Mean-field models in network inference and epidemic control

Systems that are comprised of agents and pairwise interactions between agents can be studied through the lenses of Network theory. As a general framework, Network Theory has applications in various disciplines, including Statistical Physics, Economics, and Biology. The interplay between the contact structure of a population and epidemic spreading is one of the most studied research areas in Epidemiology, where network based research has offered many breakthroughs in recent years. Since an individual based description is computationally intractable, as state spaces scale exponentially with the number of agents modelled, many mathematical approximations have been developed to describe the system in terms of low dimensional aggregate statistics, such as the average number of infected people. This thesis is focused on the application of such approximation techniques, in particular the well known mean-field models, to two key problems in Epidemiology: inference and epidemic control. In the first part of this work, the theme is the inference of network properties from the observation of outbreaks at a population-level. Typically, readily available information during an outbreak is (daily) case counts. With this in mind, a new mean-field like model is introduced to approximate epidemics on networks via Birthand-Death processes, whose rates are random variables which depend implicitly on the structure of the underlying network and disease dynamics. By using Bayesian model selection, it is possible to recover the most likely underlying network class from datasets that consist only of discrete-time observations from one single epidemic. Further, having a description in terms of Birth-and-Death processes allows to study the large N limit of the process as a one-dimensional Fokker-Planck equation, that implies an even greater reduction in dimensionality. In the second part of this thesis more standard mean-field models are adopted to perform epidemic control. The aim is to reduce the burden of an outbreak on a target population. Intervention policies may consist of one time interventions either to minimise the epidemic peak or the final size, or to maximise the average time to infection. Homogeneous mixing models are a nice tool to showcase how interventions that achieve such goals can be optimised. A network perspective is introduced to study the so-called disease-induced herd immunity: in principle, epidemics act like targeted vaccinations, preferentially immunising higher-risk individuals. This means that the herd-immunity threshold might be reached at lower levels compared to that derived from homogeneous mixing models, and this might be relevant for epidemic control. However, it is shown that the magnitude of this effect depends heavily on how both the topology of the contact network and the way non-pharmaceutical interventions are modelled. Finally, epidemic response can be thought of as a feedback process, that is, social distancing policies might be deployed depending on the observed epidemic curve, rather than being pre-determined from theoretical arguments.In this case, the goal is to maintain the epidemic at manageable levels throughout its course, by tailoring interventions that aim to be as less disruptive as possible. This possibility is investigated on a high dimensional network model, by deriving a feedback-loop control action that at its core is based on a mean-field approximation.