Fractals and Epidemic Process

The spread of an epidemic can be studied on a discrete space into small cells arranged into a d_s-dimensional regular lattice [Durett & Levin, 1994]. Each sites are occupied by healthy individuals may be infected by neighbours, after which they recover completely, they recover and are subsequently immune, or they die. Such a model is a generalisation of the differential equation approach. It corresponds to a modification of the directed percolation problem, useful to describe a large number of disordered systems in physics and chemistry. A critical concentration separate a phase where the epidemic dies out after a finite number of time steps, from a phase where the epidemic can continue forever.

In the simplest models, we assume that the vicinity, in which the infection process takes place, is a small domain surrounding the healthy individual considered. This vicinity is made up of the first layers of M = 3^ds-1 cells surrounding the central cell considered (Moore neighbourhood). The purpose of this article is to generalise the dimension of the substrate by introducing a fractal distribution of the sites. For each distribution of infected individuals in this vicinity, there is a certain probability ξ of infection. Due to the self-similarity, the infection quantities are significantly modified on fractal substrate.

The fractal distribution of the sites can be related to the spatial distribution of the epidemic vector [Meltzer, 1991]. Vector distribution is a matter of suitable habitat, which is a sum of a wide range of environmental factors (humidity, soil moisture, ground temperature, parasitic-host population density, etc..). The distribution of the sites can be also related to the genetic distribution of the susceptibility of the host population. In a herd, the laws of inheritance form a discrete and recursive system which mixes and distributes the genes of susceptibility. We can propose an aggregation model of relatives around an individual, which is based on the direct inheritance.