Statistical-mechanical analogies for space-dependent epidemics

Marcel Ovidiu Vlad, Birgitt Schönfisch and Claude Lacoursière

Physica A: Statistical Mechanics and its Applications, 1996, vol. 229, issue 3, 365-401

Abstract: 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.

Date: 1996
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/0378437195004017
Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:229:y:1996:i:3:p:365-401

DOI: 10.1016/0378-4371(95)00401-7

Access Statistics for this article

Physica A: Statistical Mechanics and its Applications is currently edited by K. A. Dawson, J. O. Indekeu, H.E. Stanley and C. Tsallis

More articles in Physica A: Statistical Mechanics and its Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().