Extremal invariant measures of the contact process ($\underline{\nu }={\delta }_0,\overline{\nu }={\lim }_{t\to \infty }{\delta }_1{P}_t$)

The graphical representation of the contact process preserves coordinatewise order. Its law from the all-infected state decreases in stochastic domination with time, and its limit is invariant by the Feller semigroup property. The all-healthy state is absorbing. Any invariant probability law lies between these two laws. Duality of the contact process gives $\overline{\nu }(\eta (x)=1)=\theta (\lambda )$ when the recovery rate is one. Consequently the two extremal laws coincide exactly when the survival probability of the contact process is zero.