Delay epidemic models determined by latency, infection, and immunity duration

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems

Background:

• Understanding infectious disease dynamics is crucial for public health.
• Previous models often simplify the complex transitions between disease states.
• Time delays in disease progression significantly influence epidemic trajectories.

Purpose of the Study:

• To introduce novel single and two-strain epidemic models using delay differential equations.
• To analyze the stability of epidemic models and understand strain competition.
• To investigate conditions leading to periodic oscillations in disease prevalence.

Main Methods:

• Formulating epidemic models as systems of delay differential equations.
• Proving the existence and positivity of model solutions.
• Reducing delay differential equations to integral equations for stability analysis.
• Investigating stationary solutions and their stability criteria.

Main Results:

• Existence and positivity of solutions for the proposed models are mathematically proven.
• Analysis of stationary solutions reveals that the strain with a higher basic reproduction number typically dominates.
• Exceeding critical basic reproduction numbers can destabilize stationary solutions, leading to periodic oscillations where both strains coexist.
• Model dynamics in oscillatory regimes depend on parameters beyond basic reproduction numbers.

Conclusions:

• Delay differential equations provide a robust framework for modeling epidemic dynamics.
• Strain competition can lead to complex, non-intuitive dynamics, including periodic oscillations.
• The findings offer insights into the spread of multi-strain infectious diseases, with implications for seasonal influenza patterns.