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. Therefore, if we denote by 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 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 , 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. For any infinite graph there exists a positive and finite critical value so that if then survival of the process starting from a finite number of particles occurs with positive probability, while if their extinction is almost certain. Note that by and the infinite monkey theorem, survival of the process is equivalent to , as , whereas extinction is equivalent to , as , and therefore, it is natural to ask about the rate at which when the process survives. Mathematical Definition If the state of the process at time is , then a site in is occupied, say by a particle, if and vacant if . The contact process is a continuous-time Markov process with state space , where 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 , where the sum is over all the neighbors of in . This means that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and