STATISTICAL-MECHANICAL ANALOGIES FOR SPACE-DEPENDENT EPIDEMICS

A statistical-mechanical model for the propagation of a space-dependen t epidemic is suggested based on the following assumptions. (1) The po pulation is made up of three types of individuals, susceptibles, infec tives and recovered. The total population size is constant and confine d in a given region of d(s)-dimensional Euclidean space (d(s) = 1, 2, 3). (2) An infected individual has a constant probability of recovery and an immune recovered individual has a constant probability of resen sibilization. (3) Any infected individual from a given neighborhood of a susceptible individual has a probability p(r) of transmitting the d isease which depends on the distance r between the two individuals. (4 ) The individuals migrate from one position to another with a rate dep ending on the displacement vector between the two positions. A statist ical-mechanical description of the epidemic spreading is given in term s of a set of grand canonical probability densities for the positions and states of the individuals. An exact nonlinear evolution equation i s derived for these probability densities and a self-consistent proced ure of solving it is suggested. An explicit computation is presented f or the case of very fast migration based on the adiabatic elimination of the coordinates of the individuals. An eikonal approximation for th e fluctuations of the different types of individuals irrespective of t heir positions is developed in the limit of very large systems with co nstant total population density resulting in a Hamilton-Jacobi equatio n for the logarithm of the probability density of the proportions of s usceptibles, infective and recovered. The deterministic equations of t he process are identified with the evolution equations for the most pr obable fluctuation paths. If the individuals are confined at discrete positions in space this approach leads to the ordinary differential eq uation description of cellular automata epidemics presented in the lit erature. A continuous space alternative description is also presented. It is shown that the logarithm of the probability density of populati on fluctuations is a Liapunov function for the deterministic evolution equations. This Liapunov function is used for developing a generalize d thermodynamic formalism of the epidemic process similar to the therm odynamic and stochastic theory of Ross, Hunt and Hunt from nonequilibr ium thermodynamics of transport processes and reaction-diffusion syste ms.