Mathematical Modelling of Social Homeostasis: Higher-Order Interactions, Temporal Networks and Adaptive Dynamics

Loneliness and social isolation have emerged as pressing public health concerns, yet their mathematical modelling remains underdeveloped. This thesis proposes a novel framework grounded in the theory of Social Homeostasis-the principle according to which individuals regulate their social behaviour around an optimal level of sociality, analogously to physiological homeostatic mechanisms. The work develops along three main directions. First, we discuss Interpolating Higher-Order Temporal Networks (IHTNs), a class of temporal network models that interpolates between dyadic and group interactions. We analyse a specific model-the \textit{r-SAD}-deriving analytical results for structural properties and for the epidemic threshold of an SIS process unfolding on the network. A key finding is the \textit{Poor-Get-Richer} effect: group interactions disproportionately increase the connectivity of low-activity nodes, democratizing the degree distribution. Second, we formalise a general framework for acute and chronic isolation, built around the concept of a Realization Adaptive Function (RAF) that governs the evolution of each agent's set point-their individually adaptive optimal sociality level. We prove several analytical properties of the system, including conditions for the emergence of isolation. Third, we combine both frameworks into a fully specified agent-based simulation. Numerical results reveal a phase transition controlled by the group interaction parameter: below a critical threshold, isolation vanishes in the long run; above it, a persistent fraction of isolated agents emerges.