Recurrent epidemic waves in a delayed epidemic model with quarantine

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems

Background:

• Epidemic models are crucial for understanding disease transmission dynamics.
• Quarantine and time delays are significant factors influencing epidemic spread.
• The behavior of epidemic models can exhibit complex dynamics, including oscillations.

Purpose of the Study:

• To analyze an epidemic model incorporating quarantine and distributed time delay.
• To determine the conditions for disease eradication versus endemicity.
• To investigate the emergence of recurrent epidemic waves through mathematical analysis and numerical simulation.

Main Methods:

• Definition and analysis of the basic reproduction number ($$).
• Stability analysis of disease-free and endemic equilibria.
• Application of Hopf bifurcation theory to identify conditions for periodic solutions.
• Numerical simulations to visualize model dynamics and parameter regions.

Main Results:

• The disease-free equilibrium is globally asymptotically stable when $$.
• An unstable disease-free equilibrium implies a unique endemic equilibrium when $$.
• Sufficient conditions for Hopf bifurcation were derived, leading to recurrent epidemic waves.

Conclusions:

• Quarantine and time delay are critical factors in the generation of recurrent epidemic waves.
• The basic reproduction number ($$) is a key determinant of epidemic persistence or extinction.
• The model provides insights into the complex dynamics of infectious disease outbreaks.