Survival probability of the contact process ($\theta (\lambda ,\mu )$)

The survival probability is the probability that the contact process begun from one infected vertex never reaches its absorbing empty state. Its critical infection rate is ${\lambda }_c(\mu )=\inf \{\lambda :\theta (\lambda ,\mu )>0\}$.

Local survival means repeated infection of a fixed vertex at arbitrarily large times in the contact process. It implies global survival, but infection can survive by moving away from every fixed vertex on an infinite graph. On a transitive graph the threshold is independent of the chosen vertex, and ${\lambda }_1\le {\lambda }_2$, with ${\lambda }_1$ the global survival threshold of the contact process.

Orient a degree-$(d+1)$ regular tree towards one end, and let height $g$ decrease along the unique parent edge and increase along the $d$ child edges. For $w_{\rho }(A)={\sum }_{x\in A}{\rho }^{g(x)}$, suppressing the occupied-target constraint only increases births, so its infinitesimal generator satisfies $L{w}_{\rho }\le [-1+\lambda ({\rho }^{-1}+d\rho )]w_{\rho }$. For $d>1$, choose $\rho =d^{-1/2}$. If $\lambda <1/(2\sqrt{d})$, the expected weight and the infection probability at each fixed site decay exponentially. The expected number of recoveries there equals its expected total occupation time, which is finite. Each infection episode ends almost surely at a recovery, so infinitely late infection is impossible. For $d=1$, take $\rho <1$ sufficiently close to one for each $\lambda <1/2$.

This is the critical infection rate for the contact process started from one infected site to avoid its absorbing empty state forever. It concerns infection somewhere in the graph at all times, rather than repeated infection of a fixed site. On a transitive graph it is independent of the starting site. Infection-arrow inclusion couples different rates monotonically.

Inside a rooted $d$-ary subtree of a degree-$(d+1)$ regular tree, pair each cell root with one distinguished child. Keep reinfection inside each pair, suppress infection back into its parent cell, and allow each child cell to be activated only once. Offspring counts of different cells are independent and identically distributed, so cell genealogies form a Galton-Watson process. A cell root has $d-1$ outgoing child-cell edges and its paired vertex has $d$. Starting at the root, the probabilities to send an arrow along one specified outgoing edge from these two vertices before internal extinction are respectivelyThey follow by first-step equations on the three nonempty pair states. The offspring mean is $M=(d-1)a+db$, andContinuity makes $M>1$ at some strictly smaller positive rate. The branching-process extinction criterion then gives positive survival probability for the suppressed process, hence for the original contact process.