Contact process (mathematics)
The contact process is a stochastic process used to model population growth on the set of sites
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.
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
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
, 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
, 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.
there exists a positive and finite critical value
then survival of the process starting from a finite number of particles occurs with positive probability, while if
Note that by reductio ad absurdum and the infinite monkey theorem, survival of the process is equivalent to
If the state of the process at time
is a finite or countable graph, usually
, and a special case of an interacting particle system.
More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site
This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and vice versa).
The contact process is a stochastic process that is closely connected to percolation theory.
Ted Harris (1974) noted that the contact process on
when infections and recoveries can occur only in discrete times
corresponds to one-step-at-a-time bond percolation on the graph obtained by orienting each edge of
in the direction of increasing coordinate-value.
A law of large numbers for the number of particles in the process on the integers informally means that for all large
Harris (1974) proved that, if the process survives, then the rate of growth of
A weak law of large numbers (that the process converges in probability) was shown by Durrett (1980).
A few years later, Durrett and Griffeath (1983) improved this to a strong law of large numbers, giving almost sure convergence of the process.
Contact processes on all integer lattices die out almost surely at the critical value.
[1] Durrett conjectured in survey papers and lecture notes during the 1980s and early 1990s regarding the central limit theorem for the Harris contact process, viz.
that, if the process survives, then for all large
[2][3][4] Durrett's conjecture turned out to be correct for a different value of
