Contact process (mathematics)

The contact process is a stochastic process used to model population growth on the set of sites ${\displaystyle S}$ of a graph in which occupied sites become vacant at a constant rate, while vacant sites become occupied at a rate proportional to the number of occupied neighboring sites. Therefore, if we denote by ${\displaystyle \lambda }$ the proportionality constant, each site remains occupied for a random time period which is exponentially distributed parameter 1 and places descendants at every vacant neighboring site at times of events of a Poisson process parameter ${\displaystyle \lambda }$ during this period. All processes are independent of one another and of the random period of time sites remains occupied. The contact process can also be interpreted as a model for the spread of an infection by thinking of particles as a bacterium spreading over individuals that are positioned at the sites of ${\displaystyle S}$, occupied sites correspond to infected individuals, whereas vacant correspond to healthy ones.

The main quantity of interest is the number of particles in the process, say ${\displaystyle N_{t}}$, in the first interpretation, which corresponds to the number of infected sites in the second one. Therefore, the process survives whenever the number of particles is positive for all times, which corresponds to the case that there are always infected individuals in the second one. For any infinite graph ${\displaystyle S}$ there exists a positive and finite critical value ${\displaystyle \lambda _{c}}$ so that if ${\displaystyle \lambda >\lambda _{c}}$ then survival of the process starting from a finite number of particles occurs with positive probability, while if ${\displaystyle \lambda <\lambda _{c}}$ their extinction is almost certain. Note that by reductio ad absurdum and the infinite monkey theorem, survival of the process is equivalent to ${\displaystyle N_{t}\to \infty }$, as ${\displaystyle t\to \infty }$, whereas extinction is equivalent to ${\displaystyle N_{t}\to 0}$, as ${\displaystyle t\to \infty }$, and therefore, it is natural to ask about the rate at which ${\displaystyle N_{t}\to \infty }$ when the process survives.