Dynamic analysis of the recurrent epidemic model.

Area of Science:

• Mathematical epidemiology
• Dynamical systems theory
• Population dynamics

Background:

• Recurrent infectious diseases pose significant public health challenges.
• Understanding disease persistence requires models that account for population structure and immunity dynamics.
• Temporary immunity and time delays are crucial factors in disease transmission.

Purpose of the Study:

• To propose and analyze a novel SIRS model incorporating age structure and temporary immunity.
• To investigate the impact of time delays on the stability of disease equilibria.
• To determine conditions for Hopf bifurcation and explore non-periodic and periodic disease behaviors.

Main Methods:

• Formulation of the SIRS model as an abstract non-densely defined Cauchy problem.
• Analysis of global stability for the disease-free equilibrium.
• Analysis of local stability for the endemic equilibrium.
• Derivation of conditions for Hopf bifurcation.

Main Results:

• Established conditions for the global stability of the disease-free equilibrium.
• Determined the local stability of the endemic equilibrium.
• Identified conditions for the existence of Hopf bifurcation, indicating potential for oscillations.
• Demonstrated that time delays significantly influence disease persistence and dynamics, allowing for both non-periodic and periodic outcomes.

Conclusions:

• The proposed age-structured SIRS model provides a robust framework for studying recurrent infectious diseases.
• Temporary immunity and time delays are critical parameters influencing disease spread and stability.
• The model predicts complex dynamics, including oscillations, underscoring the importance of incorporating these factors into epidemiological models.