Household models for endemic diseases

Abstract

The mathematical modelling of disease dynamics is now well-established, which allows us to better understand the processes affecting diseases in communities. Most of this work has focused on epidemic dynamics, in which we see a large out- break of a disease but also its extinction within a short time-frame. Of interest in this thesis will be furthering the study of endemic diseases, which persist in populations over much longer time scales. This increased time scale means that it is important to account for demography, and in particular the replenishment of susceptible individuals through births and possibly waning immunity. Another feature that has been shown to be important for modelling disease dynamics is heterogeneity in the population, and in particular household structure. This is likely to be important for endemic diseases because of the close relationship between demography and household composition as these types of diseases persist in communities for periods long enough that assuming a static population structure is not realistic. Hence we model both the disease and population dynamics as a continuous- time Markov chain, where the population of individuals is split into adults and children. These adults and children are assigned a household type, and within each household the number of adults is at most two, and the number of children is at most four. These households change through time as a consequence of demographic events such as births, deaths, children moving out of home, couples forming, separating, and migration. This demographic model is overlaid with SEIR disease dynamics, where the rate of disease transmission is strongest within households, as opposed to between. We also develop a seasonally-forced model. The so-called Gillespie algorithm is used to simulate realisations of this process but simulation can be computationally expensive. Hence we also derive a deterministic approximation, valid in the limit of a large number of households. We use this deterministic approximation extensively to analyse the models dynamics. For measles-like disease parameters, the period of the non-seasonally forced model is approximately two years, which agrees closely to the established biennial periodicity. In the model with seasonal forcing, the periodicity is more complicated, with annual periodicity in the deterministic approximation, and approximately 2-3 year periods for individual stochastic realisations. The household model without seasonal forcing is used to investigate the case of a u-like disease suddenly becoming fatal in children, adults, and both. We found that when the population has only a small proportion of susceptible individuals there is a higher chance of the disease being persistent when compared to the population with larger proportions of susceptible individuals. Significant change in household proportions are recorded initially during the first outbreak of the disease when the mortality rate increases during the infectious period of individuals. In both the models with and without seasonal forcing, a measles-like disease fades out with higher probability in the household models compared to the homogeneous models. In each of the household models, within the first five years, we see periods of approximately 2-3 years where the proportion of realisations that fade out stay essentially constant for populations of greater than 225 thousand households in the seasonally-forced model, and above 250 thousand in the non-forced model. During these periods, the proportion of realisations that have not faded out is above 0.5, which is suggestive of a critical community size between 375-500 thousand individuals in both household models. However, the household model shows greater variability than the homogeneous models as a larger proportion of realisations fadeout over the course of 15 years. The household models presented in this thesis allow a great deal of flexibility in parameterisation which means that many problems can be studied. However, as a consequence of the flexibility, the model's dynamics are evaluated at large computational expense, meaning approximations are necessary.