Stability and Hopf bifurcation analysis of an age-structured SVIRS epidemic model with temporary immunity

TL;DR

This study develops an age-structured SVIRS epidemic model that incorporates temporary immunity via recovery-age structure. It reformulates the system as a non-densely defined abstract Cauchy problem to establish existence, uniqueness, and boundedness, and derives the basic reproduction number $\mathcal{R}_0$. Stability of the disease-free and endemic equilibria is analyzed, with a detailed Hopf bifurcation analysis showing periodic dynamics can emerge as the immunity duration $\tau$ increases beyond a critical threshold $\tau_0$. Numerical simulations corroborate the theoretical predictions and highlight the critical role of immune duration in disease transmission and control strategies.

Abstract

In this paper, we investigate an SVIRS epidemic model that incorporates both temporary immunity and an age-structured recovery process. By reformulating the system as a non-densely defined abstract Cauchy problem, we establish the existence and uniqueness of solutions and derive the basic reproduction number $ \mathcal{R}_0 $. The stability of the equilibria is analyzed through the associated characteristic equations, and the occurrence of Hopf bifurcation near the endemic equilibrium is rigorously demonstrated. Our theoretical results reveal that temporary immunity plays a crucial role in shaping the stability of the endemic state. Finally, numerical simulations are carried out to verify and illustrate the analytical findings.

Figures (6)

• Figure 1: The pictorial representation of proposed model
• Figure 2: When $\Pi=5,\tau=12$, the disease-free equilibrium $E^0$ is locally asymptotically stable (LAS) for $\mathcal{R}_0 < 1$
• Figure 3: When $\Pi=20,\tau=12$, the endemic equilibrium $E^*$ is LAS for $\mathcal{R}_0 > 1$
• Figure 4: When $\Pi=20,\tau=19$, the endemic equilibrium $E^*$ is LAS for $\mathcal{R}_0 > 1$
• Figure 5: When $\Pi=20,\tau=20$, the endemic equilibrium $E^*$ is not LAS for $\mathcal{R}_0 > 1$

Theorems & Definitions (16)

• Lemma 2.1
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• Proposition 3.1
• proof
• Lemma 5.1
• proof