Estimation of safety areas for epidemic spread

TL;DR

This paper develops a probabilistic framework to estimate safety regions for epidemic spread, using a stochastic model of infection incidence and transition kernels to predict areas at risk at future times.

Contribution

It introduces a novel method for constructing safety sets in epidemic models based on random measures and ergodic assumptions, providing a way to quantify risk levels.

Findings

Proposed a method to compute safety areas with controlled risk levels.

Derived conditions for the support of infection measures and safety set properties.

Established probabilistic bounds for the intersection of safety sets with 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.