MASS-ACTION LAW VERSUS LOCAL CONTAGION DYNAMICS - A MEAN-FIELD STATISTICAL APPROACH WITH APPLICATION TO THE THEORY OF EPIDEMICS

A mean-field approach for epidemic processes with high migration is su ggested by analogy with non-equilibrium statistical mechanics. For lar ge systems a limit of the thermodynamic type is introduced for which b oth the total size of the system and the total number of individuals t end to infinity but the population density remains constant. In the th ermodynamic limit the infection rate is proportional to the product of the proportion of individuals susceptible to infection and the averag e probability of infection. The limit form of the average probability of infection is insensitive to the detailed behaviour of the fluctuati ons of the number of infectious individuals and may belong to two univ ersality classes: (1) if the fluctuation of the number of infectives i s non-intermittent it increases with the increase of the partial densi ty of infectives and approaches exponentially the asymptotic value one for large densities; (2) for intermittent fluctuations obeying a powe r-law scaling the average probability of infection also displays a sat uration effect for large densities of infectives but the asymptotic va lue one is approached according to a power law rather than exponential ly. For low densities of infectives both expressions for the average p robability of infection are linear functions of the proportion of infe ctives and the infection rate is given by the mass-action law.