Stochastic epidemic models in heterogeneous communities

Sammanfattning: The aim of Paper I is to explain where randomness should be taken into account when modelling epidemic spread, i.e. when a stochastic model is preferable to a deterministic counterpart. Two examples are used to show that the probability of a large outbreak and the initial growth rate of the epidemic are affected by randomness in infectious period and latent period. It follows that the basic reproduction number is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. In paper II we analyse an open population stochastic epidemic S-I-S model.  That is, individuals in the population move between the states of infectiousness and susceptibility, and enter of leave the population through birth and death. An approximate expression for the outbreak probability is derived using a coupling argument. It is proved that the number of infectives and susceptibles close to quasi-stationarity behaves like an Ornstein-Uhlenbeck process, for an exponentially distributed time before going extinct. In Paper III we analyse an estimator, based on martingale methods, of the Malthusian parameter, which determines the speed of epidemic spread. Asymptotic properties of the estimator are obtained, and compared to the results from simulations. The advantage of the estimator is that it may use any proportion of the information contained in the epidemic curve, in contrast to the more common simpler estimators. In paper IV a social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights. The aim is to analyse how individual heterogeneity in susceptibility and infectivity affects the basic reproduction number, but also the size and probability of a major outbreak. The main qualitative conclusion is that the basic reproduction number gets larger as the community becomes more heterogeneous.