# Stochastic epidemic SEIRS models with a constant latency period

## TL;DR

This paper analyzes the stability of deterministic and stochastic SEIRS epidemic models with a fixed latency period, providing conditions for disease-free and coexistence equilibrium stability, and examining the effects of stochastic fluctuations.

## Contribution

It introduces stability conditions for both deterministic and stochastic SEIRS models with delay, including new results on stochastic stability using Lyapunov methods.

## Key findings

- Conditions for stability of disease-free equilibrium in deterministic models
- Stability criteria for coexistence equilibrium with delay
- Concentration results for stochastic fluctuations and stability under randomness

## Abstract

In this paper we consider the stability of a class of deterministic and stochastic SEIRS epidemic models with delay. Indeed, we assume that the transmission rate could be stochastic and the presence of a latency period of $r$ consecutive days, where $r$ is a fixed positive integer, in the "exposed" individuals class E. Studying the eigenvalues of the linearized system, we obtain conditions for the stability of the free disease equilibrium, in both the cases of the deterministic model with and without delay. In this latter case, we also get conditions for the stability of the coexistence equilibrium. In the stochastic case we are able to derive a concentration result for the random fluctuations and then, using the Lyapunov method, that under suitable assumptions the free disease equilibrium is still stable.