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

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.

Figures (14)

• Figure 1: Schematic diagram illustrating the transmission dynamics between population groups.
• Figure 2: The graph of the function $A$ shows that it has a single simple positive root for the given parameters: $\alpha = 130$, $\sigma = 0.5$, $\beta= 0.039$, $\mu= 0.045$, $q_1= 120$, $q_2= 90$, $\lambda= 0.002$, $\delta=140$, $\rho=5$, $b= 0.52$, $\gamma = 0.1$. The basic reproduction number is computed and we found $R_0=3.1080>1$.
• Figure 3: The graph of the function $A$ shows that it has one double positive root and one simple positive root using the parameters: $\alpha = 177.46$, $\sigma = 0.5$, $\beta= 0.04$, $\mu= 0.045$, $q_1= 509.99856$, $q_2= 30$, $\lambda = 0.00200000043$, $\delta = 140.0001$, $\rho = 0.2$, $b= 0.52$, $\gamma = 0.1$. We found $R_0=3.1877>1$.
• Figure 4: The graph of the function $A$ shows the existence of three positive real roots for the given parameters: $\alpha = 270$, $\sigma = 0.5$, $\beta = 0.04$, $\mu= 0.045$, $q_1= 510$, $q_2= 30$, $\lambda = 0.002$, $\delta = 140$, $\rho= 0.2$, $b= 0.52$, $\gamma = 0.1$. The basic reproduction number $R_0=3.1877>1$.
• Figure 5: The graph of the equation $K\left( x\right) =x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+\left( r_{2}x^{2}+r_{1}x+r_{0}\right) \left( 1-\cos \left( \sqrt{x}L\right) \right)=0.$ We observe that $x_{-}^{\ast }=0.3599$ and $x_{+}^{\ast }=0.5212$.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof