Statistical-mechanical analogies for space-dependent epidemics

A statistical-mechanical model for the propagation of a space-dependent epidemic is suggested based on the following assumptions. (1) The population is made up of three types of individuals, susceptibles, infectives and recovered. The total population size is constant and confined in a given region of ds-dimensional Euclidean space (ds = 1, 2, 3). (2) An infected individual has a constant probability of recovery and an immune recovered individual has a constant probability of resensibilization. (3) Any infected individual from a given neighborhood of a susceptible individual has a probability p(r) of transmitting the disease which depends on the distance r between the two individuals. (4) The individuals migrate from one position to another with a rate depending on the displacement vector between the two positions. A statistical-mechanical description of the epidemic spreading is given in terms of a set of grand canonical probability densities for the positions and states of the individuals. An exact nonlinear evolution equation is derived for these probability densities and a self-consistent procedure of solving it is suggested. An explicit computation is presented for the case of very fast migration based on the adiabatic elimination of the coordinates of the individuals. An eikonal approximation for the fluctuations of the different types of individuals irrespective of their positions is developed in the limit of very large systems with constant total population density resulting in a Hamilton-Jacobi equation for the logarithm of the probability density of the proportions of susceptibles, infective and recovered. The deterministic equations of the process are identified with the evolution equations for the most probable fluctuation paths. If the individuals are confined at discrete positions in space this approach leads to the ordinary differential equation description of cellular automata epidemics presented in the literature. A continuous space alternative description is also presented. It is shown that the logarithm of the probability density of population fluctuations is a Liapunov function for the deterministic evolution equations. This Liapunov function is used for developing a generalized thermodynamic formalism of the epidemic process similar to the thermodynamic and stochastic theory of Ross, Hunt and Hunt from nonequilibrium thermodynamics of transport processes and reaction-diffusion systems.