Epidemic processes on adaptive temporal networks

The emergence of new infectious diseases and the persistence of old ones are a major concern for public health. Understanding the fundamental mechanisms driving the spread of epidemics is crucial in order to develop effective control and containment measures. An essential role is played by the mathematical modelling of epidemic processes, based on the detailed knowledge of the structure of the social interactions. In modelling the spread of epidemics, two levels of coupling between epidemic processes and social interactions must be taken into account. On the one hand, human interactions are continuously rearranged over time, producing a social dynamics which deeply affects the epidemic process. On the other hand, the presence of an epidemic induces adaptive behaviours with which the population responds to the spread of the pathogen, modifying the social dynamics. A powerful paradigm for considering both these levels of coupling consists in the theory of adaptive temporal networks, in which social interactions are represented by a time-varying network whose evolution is coupled to an epidemic process. In this thesis, we deal with epidemic processes on adaptive temporal networks, focusing on activity-driven networks, an empirically validated class of networks whose dynamics is determined by the propensity of the nodes to engage interactions over time. These networks can be treated both with rigorous analytical approaches and numerical techniques, allowing the formulation of models for the characterization of the basic mechanisms of adaptive behaviours. We develop a general formalism for adaptive activity-driven networks coupled to epidemic processes, assuming a change in the nodes activity and attractiveness based on their health status. The epidemic threshold can be estimated analytically, unveiling the crucial role of correlations in the behaviour of individuals between the susceptible and the infected state. The model allows to describe several adaptive behaviours of populations exposed to epidemics, including behaviours observed in the early stages of the COVID-19 pandemic. Inspired by these behavioural changes, we implement two different types of quarantine, comparing their effectiveness and showing the impact of timing in the adoption of measures. The adaptive formalism proposed can be suitably modified to describe even more complex adaptive behaviours, such as contact tracing which is crucial for controlling SARS-CoV-2 diffusion without disrupting societal activities. We implement contact tracing in its manual (interview-based), digital (app-based) and hybrid protocols. The model highlights an intrinsic difference in contacts exploration: manual tracing performs a stochastic sampling (annealed), while digital tracing performs a sampling localized on a subpopulation (quenched). Because of this, the manual tracing is robustly more effective than the digital one, even assuming the same probability of tracing a contact. This difference, previously overlooked, is further amplified by the presence of heterogeneity in the individuals behaviour, i.e. superspreaders. Moreover, in active populations a key property of social interactions is their higher-order nature, due to the formation of social groups and gatherings. In the presence of epidemics, large gatherings can generate superspreading events, thus they must be addressed by control strategies. We implement an epidemic model for the diffusion of SARS-CoV-2 on simplicial adaptive activity-driven networks, in which the interactions are organized in simplices and the tracing is implemented on gatherings. Beside forward and backward tracing, a new tracing mechanism is active in gatherings: the sideward tracing, which occurs laterally exploiting the simplicial structure of interactions. We unveil the relevance of the sideward mechanism in tracing large gatherings, especially in the presence of strategies targeted on them. We implement our model on an empirical dataset of gatherings in a University, suggesting the optimal size of groups to be traced to reach the epidemic control, avoiding the University closure.