Periodic solutions and bifurcation in an epidemic model with birth pulses

Area of Science:

• Mathematical Biology
• Epidemiology
• Dynamical Systems

Background:

• Epidemic models are crucial for understanding disease transmission dynamics.
• Periodic events, like births, and population fluctuations can significantly impact disease spread.
• Analyzing the stability of disease-free and endemic states is essential for public health interventions.

Purpose of the Study:

• To analytically and numerically investigate the dynamical behavior of an epidemic model with birth pulses and a varying population.
• To determine the conditions for the existence and stability of both infection-free and endemic periodic solutions.
• To derive conditions for the bifurcation of positive periodic solutions.

Main Methods:

• Utilized discrete maps to represent the model's dynamics.
• Applied the center manifold theorem for stability analysis.
• Employed bifurcation theory to identify conditions for new solution branches.

Main Results:

• Derived conditions for the existence of bifurcations leading to positive periodic solutions.
• Demonstrated good agreement between theoretical predictions and numerical simulations.
• Illustrated phase portraits, periodic solutions, and bifurcation diagrams.

Conclusions:

• The study provides a comprehensive analysis of the complex dynamics in an epidemic model with periodic forcing and population variability.
• The findings offer insights into the conditions that can lead to sustained disease presence or eradication.
• The validated analytical framework can be applied to other population dynamics models.