Abstract
Between pandemics, the influenza virus exhibits periods of incremental evolution via a process known as antigenic drift. This process gives rise to a sequence of strains of the pathogen that are continuously replaced by newer strains, preventing a build up of immunity in the host population. In this paper, a parsimonious epidemic model is defined that attempts to capture the dynamics of evolving strains within a host population. The ‘evolving strains’ epidemic model has many properties that lie in-between the Susceptible–Infected–Susceptible and the Susceptible–Infected–Removed epidemic models, due to the fact that individuals can only be infected by each strain once, but remain susceptible to reinfection by newly emerged strains. Coupling results are used to identify key properties, such as the time to extinction. A range of reproduction numbers are explored to characterise the model, including a novel quasi-stationary reproduction number that can be used to describe the re-emergence of the pathogen into a population with ‘average’ levels of strain immunity, analogous to the beginning of the winter peak in influenza. Finally the quasi-stationary distribution of the evolving strains model is explored via simulation.
Highlights
Epidemic models have become important tools for understanding, predicting and developing mitigation strategies for public health planners dealing with infectious diseases
In this paper we develop a parsimonious epidemic model that describes the transmission dynamics of a multistrain pathogen with evolutionary dynamics similar to the influenza A virus evolving via antigenic drift
To characterise the dynamics of the models, we look to a number of key statistics which are related to the commonly used basic reproduction number, R0, that illustrates whether or not an epidemic is likely to infect a large proportion of the population
Summary
Epidemic models have become important tools for understanding, predicting and developing mitigation strategies for public health planners dealing with infectious diseases. In this paper we develop a parsimonious epidemic model that describes the transmission dynamics of a multistrain pathogen with evolutionary dynamics similar to the influenza A virus evolving via antigenic drift. Meehan et al [3] analyse multi-strain epidemic models with mutation between strains within an ODE framework Since their focus is on drug-resistance, they do not consider the effect of immunity. To capture within a mathematical model the complex processes driving the evolution of the influenza virus is extremely challenging due to the interactions between host immunity and viral evolution [8]. In this paper we define a novel epidemic model with countably infinite, evolving strains that sits between the traditional susceptible– infected–susceptible (SIS) and susceptible–infected–removed (SIR) epidemic models, in that each individual may be infected many times with the pathogen, but only once by a strain.
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