On the time to extinction for a two-type version of Bartlett’s epidemic model

We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible → infectious → recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.