Aggregation and optimisation in epidemiological models of heterogeneous populations

Abstract

This thesis is concerned with the analysis of models of heterogeneous populations in infectious disease epidemiology. Special considerations are made with respect to variables arising from the aggregation of heterogeneous variables. We analyse the asymptotic behaviour, steady states, and stability of simple heterogeneous SI-, SIS-, and SIR-models with parametric heterogeneity, which are described by an infinite dimensional system of ODEs. As for homogeneous models, we are able to define a basic reproduction number which can be used as an indicator for the existence of endemic steady states and stability of disease free steady states. In some cases a finite dimensional ODE system for the aggregated variables can be formulated, which simplifies both analysis and practical calculations. For SIS-models we also consider the influence of heterogeneity on early warning signs for critical transitions. We develop a stochastic model to incorporate fluctuation effects and the random import of the disease into the population. We analyse the influence of heterogeneity on warning signs for critical transitions. This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. Using numerical simulations, we show that known scaling laws for early warning signs no longer hold true for heterogeneous models. We identify various different ways in which heterogeneity can influence these scaling laws. This is of importance if one wants to interpret potential warning signs for disease outbreaks. One obstacle to applying heterogeneous models in practice is that in order for the equations to be well defined it is necessary to have knowledge of the initial conditions for the distributed heterogeneous variables. This information is in many cases not available. However, the variables of interest are often not the heterogeneous variables, but their aggregated counterparts. We therefore develop set-membership estimation techniques for these aggregated variables under the assumption that the initial conditions for the heterogeneous variables are only partially known. By numerically solving certain optimisation problems we are able to calculate these estimations. Furthermore, we consider optimal control problems for heterogeneous systems. For models with parametric heterogeneity, we show by example how aggregation techniques can in certain cases be used to reduce the infinite dimensional problem to a finite dimensional one, for which the well developed standard optimal control theory can be applied. We also develop a version of Pontryagin's maximum principle for heterogeneous systems that include aggregated variables. We do this not in the framework of parametric heterogeneity, but more generally for size structured PDEs.