Limits of a multi-patch SIS epidemic model

Area of Science:

• Epidemiology
• Mathematical Biology
• Statistical Mechanics

Background:

• Stochastic SIS models are crucial for understanding epidemic dynamics.
• Analyzing large-scale population dynamics requires considering both local and global contact patterns.
• The behavior of complex systems often simplifies in limiting cases.

Purpose of the Study:

• To analyze the limit behavior of a stochastic SIS epidemic model with a large number of sites (M) and individuals per site (N).
• To determine the long-term distribution of infected individuals across sites.
• To establish the existence and uniqueness of solutions governing the epidemic's spread.

Main Methods:

• Stochastic SIS model formulation for a multi-site population.
• Analysis of system behavior in the limit as M and N approach infinity.
• Investigation of two distinct limit procedures based on the order of M and N increase.
• Derivation and analysis of a partial differential equation (PDE) in weak form.

Main Results:

• The limiting distribution of infected individuals across sites is a probability measure.
• The evolution of this distribution is governed by the weak form of a PDE.
• Existence and uniqueness of solutions for the PDE were demonstrated.
• The infected distribution converges to a Dirac measure at a specific equilibrium value, x(*).

Conclusions:

• The long-term epidemic distribution in a large, multi-site population stabilizes.
• This stable state corresponds to the equilibrium of a simplified single-site SIS model.
• The findings provide insights into the macroscopic behavior of epidemics in spatially structured populations.