev & Ferguson 2008). We study model variants with and without transient straintranscending immunity of the type first postulated in Gupta et al (1998) to explore to what extent such immunity is necessary to reproduce influenza-like evolutionary dynamics

MODEL DEFINITION

SYSTEM DYNAMICS

Findings

CONCLUSIONS