[Paper Review] Multi-patch epidemic models with general exposed infectious periods

Guodong Pang, Étienne Pardoux|arXiv (Cornell University)|Jun 25, 2020

COVID-19 epidemiological studies22 references4 citations

TL;DR

This paper develops multi-patch epidemic models with general exposed and infectious period distributions, allowing for Markovian migration across patches in all disease states. It establishes a functional law of large numbers (FLLN) yielding Volterra integral equations and a functional central limit theorem (FCLT) with stochastic Volterra equations driven by Brownian motions and Gaussian processes, providing a rigorous diffusion approximation for complex, heterogeneous epidemic dynamics with non-exponential latency and infectious periods.

ABSTRACT

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as ``from distance from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.

Motivation & Objective

• To model epidemic spread across multiple patches with general distributions for exposed and infectious periods, beyond exponential assumptions.
• To incorporate migration of individuals in all disease states (susceptible, exposed, infectious, recovered) between patches via Markovian processes.
• To derive a functional law of large numbers (FLLN) that characterizes the macroscopic behavior of the epidemic process.
• To establish a functional central limit theorem (FCLT) for the diffusion scaling of the epidemic process.
• To provide a rigorous stochastic limit theory for multi-patch models with non-Markovian infection durations and spatial mixing.

Proposed method

• Formalize a multi-patch stochastic epidemic model with Markovian migration in all disease states: susceptible, exposed, infectious, recovered.
• Assume general (non-memoryless) distributions for exposed and infectious periods, independent of migration dynamics.
• Apply martingale functional central limit theorem techniques to derive the FCLT for the rescaled process.
• Derive the FLLN limit as a system of Volterra integral equations driven by patch-specific transmission rates and inter-patch transmission kernels.
• Characterize the FCLT limit as a system of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with explicit covariance structure.
• In the special case of deterministic exposed and infectious periods, the FLLN reduces to a system of delay differential equations (DDEs).

Experimental results

Research questions

• RQ1 How does the inclusion of general distributions for exposed and infectious periods affect the macroscopic dynamics of a multi-patch epidemic model?
• RQ2 What is the limiting behavior of the epidemic process in the law of large numbers scaling when migration and infection dynamics are coupled across patches?
• RQ3 How does the diffusion scaling of the epidemic process behave under general infection duration distributions and Markovian migration?
• RQ4 What is the covariance structure of the fluctuation process in the functional central limit theorem for such models?
• RQ5 Under what conditions does the FLLN limit reduce to a system of delay differential equations?

Key findings

• The functional law of large numbers (FLLN) limit is characterized by a system of Volterra integral equations that describe the mean-field dynamics of the epidemic across patches.
• In the case of deterministic exposed and infectious periods, the FLLN limit reduces to a system of delay differential equations (DDEs), extending classical SIR-type models to multi-patch settings.
• The functional central limit theorem (FCLT) limit is given by a system of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes.
• The covariance structure of the limiting Gaussian process is explicitly derived, depending on the patch-specific transmission rates and migration intensities.
• The model allows for long-range transmission from distant patches, with transmission rates depending on the distance between patches.
• The results provide a rigorous diffusion approximation for multi-patch epidemic models with non-exponential infection periods, enabling statistical inference and simulation under complex transmission and mobility patterns.