Partial immunization processes
A partial immunization process (PIP) on a graph \(G\) with set of vertices \(V\) is a continuous-time Markov process with state-space \(\{-1,0,1\}^V\). So each vertex can be in the states \(-1\) (never infected), \(1\) (infected) or \(0\) (previously infected, partially immunized). Each vertex \(v\) flips according to the following rules: \(-1\)-\(1\) at rate \(\lambda_n\cdot i\), \(0\)-\(1\) at rate \(\lambda_o\cdot i\), \(1\)-\(0\) at rate 1, where \(i\) is the number of infected neighbors of \(v\) in \(G\), \(\lambda_o\), \(\lambda_n\) are some fixed positive numbers. The PIP \(\eta_t^O\) starts from some origin vertex \(O\) (i.e. \(\eta_0^O(O)=1\), \(\eta_0^O(v)=-1\) for all \(v\not=0\)). It is said that \(\eta_t^O\) strongly survives if \(\text{Pr}\{\forall T \exists t>T \text{ with }\eta_t^O(O)=1\}>0\). The author describes assumptions on \((\lambda_o,\lambda_n)\) under which \(\eta^O_t\) survives for \(G={\mathbb{Z}}^d\) (regular lattice) and \(G={\mathbb{T}}_d\) (regular tree). E.g. on \({\mathbb{Z}}^d\), \(\eta^O_t\) survives strongly if \(\lambda_o>\lambda_c\) and \(\lambda_n>0\), where \(\lambda_c\) is a critical value of the contact process on \({\mathbb{Z}}^d\). (A contact process is PIP with \(\lambda_o=\lambda_n\).) If \(\lambda_o<\lambda_c\), then PIP does not survive.
scientific article; zbMATH DE number 2005173
scientific article