[Paper Review] Analysis of A Spatially Inhomogeneous Stochastic Partial Differential Equation Epidemic Model

Dang H. Nguyen, Nhu N. Nguyen+1 more|arXiv (Cornell University)|Dec 8, 2018

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine34 references3 citations

TL;DR

This paper introduces a novel spatially inhomogeneous stochastic partial differential equation (SPDE) model for epidemic dynamics, incorporating spatial variation and environmental noise via multiplicative white noise. It establishes existence and uniqueness of positive mild solutions and derives sufficient conditions for disease extinction or permanence, with a key result showing that extinction occurs when the infection rate is below a threshold involving the recovery rate and noise intensity.

ABSTRACT

This work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, existence and uniqueness of solutions of the underlying SPDEs are examined. Then definitions of permanence and extinction are given. Certain sufficient conditions are provided for the permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.

Motivation & Objective

• To develop a new class of epidemic models using stochastic partial differential equations (SPDEs) to capture spatial heterogeneity and environmental noise.
• To extend classical SIR models beyond deterministic, mean-field frameworks by incorporating spatial diffusion and random perturbations.
• To analyze long-term behavior—specifically extinction and permanence—of the epidemic process under stochastic spatial dynamics.
• To provide rigorous mathematical foundations for SPDE-based epidemic modeling, including existence, uniqueness, and positivity of solutions.
• To open new research avenues in stochastic spatial epidemiology using SPDEs and lay groundwork for future extensions with regime-switching or Lévy noise.

Proposed method

• Formulates a system of SPDEs for SIR-type dynamics with spatially varying coefficients: diffusion, recruitment, death, and infection rates.
• Models environmental noise using multiplicative Itô noise terms: $ S(t,x)\,dW_1(t,x) $ and $ I(t,x)\,dW_2(t,x) $, representing stochastic perturbations in susceptible and infected populations.
• Defines and proves existence and uniqueness of positive mild solutions in a suitable Banach space $ E $, ensuring non-negativity of population densities.
• Applies semigroup theory and stochastic calculus in infinite dimensions to analyze the SPDE system and derive long-time behavior results.
• Uses Lyapunov-type analysis and ergodic-type arguments to derive sufficient conditions for extinction and permanence in terms of model parameters.
• Considers a special case with space-homogeneous coefficients and Brownian motion noise to illustrate theoretical results with explicit extinction/permanence thresholds.

Experimental results

Research questions

• RQ1 Under what conditions does the infected population go extinct in a spatially inhomogeneous SPDE epidemic model with environmental noise?
• RQ2 What conditions ensure the long-term persistence (permanence) of the infected population in the presence of spatial variation and stochastic perturbations?
• RQ3 How do spatial diffusion and multiplicative noise interact to influence the threshold between extinction and persistence in epidemic dynamics?
• RQ4 Can the classical SIR threshold criterion be adapted to stochastic spatial models, and how does noise modify this threshold?
• RQ5 What mathematical techniques are required to establish existence, uniqueness, and positivity of solutions for SPDE epidemic models?

Key findings

• The SPDE system admits a unique positive mild solution almost surely for any non-negative initial data in the space $ E $, ensuring biologically meaningful population densities.
• Extinction of the infected population occurs almost surely when $ \alpha < \mu_2 + \frac{1}{2}\sigma_2^2 $, where $ \alpha $ is the infection rate and $ \sigma_2^2 $ is the noise intensity on the infected class.
• Permanence of the infected population is guaranteed when $ \alpha > \mu_2 + \frac{1}{2}\sigma_2^2 $, provided the initial data satisfy $ \int_{\mathcal{O}} -\ln I_0(x)\,dx < \infty $.
• The extinction and permanence thresholds are sharp in the sense that the condition for permanence is almost necessary, mirroring results from SIS models.
• The model framework allows for extension to regime-switching SPDEs and Lévy-driven noise, suggesting broader applicability beyond Brownian motion.