Contact process

In the Contact process, infected vertices recover at rate $\mu$, while each oriented nearest-neighbour edge transmits infection at rate $\lambda$. The empty configuration is absorbing.

Consider the Contact process started from a finite infected set $A$ on a degree-$(d+1)$ regular tree, with $d\ge 2$, recovery rate one and per-oriented-edge infection rate $\lambda >0$. Choose $0<\rho <1$ satisfying $\rho \lambda (d-1)\ge 1$. The bounded functional $f(A)={\rho }^{|A|}$ has generatorRecoveries decrease the infected count by one, whereas infections across the edge boundary increase it by one; summing these changes gives the formula.

A finite nonempty set induces a forest. If it has $c(A)$ connected components, edge counting gives $|{\partial }_{E}A|=(d-1)|A|+2c(A)$. Therefore the stated parameter condition makes $Lf\le 0$. At the absorbing empty state its generator is zero. Localizing the finite-particle process and using boundedness yields that ${\rho }^{|A_t|}$ is a supermartingale.

On extinction by time $t$, its value is one, so ${\mathbb{P}}_A(T_{\varnothing }\le t)\le {\mathbb{E}}_A{\rho }^{|A_t|}\le {\rho }^{|A|}$. Letting $t\to \infty$ proves the extinction bound ${\mathbb{P}}_A(T_{\varnothing }<\infty )\le {\rho }^{|A|}$. A strict choice $\rho <1$ exists when $\lambda (d-1)>1$, giving a sufficient survival condition from every nonempty finite seed. The range of $\rho$ is essential to both boundedness and the sign argument.

For the Contact process on a countable undirected bounded-degree graph with recovery rate one and symmetric edge infection rate $\lambda$, the all-healthy law $\underline{\nu }={\delta }_0$ is invariant. The graphical representation preserves coordinatewise order. Since every configuration is below the all-infected configuration, the laws ${\delta }_1{P}_t$ decrease in stochastic domination as $t$ grows. Their weak limit $\overline{\nu }$ exists and is invariant by the Feller property.

Every invariant probability law lies between $\underline{\nu }$ and $\overline{\nu }$ in stochastic order. These are the least and greatest invariant laws; classifying all extremal invariant laws requires further hypotheses. The greatest law is also an extreme point: in a convex decomposition into invariant laws, both components are dominated by it, and equality of all increasing finite-cylinder expectations forces both to equal it.

Self-duality identifies the one-site density with survival from that site:On a transitive graph the right side is independent of $x$ and is conventionally written $\theta (\lambda )$. On a general graph its site dependence must be retained. The two bounding invariant laws coincide exactly when ${\theta }_x(\lambda )=0$ for every site: countability then forces the upper law to assign probability one to the empty configuration. The lattice version of the maximal-law and duality construction is given by the two basic contact-process theorems in Rick Durrett's survey.^[ref]

With recovery rate one, the number of infected sites in a nearest-neighbour contact process on ${\mathbb{Z}}^d$ has death rate equal to its size and birth rate at most $2d\lambda$ times its size. It is dominated by a linear birth-death branching process with these per-particle rates. In particular, from one infection its expected size is at most $e^{(2d\lambda -1)t}$, so its Survival probability is zero when $2d\lambda <1$. This proves the displayed lower bound on the critical rate.

Place recovery marks at each vertex according to independent rate-$\mu$ Poisson processes and infection arrows on each oriented nearest-neighbour edge according to independent rate-$\lambda$ Poisson processes. A vertex is infected at time $t$ exactly when a time-directed path from an initially infected vertex reaches it while following arrows and avoiding recovery marks.

For the one-dimensional contact process with recovery rate one and per-neighbour infection rate $\lambda$, use parity sites $(x,n)$ with $x+n$ even. Call a site good if $x$ has no recovery in $[(n-1)\delta ,(n+1)\delta ]$ and sends an arrow to each neighbour during $[n\delta ,(n+1)\delta ]$. The goodness indicators are independent: recovery intervals for successive parity sites at the same spatial point only touch at endpoints, and all outgoing-arrow intervals and clocks are disjoint. Their common probability is the displayed $q$. A path of good sites carries infection from each time layer to the next, because both its source and target avoid recovery during the transmitting interval. Taking $\delta ={\lambda }^{-1/2}$ makes $q\to 1$. The incoming-edge coupling of site and bond percolation and $p_c^{\mathrm{bond}}<1$ imply $p_c^{\mathrm{site}}<1$, so this proves a finite critical infection rate.

Map a site $x\in {\mathbb{Z}}^d$ to its height $h(x)={\sum }_i{x}_i$. Maintain one infected representative in height $n$ for every infected site of an auxiliary one-dimensional contact process. The representative's death makes that auxiliary site recover. Its $d$ outgoing arrows to each neighbouring height, each of rate $\lambda$, make an auxiliary infection at total rate $d\lambda$, using the arrow's target as the new representative. Arrows to an already infected auxiliary height do nothing there. Distinct active representatives use distinct death and outgoing-arrow clocks, so their conditional rates are exactly those of the one-dimensional process. Every auxiliary infection has a live representative upstairs, proving the survival comparison.

Reachability in the graphical representation gives

Time reversal of the Poisson graphical representation, together with reversal of every infection arrow, gives

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.