| Summary: | The rate at which individuals in a population mix with one another can have a large impact on how far a disease spreads among that population, as well as what fraction of the population needs to be immune from infection in order to protect the remaining susceptible population from a major outbreak. In this thesis we consider both deterministic and stochastic SEIR (susceptible - exposed - infectious -recovered) epidemic models. We impose a household structure on the population, so that individuals mix globally with the population at large and, at a higher rate, locally with members of their household. We also consider an extension of this model in which individuals are typed, making global contacts at different rates dependent on their type. We investigate herd immunity for these models, providing a more realistic insight than the standard epidemic model in which all individuals in the population mix at the same rate.
The disease-induced herd immunity level hD is the fraction of the population that must be infected by an epidemic to ensure that a new epidemic among the remaining susceptible population is not supercritical. For a homogeneously mixing population hD equals the classical herd immunity level hC, which is the fraction of the population that must be vaccinated in advance of an epidemic so that the epidemic is not supercritical. A detailed comparison of hD and hC is given for the households model, where we also define an approximation h˜D of hD which is more amenable to analysis. It is found that hD > hC unless the household size variability is sufficiently large, in contrast to other models with heterogeneous mixing of individuals, in which hD < hC typically occurs.
We obtain the asymptotic variance for hD as the population size goes to infinity, using a Gaussian approximation. We then consider a model with individual types and household structure, deriving several reproduction numbers and a central limit theorem for the final outcome under the assumption of proportionate global mixing, which we show greatly simplifies these calculations and results. We provide comparison of hD and hC when these individual types correspond to activity levels, showing that the ordering of these herd immunity levels is strongly dependent on the distribution of the individuals of each activity level among the households.
Finally, we consider the impact of global restrictions on disease-induced herd immunity in a model with household structure and types of individuals. We extend the approximation hD to account for local infection being increased during times of global restrictions. We then consider a scenario in which two supercritical epidemics can occur, the first with constant control measures, and find an optimal control such that the number of individuals ever infected across the two epidemics is minimised.
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