the contact process
The contact process is the canonical interacting particle system model of an epidemic or invasion spreading on a network. Each site x of Z^d (or a graph) is either healthy (eta(x) = 0) or infected (eta(x) = 1). An infected site recovers at rate 1, returning to healthy with no immunity; a healthy site becomes infected at a rate lambda times the number of its currently infected neighbours. The single parameter lambda is the infection rate, the relative strength of contagion versus recovery. It is the spatial analogue of the SIS (susceptible-infected-susceptible) epidemic, and the question it poses is the central question of epidemiology: starting from a single infected site, does the infection die out, or can it survive and spread forever?
Formally the flip rates are c(x, eta) = 1 if eta(x) = 1 (recovery), and c(x, eta) = lambda * (number of infected neighbours of x) if eta(x) = 0 (infection). The all-healthy configuration is absorbing — once everyone is healthy, nothing can restart the infection — so the empty state is always a trap and the delta measure on it is always invariant. The contact process is attractive (monotone): more infection now leads to more infection later, which permits the basic coupling and a graphical construction with infection arrows (Poisson rate lambda along each oriented edge) and recovery marks (Poisson rate 1 at each site). It is dual to itself in a self-dual, monotone sense: the probability that an infection started from set A reaches set B by time t equals the probability that one started from B reaches A — formally P(eta_t^A intersect B nonempty) = P(eta_t^B intersect A nonempty), which reduces survival questions to the behaviour of a single growing cluster. The contact process is not exactly solvable; its analysis rests on monotonicity, subadditivity, duality and percolation comparisons rather than closed formulas.
The contact process matters because it is the cleanest rigorous model exhibiting a genuine phase transition in a stochastic spatial epidemic: there is a critical value lambda_c (depending on dimension and graph) below which the infection from any finite seed dies out almost surely, and above which it survives with positive probability. It is also the standard testing ground for the complete convergence theorem, which describes the long-run law for any starting configuration. An honest caveat: lambda_c is known only by bounds and numerics, not in closed form (in d = 1 it is approximately 1.649), and behaviour at exactly lambda_c is itself a deep result — the contact process dies out at criticality (Bezuidenhout-Grimmett), which was a major theorem, not an obvious fact.
In d = 1, start with a single infected site at the origin. If lambda < lambda_c (approximately 1.649) the infection almost surely dies out in finite time. If lambda > lambda_c, with positive probability the infected region grows roughly linearly, the leftmost and rightmost infected sites drifting apart at a deterministic speed (a shape theorem), and conditioned on survival the interior settles into the nontrivial upper invariant measure.
One parameter, two fates: below lambda_c the epidemic dies out; above it survives and spreads at a positive speed.
The empty (all-healthy) state is absorbing, so once the infection reaches it the process is stuck — and lambda_c is not known in closed form. A subtle but proven fact: at exactly lambda = lambda_c the contact process dies out (Bezuidenhout-Grimmett), so there is no surviving critical phase.