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Chemotaxis-inspired PDE model for airborne infectious disease transmission: analysis and simulations

Pierluigi Colli, Gabriela Marinoschi, Elisabetta Rocca, Alex Viguerie

TL;DR

This work introduces a nonlinear, chemotaxis-inspired flux into a reaction–diffusion PDE framework for airborne infectious disease spread, coupling susceptible, infected, and removed densities $s,i,r$ on a bounded domain. The key novelty is a flux $\mathbf{j}=-D_i\nabla i + \mathcal{F}(i)s$ with $\mathcal{F}(i)=\mathcal{F}_i\frac{i}{1+i/C_0}$ that drives transmission along regions of higher susceptibility, yielding rich spatial dynamics including long-range propagation and localized hotspots. The authors prove well-posedness, positivity, and asymptotic convergence results, analyze a reduced diffusion–vanishing limit, and demonstrate through Lombardy simulations that chemotaxis improves qualitative agreement with surveillance data compared to purely diffusive models, without requiring time- or space-varying parameters. They also reveal nonmonotone effects of chemotaxis on transmission and discuss implications for intervention planning and future model extensions, such as additional compartments and control strategies.

Abstract

Partial differential equation (PDE) models for infectious disease have received renewed interest in recent years. Most models of this type extend classical compartmental formulations with additional terms accounting for spatial dynamics, with Fickian diffusion being the most common such term. However, while diffusion may be appropriate for modeling vector-borne diseases, or diseases among plants or wildlife, the spatial propagation of airborne diseases in human populations is heavily dependent on human contact and mobility patterns, which are not necessarily well-described by diffusion. By including an additional chemotaxis-inspired term, in which the infection is propagated along the positive gradient of the susceptible population (from regions of low- to high-density of susceptibles), one may provide a more suitable description of these dynamics. This article introduces and analyzes a mathematical model of infectious disease incorporating a modified chemotaxis-type term. The model is analyzed mathematically and the well-posedness of the resulting PDE system is demonstrated. A series of numerical simulations are provided, demonstrating the ability of the model to naturally capture important phenomena not easily observed in standard diffusion models, including propagation over long spatial distances over short time scales and the emergence of localized infection hotspots

Chemotaxis-inspired PDE model for airborne infectious disease transmission: analysis and simulations

TL;DR

This work introduces a nonlinear, chemotaxis-inspired flux into a reaction–diffusion PDE framework for airborne infectious disease spread, coupling susceptible, infected, and removed densities on a bounded domain. The key novelty is a flux with that drives transmission along regions of higher susceptibility, yielding rich spatial dynamics including long-range propagation and localized hotspots. The authors prove well-posedness, positivity, and asymptotic convergence results, analyze a reduced diffusion–vanishing limit, and demonstrate through Lombardy simulations that chemotaxis improves qualitative agreement with surveillance data compared to purely diffusive models, without requiring time- or space-varying parameters. They also reveal nonmonotone effects of chemotaxis on transmission and discuss implications for intervention planning and future model extensions, such as additional compartments and control strategies.

Abstract

Partial differential equation (PDE) models for infectious disease have received renewed interest in recent years. Most models of this type extend classical compartmental formulations with additional terms accounting for spatial dynamics, with Fickian diffusion being the most common such term. However, while diffusion may be appropriate for modeling vector-borne diseases, or diseases among plants or wildlife, the spatial propagation of airborne diseases in human populations is heavily dependent on human contact and mobility patterns, which are not necessarily well-described by diffusion. By including an additional chemotaxis-inspired term, in which the infection is propagated along the positive gradient of the susceptible population (from regions of low- to high-density of susceptibles), one may provide a more suitable description of these dynamics. This article introduces and analyzes a mathematical model of infectious disease incorporating a modified chemotaxis-type term. The model is analyzed mathematically and the well-posedness of the resulting PDE system is demonstrated. A series of numerical simulations are provided, demonstrating the ability of the model to naturally capture important phenomena not easily observed in standard diffusion models, including propagation over long spatial distances over short time scales and the emergence of localized infection hotspots
Paper Structure (8 sections, 6 theorems, 167 equations, 6 figures, 2 tables)

This paper contains 8 sections, 6 theorems, 167 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Standard flux choice: Fickian diffusion
  4. Flux definition 2: Chemotaxis-inspired
  5. Mathematical analysis of the chemotaxis-inspired model
  6. Analysis of a reduced system
  7. Numerical simulations
  8. Conclusions

Key Result

Theorem 3.1

Under the assumptions (niu)-(ci-cond) there exists a triplet satisfying the following notion of weak solution to system (eq-s1)-(ci1): for all $v\in L^2(0,T;V)$. Moreover, it has the properties whence it turns out that the solution $(s_{ \if@compatibility \mathchar"0122 {} \mathchar"0122 },i_{ \if@compatibility \mathchar"0122 {} \mathchar"0122 },r_{ \if@compatibility \mathchar"0122

Figures (6)

  • Figure 1: Left: observed cumulative infections across five relevant provinces in Lombardy (data and figure retrived from Lab24). Right: geographic locations of each province.
  • Figure 2: Cumulative simulated incidence in the provinces of Bergamo, Brescia, Lodi, Milano, and Cremona over 30 days, for $\if@compatibility \mathchar"0117 {} \mathchar"0117 _i$=1.0 km$^{2} \cdot$ Days$^{-1}$ and varying values of $\if@compatibility \mathchar"0116 {} \mathchar"0116 _i$ and $C_0$.
  • Figure 3: Cumulative simulated incidence in the provinces of Bergamo, Brescia, Lodi, Milano, and Cremona over 30 days, for $\if@compatibility \mathchar"0117 {} \mathchar"0117 _i$=2.5 km$^{2} \cdot$ Days$^{-1}$ and varying values of $\if@compatibility \mathchar"0116 {} \mathchar"0116 _i$ and $C_0$.
  • Figure 4: Left-to-right, top-to-bottom: The infected compartment on days 1, 5, 10, and 15 for $\if@compatibility \mathchar"0117 {} \mathchar"0117 _i=1.0$, $\if@compatibility \mathchar"0116 {} \mathchar"0116 _i=0$
  • Figure 5: Left-to-right, top-to-bottom: The infected compartment on days 1, 5, 10, and 15 for $\if@compatibility \mathchar"0117 {} \mathchar"0117 _i=1.0$, $\if@compatibility \mathchar"0116 {} \mathchar"0116 _i=0.01$
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Theorem 4.1