Chemotaxis-inspired PDE models of airborne infectious disease transmission: epidemiologically-motivated mathematical and numerical analyses

TL;DR

This work extends PDE-based infectious disease modeling by incorporating chemotaxis-inspired flux toward higher susceptible density, yielding a spatially aware reproduction number $R_0$ that captures population-density effects on transmission. It develops a rigorous variational framework with existence, uniqueness, and conservation results, and analyzes stability and local growth through a Peclet-type indicator, along with a spatially dependent $R_0$ bound. Numerically, the authors implement a fully implicit, stabilized finite element scheme with streamline-diffusion to handle convection-dominated transport, and demonstrate two regional case studies (Lombardy and Georgia) where the chemotaxis model qualitatively better matches observed spatiotemporal patterns than pure diffusion. The findings suggest spatial heterogeneity and directed transport toward high-susceptible regions can drive hotspot formation and nonlocal-like spread, with practical implications for surveillance and intervention planning, and point to avenues for quantitative calibration and method development.

Abstract

Partial differential equation (PDE) models for infectious diseases, while less common than their ordinary differential equation (ODE) counterparts, have found successful applications for many years. Such models are typically of reaction-diffusion type, and model spatial propagation as a diffusive process. However, given the complex nature of human mobility, such models are limited in their ability to describe airborne infectious diseases in human populations. Recent work has advocated for the inclusion of an additional chemotaxis-type term as an alternative; spatial propagation of infection fronts is assumed additionally to flow from low-to-high concentrations of susceptible populations. The present work extends the study of such models by providing an epidemiologically interpretable analysis, directly connecting model behavior to information readily available to policymakers. In particular, we derive a spatially-aware basic reproduction number, which accounts for spatial heterogeneity in population density. Furthermore, we discuss several important aspects concerning the numerical solution of the model, including the introduction of a stabilization scheme. Finally, we perform a series of simulation studies in the Italian region of Lombardy (severely affected by the COVID-19 outbreak in 2020) and in the US state of Georgia, in which we demonstrate the model's potential to better capture important spatiotemporal dynamics observed in real-world data compared to pure reaction-diffusion models.

Figures (28)

• Figure 1: Initial mesh configuration for Lombardy simulation.
• Figure 2: Initial conditions for susceptible (left) and infected (center) compartments, Lombardy simulation. Location of important provinces for the simulation study (right).
• Figure 3: Left: observed cumulative infections across five relevant provinces in Lombardy (data from Lab24). Right: geographic locations of each province.
• Figure 4: Days 5 (top-left), 10 (top-right), 15 (bottom-left) and 25 (bottom-right) of the Lombardy simulation for $\mu_i=0.0$ and $\nu_i=1.0$ km$^2 \cdot$ Days $^{-1}$.
• Figure 5: Days 5 (top-left), 10 (top-right), 15 (bottom-left) and 25 (bottom-right) of the Lombardy simulation for $\mu_i=0.01$ and $\nu_i=1.0$ km$^2 \cdot$ Days $^{-1}$.