A DELAYED SEIQR EPIDEMIC MODEL WITH VACCINATION: STABILITY ANALYSIS, PARAMETER ESTIMATION, AND HOPF BIFURCATION DYNAMICS

Abstract

This paper investigates a delayed SEIQR (Susceptible–Exposed–Infectious–Quarantined–Recovered) epidemic model incorporating both vaccination of susceptible individuals and a time delay representing the latency period from exposure to infectiousness. We derive the basic reproduction number R₀ and conduct a thorough stability analysis of disease-free and endemic equilibria using the theory of characteristic equations and Lyapunov functions. It is established that when R₀ < 1, the disease-free equilibrium is globally asymptotically stable; when R₀ > 1, a unique endemic equilibrium exists and is locally asymptotically stable for small delays. We demonstrate that the time delay can destabilize the endemic equilibrium and induce Hopf bifurcation, leading to periodic oscillations in disease prevalence. Explicit conditions on the delay parameter τ are derived for the existence of purely imaginary eigenvalues. The direction and stability of the Hopf bifurcation are analyzed using the center manifold theorem and normal form theory. Parameter estimation is performed via nonlinear least squares fitting to publicly available COVID-19 incidence data from Pakistan (2020–2022). Numerical simulations validate the analytical findings and illustrate the rich dynamics produced by interactions between vaccination coverage, quarantine efficacy, and time delay.

Keywords: Delay differential equations; SEIQR model; Hopf bifurcation; basic reproduction number; vaccination; quarantine; parameter estimation; COVID-19.