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Asymptotic analysis of a stochastic SVEIS epidemic model using Black-Karasinski process

Lahcen Khammich, Driss Kiouach

TL;DR

The paper analyzes a stochastic SVEIS epidemic model in which transmission is perturbed by a Black-Karasinski process. Using Lyapunov-function techniques and Ito calculus, it derives two thresholds, $R_0^s$ for persistence and $R_0^e$ for extinction. It proves the existence of an ergodic stationary distribution when $R_0^s>1$ and shows exponential extinction when $R_0^e<1$, illustrating that random fluctuations can promote outbreaks. These results provide rigorous criteria for long-run disease behavior under stochastic perturbations and offer insight into noise-driven outbreak risk for public health planning.

Abstract

In this paper, we present a stochastic SVEIS epidemic model perturbed by a Black-Karasinski process. Using a Lyapunov functional approach, we derive a sufficient condition, Rs0>1 for the existence of a stationary distribution, which indicates disease persistence. Additionally, we theoretically demonstrate that the disease will die out at an exponential rate if Re0<1 . Our results show that random fluctuations will facilitate disease outbreak.

Asymptotic analysis of a stochastic SVEIS epidemic model using Black-Karasinski process

TL;DR

The paper analyzes a stochastic SVEIS epidemic model in which transmission is perturbed by a Black-Karasinski process. Using Lyapunov-function techniques and Ito calculus, it derives two thresholds, for persistence and for extinction. It proves the existence of an ergodic stationary distribution when and shows exponential extinction when , illustrating that random fluctuations can promote outbreaks. These results provide rigorous criteria for long-run disease behavior under stochastic perturbations and offer insight into noise-driven outbreak risk for public health planning.

Abstract

In this paper, we present a stochastic SVEIS epidemic model perturbed by a Black-Karasinski process. Using a Lyapunov functional approach, we derive a sufficient condition, Rs0>1 for the existence of a stationary distribution, which indicates disease persistence. Additionally, we theoretically demonstrate that the disease will die out at an exponential rate if Re0<1 . Our results show that random fluctuations will facilitate disease outbreak.

Paper Structure

This paper contains 5 sections, 3 theorems, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of the global solution
  3. Stationary distribution
  4. Extinction
  5. Conclusion

Key Result

Theorem 1

For any initial condition $(S(0), V(0), E(0), I(0),z(0)) \in \mathbb{R}_+^4 \times \mathbb{R}$ the system 3 admits a unique global solution almost surely , and the solution remains forever in the invariant set: .

Figures (1)

  • Figure 1: Schematic diagram for SEIV model

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof