Infection dynamics in dynamical random environments

Abstract

In this thesis, we investigate several variations of the contact process in dynamical random environments acting on the vertices and edges of an underlying graph $G$. We consider both Markovian environments and non-Markovian environments driven by renewal processes. In the Markovian setting, we study a large class of environment processes on the $d$-dimensional integer lattice with a finite local state space. Our main focus is the asymptotic shape of the set $\bfH_t$ of vertices that have been infected up to time $t$. Under suitable exponential estimates on the growth of the infection region and the extinction time, we prove an asymptotic shape theorem under the assumption that the entire process is \textit{worst-case-monotone}. This assumption is weaker than the classical monotonicity condition commonly used for variants of the contact process. If the environment consists of independent Markov processes at every vertex and edge, we show that this assumption can be omitted. We verify the required growth conditions for several types of random environments using a dynamical renormalisation approach and, in the monotone case with independent vertex and edge updates, obtain the asymptotic shape theorem throughout the entire supercritical region. We further use the renormalisation scheme to establish extinction of the critical process and complete convergence. In the non-Markovian setting, we consider a contact process in a vertex environment driven by renewal processes, motivated by the phenomenon of dormancy. Vertices can be either active or dormant, with dormant-to-active transitions governed by renewal processes with heavy-tailed interarrival distributions. We further assume that recovery is only possible in the active state. If infections between dormant individuals are impossible and the interarrival distribution has approximately power-law tails with exponent $\alpha\in(0,1)$, we show that the infection grows at most logarithmically, in contrast to the linear growth observed in the Markovian setting. On the other hand, if infections between dormant individuals are possible, we show that the process can survive with positive probability even on finite graphs under suitable conditions.