On the Critical Behavior at the Lower Phase Transition of the Contact   Process
  We present general results for the contact process by a method which applies to all transitive graphs of bounded degree, including graphs of exponential growth. The model's infection rates are varied through a control parameter, for which two natural transition points are defined as: i. $\lambda_T$, the value up to which the infection dies out exponentially fast if introduced at a single site, and ii. $\lambda_H$, the threshold for the existence of an invariant measure with a non-vanishing density of infected sites. It is shown here that for all transitive graphs the two thresholds coincide. The method, which proceeds through partial differential inequalities for the infection density, yields also generally valid bounds on two related critical exponents. The main results discussed here were established by Bezuidenhout and Grimmett \cite{BG2} in an extension to the continuous-time process of the discrete-time analysis of Aizenman and Barsky \cite{AB}, and of the partially similar results of Menshikov \cite{M}. The main novelty here is in the direct derivation of the partial differential inequalities by an argument which is formulated for the continuum. 
Michael Aizenman and Paul Jung