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arXiv:1610.04301 (Published 2016-10-14)
On an epidemic model on finite graphs
Comments: 25 pages. arXiv admin note: text overlap with arXiv:1609.08738Categories: math.PRWe study a system of random walks, known as the frog model, starting from an independent Poisson($\lambda$) particle's profile with one additional active particle planted at some vertex $\mathbf{o}$ of a finite connected simple graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. Initially, only the particles occupying $\mathbf{o}$ are active. Active particles perform $t \in \mathbb{N} \cup \{\infty \}$ steps of the walk they picked before vanishing and activate all inactive particles they hit. This system is often taken as a model for the spread of an epidemic over a population. Let $\mathcal{R}_t$ be the set of vertices which are visited by the process, when active particles vanish after $t$ steps. We study the susceptibility of the process on the underlying graph, defined as the random quantity $\mathcal{S}(\mathcal{G}):=\inf \{t:\mathcal{R}_t=\mathcal{V} \}$, the time it takes for the entire population get infected. We consider the cases that the underlying graph is either a regular expander or a $d$-dimensional torus of side length $n$ (for all $d \ge 1$) and determine asymptotic bounds for $\mathcal{S} $ up to a constant factor.
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Local and global survival for nonhomogeneous random walk systems on Z
Comments: 24 pages, 4 figures, all major results have been largely improvedCategories: math.PRWe study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site $n \ge 1$. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left jump probability $l_n$. We give conditions for global survival, local survival and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability $p_n \in [0, 1]$. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases $1/2-l_n \sim \pm 1/n^\alpha$, $p_n = 1$ and $1_n-1/2 \sim \pm 1/n^\alpha$, $1-p_n \sim 1/n^\beta$ (where $\alpha,\beta > 0$).