Delay-driven phase transitions in an epidemic model on time-varying networks

Area of Science:

• Complex systems
• Network science
• Mathematical epidemiology

Background:

• Networked systems often exhibit dynamic interactions, posing challenges for analysis and control.
• Epidemic modeling on static networks is well-established, but time-varying interactions require advanced methods.

Purpose of the Study:

• To investigate epidemic processes on complex networks with time-varying interactions and time delays.
• To analyze the impact of time delays on system stability and pattern formation.
• To determine critical conditions for phase transitions in delayed, time-varying epidemic models.

Main Methods:

• Development of an averaging theorem to approximate the delayed time-varying system with autonomous differential equations.
• Analysis of system evolution and determination of critical time delays using Hopf bifurcation theory.
• Numerical simulations on periodically, blinking, and quasi-periodically time-varying networks.

Main Results:

• An averaging theorem was established for analyzing delayed, time-varying systems.
• A critical time delay was identified, beyond which endemic equilibrium becomes unstable, leading to oscillations via Hopf bifurcation.
• Numerical simulations confirmed theoretical findings across different time-varying network types.

Conclusions:

• Time delays play a crucial role in the stability and dynamics of epidemic processes on time-varying networks.
• Time delays can facilitate pattern formation, such as Turing patterns, by extending the network frequency range.
• The findings provide insights into controlling and predicting epidemic spread in dynamic environments.