How delay, isolation and vaccination shape epidemic waves: a bifurcation approach in mathematical epidemiology

TL;DR

This paper develops a bifurcation-based mathematical model to analyze how delays in vaccination and isolation influence epidemic waves, revealing conditions for stability and periodic outbreaks.

Contribution

It introduces an SQIR-V model incorporating nonlinear infection rates and delays, providing new insights into epidemic wave formation and control strategies.

Findings

Vaccination and isolation delays significantly affect epidemic dynamics.

The model predicts the emergence of periodic epidemic waves via Hopf bifurcation.

Global stability of the disease-free state is confirmed when R0 ≤ 1.

Abstract

This research paper introduces an SQIR-V epidemic model to investigate the transmission of infectious diseases. Particular attention is paid to the roles of vaccination and quarantine (incorporating physical distancing interventions) in protecting susceptible individuals. The model features nonlinear transition rates that depend on the history of infection, allowing the emergence of periodic solutions. We calculate the basic reproduction number, R 0 , and analyze the local asymptotic stability of the equilibrium points. Additionally, we demonstrate that the diseasefree equilibrium is globally asymptotically stable when R 0 $\le$ 1. The study further explores the existence of periodic solutions through a Hopf bifurcation, showing the occurrence of epidemic waves. A condition was derived to determine the direction of the crossing of the imaginary axis. We finish by presenting some numerical simulations to illustrate how vaccination and isolation delays influence disease dynamics. Those findings highlight potential areas for further research and validation.