A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis

Area of Science:

• Mathematical Epidemiology
• Dynamical Systems Theory
• Public Health Modeling

Background:

• Epidemic transmission dynamics are complex, influenced by factors like social behaviors, natural conditions, and control measures.
• Mathematical models are crucial for understanding disease spread, predicting outbreaks, and evaluating intervention strategies.

Purpose of the Study:

• To develop and analyze a deterministic mathematical model for epidemic transmission incorporating time delays and nonlinear incidence/treatment rates.
• To investigate the stability of disease-free and endemic equilibria and identify conditions for disease persistence or eradication.

Main Methods:

• A compartmental model dividing the population into susceptible, infectious, and recovered individuals was formulated.
• Mathematical analysis of the delayed differential equations to determine local and global stability of equilibria.
• Investigation of Hopf bifurcation to understand the emergence of oscillations in disease prevalence.

Main Results:

• The disease-free equilibrium (DFE) is locally and globally asymptotically stable when the basic reproduction number (R0) is less than 1.
• An endemic equilibrium exists and can be locally and globally asymptotically stable under specific conditions (R0 > 1).
• Hopf bifurcation analysis indicates conditions under which the endemic equilibrium can lead to sustained oscillations in epidemic dynamics.

Conclusions:

• The model demonstrates that low basic reproduction numbers (<1) ensure disease eradication, while higher numbers (>1) can lead to persistent endemic states.
• Time delays and nonlinear functional responses significantly influence epidemic dynamics, potentially leading to complex behaviors like oscillations.
• The findings provide insights into epidemic control strategies by highlighting the critical role of the basic reproduction number and bifurcation phenomena.