Dynamics of an epidemic model with relapse over a two-patch environment.

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems

Background:

• Infectious disease modeling often simplifies recovery dynamics.
• Understanding population mobility is crucial for disease spread.
• Relapse and fixed recovery periods require specific modeling approaches.

Purpose of the Study:

• To develop a delay differential equation model for infectious diseases in a two-patch environment.
• To analyze the impact of relapse and individual mobility on disease dynamics.
• To investigate how different travel rate scenarios affect epidemic thresholds and equilibria.

Main Methods:

• Derivation of a delay differential equation model with fixed relapse time and non-local mobility terms.
• Analysis of model dynamics under two scenarios: irreducible and reducible travel rate matrices.
• Calculation of global threshold dynamics using principal eigenvalues for irreducible matrices.
• Exploration of explicit results for special cases within reducible matrices.

Main Results:

• Established global threshold dynamics based on travel rates and relapse parameters.
• Demonstrated that travel rates of recovered/infectious individuals have different impacts compared to susceptible individuals.
• Identified that a boundary equilibrium is possible under reducible travel rate scenarios, but not under irreducible ones.

Conclusions:

• The developed model captures complex epidemic dynamics including relapse and spatial mobility.
• Travel rates play a critical role in disease transmission, with distinct effects depending on the population group (susceptible vs. infectious/recovered).
• The reducibility of travel matrices significantly alters the possible epidemic outcomes, allowing for boundary equilibria not seen in irreducible cases.