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Stability Analysis of a Diffusive SVIR Epidemic Model with Distributed Delay, Imperfect Vaccine and General Incidence Rate

Achraf Zinihi, Mostafa Tahiri, Moulay Rchid Sidi Ammi

TL;DR

This work analyzes a reaction-diffusion SVIR epidemic model that incorporates distributed delay and a general incidence rate, with imperfect vaccination. The authors establish well-posedness and global existence, derive the basic reproduction number $R_0$, and prove global asymptotic stability of the disease-free equilibrium for $R_0<1$ and of the endemic equilibrium for $R_0>1$ using Lyapunov functionals and LaSalle's invariance principle. They characterize the disease-free and endemic equilibria explicitly and demonstrate the theoretical results with 1D numerical simulations that align with the analytic predictions. The study provides insights into how spatial diffusion, delay, and vaccination interact to shape long-term disease dynamics, with implications for control strategies in heterogeneous landscapes.

Abstract

In this chapter, we consider a reaction-diffusion SVIR infection model with dis-tributed delay and nonlinear incidence rate. The wellposedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.

Stability Analysis of a Diffusive SVIR Epidemic Model with Distributed Delay, Imperfect Vaccine and General Incidence Rate

TL;DR

This work analyzes a reaction-diffusion SVIR epidemic model that incorporates distributed delay and a general incidence rate, with imperfect vaccination. The authors establish well-posedness and global existence, derive the basic reproduction number , and prove global asymptotic stability of the disease-free equilibrium for and of the endemic equilibrium for using Lyapunov functionals and LaSalle's invariance principle. They characterize the disease-free and endemic equilibria explicitly and demonstrate the theoretical results with 1D numerical simulations that align with the analytic predictions. The study provides insights into how spatial diffusion, delay, and vaccination interact to shape long-term disease dynamics, with implications for control strategies in heterogeneous landscapes.

Abstract

In this chapter, we consider a reaction-diffusion SVIR infection model with dis-tributed delay and nonlinear incidence rate. The wellposedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
Paper Structure (7 sections, 4 theorems, 58 equations, 2 figures, 1 table)

This paper contains 7 sections, 4 theorems, 58 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Basic properties of the model
  4. Equilibria and the basic reproduction number
  5. Global stability of equilibria
  6. Numerical results
  7. Conclusion and discussion

Key Result

Theorem 1

For any initial value function in $\mathbb{X}$, system E2.1 has a unique non-negative solution $\left(S(t, x), V(t, x), I(t, x), R(t, x)\right)$ for all $(t,x) \in [0, \infty)\times\Omega$.

Figures (2)

  • Figure 1: Numerical results of \ref{['E2.1']}-\ref{['E2.2']} when $\mathcal{R}_0 < 1$.
  • Figure 2: Numerical results of \ref{['E2.1']}-\ref{['E2.2']} when $\mathcal{R}_0 > 1$.

Theorems & Definitions (8)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof