Statistical mechanical model for growth and spread of contagions under gauged population confinement

We construct a stochastic process model for cascading families of time series descriptive of the initial development and then controlled evolution of a general epidemic/pandemic phenomenon. A distinguishing feature of the model is the effect of the spatial position of individual infections amongst other appropriate characteristics. The model naturally reproduces regime transitions representative of shifts from full community to localized contagions as phase transitions along time. More specifically, the model is defined as a growing family of renewal processes with an exponential time proliferation. The core part of the model consists of a renewal process of non-independent events that has been shown (Velázquez and Robledo (2011)) to be analogous to a statistical–mechanical thermal system capable of undergoing thermodynamic phase transitions. An external control parameter, akin to temperature, affects the spread of contagions, as quarantine is enforced on agents. When the thermal analog of the model is particularized to the so-called Hamiltonian Mean Field Model the space and time properties of the stochastic process can be solved explicitly in full detail. We show quantitative agreement with time series data from the recent COVID19 pandemic. We discuss additional modeling that could make this idealized construction perhaps closer to useful applications.