Estimation of safety areas for epidemic spread

Abstract

In this work we study safety areas in epidemic spred. The aim of this work is, given the evolution of epidemic at time $t$, find a safety set at time $t+h$. This is, a random set $K_{t+h}$ such that the probability that infection reaches $K_{t+h}$ at time $t+h$ is small. More precisely, inspired on the study of epidemic spread, we consider a model in which the measure $\mu_n(A)$ is the incidence -density of infectives individuals- in the set $A$, at time $n$ and $$\mu_{n+1}(A)(\omega)=\int_S{\pi_{n+1}(A;s)(\omega)\mu_n(ds)(\omega)}, {for any Borel set} A, $$ with random transition kernels of the form $$\pi_n(.;.)(\omega)=\Pi(.;.)(\xi_n(\omega),Y_n(\omega)),$$ where $\xi$, $Y$ satisfy some ergodic conditions. The support of $\mu_n$ is called $S_n$. We also assume that $S_0$ is compact with regular border and that for any $x,y$ the kernel $\Pi(.;.)(x,y)$ has compact support. A random set $K_{n+1}$ is a safety area of level $\alpha$ if: [{$i$)}] $K_{n+1}$ {\rm is a function of} $S_0, S_1, ...,S_n.$ [{$ii$)}] $P(K_{n+1} \cap S_{n+1} \neq \emptyset)\leq \alpha.$ We present a method to find these safety areas and some related results.