Bifurcation Analysis of a Delayed Infection Model with General Incidence Function

Area of Science:

• Mathematical epidemiology
• Dynamical systems theory
• Infectious disease modeling

Background:

• Infectious disease models are crucial for understanding disease spread.
• Time delays and general incidence functions significantly impact model dynamics.
• Stability analysis of disease equilibria is essential for public health insights.

Purpose of the Study:

• To formulate and analyze a mathematical infection model incorporating time delays and general incidence.
• To investigate the occurrence and characteristics of Hopf bifurcations.
• To explore the impact of different functional incidence types on model stability and dynamics.

Main Methods:

• Theoretical analysis of the formulated infection model.
• Bifurcation analysis using time delay as the bifurcation parameter.
• Numerical simulations employing four distinct functional incidence types (bilinear, saturation, Beddington-DeAngelis, Hattaf-Yousfi).

Main Results:

• The positive equilibrium of the model can lose stability, leading to Hopf bifurcations.
• The direction of Hopf bifurcations and stability of periodic solutions were determined.
• Numerical simulations demonstrated rich dynamics, including bifurcations and chaotic solutions, across different incidence functions.

Conclusions:

• Time delays are critical in driving complex dynamics in infection models.
• The choice of incidence function significantly influences the stability and emergent behaviors of disease models.
• The study highlights the potential for rich and sometimes chaotic disease dynamics in epidemiological systems.