Four-compartment epidemic model with retarded transition rates

Area of Science:

• Epidemiology
• Mathematical Biology
• Statistical Physics

Background:

• Existing epidemic models often lack memory effects, limiting their ability to capture complex disease dynamics.
• Understanding disease transmission requires models that account for variable incubation, infectiousness, and recovery periods.

Purpose of the Study:

• To develop and analyze a novel Susceptible-Incubated-Infectious-Recovered-Susceptible (S-C-I-R-S) epidemic model incorporating memory effects.
• To investigate the impact of non-exponential waiting times on epidemic dynamics using fractional calculus.
• To validate the macroscopic model with a microscopic random-walker simulation.

Main Methods:

• Derivation of memory evolution equations using fractional calculus for the macroscopic S-C-I-R-S model.
• Analysis of endemic equilibrium, its existence, and stability, including Hopf bifurcations.
• Implementation of a multiple random-walker simulation to model disease spread microscopically.

Main Results:

• Fractional ordinary differential equations describe epidemic evolution with fat-tailed waiting-time distributions.
• Formulas for endemic equilibrium and conditions for its existence were derived.
• Macroscopic model predictions showed high accuracy when compared with microscopic simulation results.

Conclusions:

• The S-C-I-R-S model with memory effects provides a more realistic framework for studying epidemic dynamics.
• Fractional calculus effectively captures long-waiting time phenomena in disease spread.
• The random-walker approach serves as a valid microscopic basis for the macroscopic S-C-I-R-S model, applicable to various epidemic scenarios.