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

Technical Summary: Chemotaxis-Inspired PDE Models of Airborne Infectious Disease Transmission

Problem Statement
Mathematical modeling of infectious diseases has traditionally relied on Ordinary Differential Equation (ODE) compartmental models (e.g., SIR), which lack explicit spatial dynamics. While Partial Differential Equation (PDE) models have been proposed to address spatial propagation, they typically treat disease spread as a Fickian diffusive process. In this standard framework, infection propagates from regions of high infection density to regions of low infection density. However, the authors argue that this assumption is insufficient for airborne diseases in human populations. Empirical data, particularly from the COVID-19 pandemic, suggests that transmission correlates strongly with population density, often moving from areas of lower susceptible density to areas of higher susceptible density (e.g., toward urban centers). Pure diffusion models fail to capture this "up-gradient" movement and the resulting formation of infection hotspots in dense population centers. Furthermore, existing PDE approaches often struggle to provide epidemiologically interpretable quantities, such as a spatially-aware basic reproduction number ($R_0$), which are crucial for public health decision-making.

Methodology
The paper extends a previously introduced chemotaxis-inspired PDE model [42] for airborne infectious diseases. The model utilizes a system of reaction-diffusion equations with an added chemotaxis term to describe the dynamics of susceptible ($s$), infected ($i$), and recovered ($r$) populations.

• Mathematical Formulation: The core innovation lies in the flux term $j$ for the infected population. While a standard diffusion term ($-\nu_i \nabla i$) accounts for random movement, the model introduces a chemotaxis term ($\mu_i \chi(i) \nabla s$). This term posits that the disease propagates along the positive gradient of the susceptible population density, effectively moving toward areas with higher concentrations of susceptible individuals. The function $\chi(i)$ acts as a saturation capacity term to prevent unbounded growth.
• Mathematical Analysis: The authors perform a rigorous mathematical analysis of the system, establishing the existence and uniqueness of weak solutions. A primary focus is the derivation of a spatially-dependent basic reproduction number. By analyzing the stability of the infection population in the $L^2$ norm, the authors derive a modified $R_0$ that incorporates the Laplacian of the susceptible population ($\Delta s$). This formulation suggests that local population density concavity (specifically, local maxima where $\Delta s < 0$) significantly influences transmission rates.
• Numerical Solution: The paper addresses the computational challenges of solving this system, particularly the potential for numerical instability due to convection-like dynamics introduced by the chemotaxis term. The authors employ a Finite Element Method (FEM) for spatial discretization and a Backward Differentiation Formula (BDF2) for temporal discretization. To ensure stability and mass conservation, they utilize a fully implicit time-stepping scheme and introduce a streamline-diffusion stabilization technique to handle sharp gradients.

Key Contributions

1. Epidemiologically Interpretable Analysis: The paper derives a spatially-aware basic reproduction number ($R_0$) that accounts for spatial heterogeneity in population density. Unlike standard ODE $R_0$, this metric depends on the local Laplacian of the susceptible population, providing a theoretical basis for why infection hotspots form in dense urban areas.
2. Numerical Stabilization: The authors introduce and discuss a stabilization scheme (streamline-diffusion) necessary for solving the chemotaxis-driven PDE system, addressing the increased computational complexity compared to pure diffusion models.
3. Comparative Simulation Studies: The model is tested in two distinct geographic regions: Lombardy, Italy, and the state of Georgia, USA. The simulations compare the chemotaxis model against pure reaction-diffusion models using time-independent parameters.

Results
The simulation studies demonstrate that the chemotaxis model captures spatiotemporal dynamics observed in real-world COVID-19 surveillance data more accurately than pure diffusion models, even without time-varying parameters.

• Lombardy, Italy: Pure diffusion models failed to replicate the rapid spread of the epidemic into major urban centers like Milan, Bergamo, and Brescia, keeping the infection artificially concentrated in the initial outbreak zones (Lodi and Cremona). In contrast, the chemotaxis model successfully simulated the migration of the infection front toward high-density population centers, reproducing the observed "hotspots" in Bergamo and Milan. The model correctly identified that areas with high negative Laplacian values in the susceptible population (local density maxima) were particularly vulnerable.
• Georgia, USA: In a state-level simulation, pure diffusion models struggled to generate transmission in major population centers outside the Atlanta metro area (e.g., Chatham, Richmond, Bibb counties) while overestimating transmission in lower-density suburban areas (e.g., Forsyth, Cherokee). The chemotaxis model corrected these discrepancies, significantly increasing transmission in the under-simulated population centers and reducing it in the over-simulated low-density areas, aligning more closely with observed mortality trends.

Significance and Claims
The paper claims to provide a "proof-of-concept" that chemotaxis-inspired PDE models offer a superior framework for describing airborne infectious disease transmission in human populations compared to standard diffusion models. The authors emphasize that their work bridges the gap between complex mathematical modeling and practical epidemiological application by deriving interpretable quantities (the spatial $R_0$) that link model behavior to population density patterns.

The significance of the work lies in its ability to explain and predict the formation of infection hotspots driven by spatial population gradients without requiring complex, time-dependent parameter tuning. The authors modestly note that the current study is qualitative; the model is not intended as a definitive predictive tool for COVID-19 without incorporating additional complexities such as age structure, vaccination, and mobility policies. However, the results suggest that incorporating chemotactic drift is a necessary step for accurately modeling the spatial propagation of airborne diseases in heterogeneous human populations. Future work is proposed to extend these models to more complex compartmental structures and to develop higher-order stabilization schemes.