Epidemics with recovery in \(D=2\)
A mathematical model of the spread of epidemics (or forest fires) is considered. This model is constructed as an interacting random process on the lattice \(Z^ 2\), where only nearest neighbors interact. The probability distributions of the model are defined on the function space \((0,1,2)^{Z^ 2}\), where for each site z, \(z\in Z^ 2\), the states 0,1,2 have the following epidemic interpretation: state 0\(=healthy\) (alive), \(1=\inf ected\) (on fire), and \(2=removed=immune\) or dead (burnt). The random process dynamics in continuous time are defined by the following postulates: 1. An infected (burning) tree sends out the infection (sparks) at rate \(\alpha\). 2. An infection (spark) emitted from \(x\in Z^ 2\) attacks one of the four nearest neighbors chosen at random with equal probabilities. 3. If the infection (spark) hits a live tree, the tree catches it immediately and begins to emit the infection (sparks). The tree burns for an exponentially distributed amount of time with mean 1, then burns out. 4. Burnt sites return to life at rate \(\beta\). For \(\beta =0\) it is known that there is a critical value \(\alpha_ c\) such that if \(\alpha >\alpha_ c\) and we start with one infected tree, then the infection has positive probability of not going out [\textit{J. T. Cox}, and the first author, Stochastic Processes Appl. 30, No.2, 171-191 (1988; Zbl 0667.92016)]. The article is devoted to the proof of the conjecture that if \(\beta >0\) and \(\alpha >\alpha_ c\), then there exists a nontrivial stationary distribution on \((0,1,2)^{Z^ 2}\).
scientific article; zbMATH DE number 4213804