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Waterborne epidemics via a new coupled SIR--Pathogen--Navier-Stokes system: Mathematical modeling, nonlinear analysis and numerical simulation

Mohamed Mehdaoui, Yassine Ouzrour

TL;DR

The paper tackles spatially distributed waterborne epidemics by coupling epidemiological dynamics with environmental transport and fluid flow in a novel SIRPNS framework. It analyzes the well-posedness of the system by proving the existence of global biologically feasible weak solutions via a Faedo–Galerkin approach and establishes uniqueness in 2D under Lipschitz conditions, complemented by a semi-implicit finite element numerical scheme. Key contributions include a rigorous mathematical foundation for the coupled system and qualitative numerical demonstrations showing how environmental reservoirs and hydrodynamic feedback shape infection spread, persistence, and decline. The findings have practical implications for understanding and predicting waterborne outbreak patterns and for developing intervention strategies that account for fluid-mediated transport and viscosity feedback.

Abstract

Water-borne diseases are still a major public health concern, as there are circumstances under which water could act as a carrier of the pathogen, extending their modeling beyond direct contact between hosts. In the present work, we introduce a new mathematical framework, coupling epidemiological dynamics with fluid motion, in order to understand the spatial spread of such an infection. Our model couples the classical Susceptible-Infected-Recovered (SIR) model with the Navier-Stokes equations describing the motion of fluids, which enhances the existing literature by simultaneously taking into account two aspects: the pathogen being transported by the water currents and the dependence of the effective viscosity of the fluid on the pathogen concentration. We apply the Faedo-Galerkin method and compactness arguments to prove the existence of a global, biologically feasible solution to the coupled SIR--Pathogen--Navier-Stokes (SIRPNS) system. Additionally, we investigate the uniqueness of such solutions in the two-dimensional case. Finally, by constructing a numerical scheme based on the semi-implicit scheme in time and the finite element method in space, we run several numerical simulations to show how infection dispersal, environmental contamination, and hydrodynamic feedback together govern the spatial dynamics, persistence, and eventual decline of waterborne epidemics.

Waterborne epidemics via a new coupled SIR--Pathogen--Navier-Stokes system: Mathematical modeling, nonlinear analysis and numerical simulation

TL;DR

The paper tackles spatially distributed waterborne epidemics by coupling epidemiological dynamics with environmental transport and fluid flow in a novel SIRPNS framework. It analyzes the well-posedness of the system by proving the existence of global biologically feasible weak solutions via a Faedo–Galerkin approach and establishes uniqueness in 2D under Lipschitz conditions, complemented by a semi-implicit finite element numerical scheme. Key contributions include a rigorous mathematical foundation for the coupled system and qualitative numerical demonstrations showing how environmental reservoirs and hydrodynamic feedback shape infection spread, persistence, and decline. The findings have practical implications for understanding and predicting waterborne outbreak patterns and for developing intervention strategies that account for fluid-mediated transport and viscosity feedback.

Abstract

Water-borne diseases are still a major public health concern, as there are circumstances under which water could act as a carrier of the pathogen, extending their modeling beyond direct contact between hosts. In the present work, we introduce a new mathematical framework, coupling epidemiological dynamics with fluid motion, in order to understand the spatial spread of such an infection. Our model couples the classical Susceptible-Infected-Recovered (SIR) model with the Navier-Stokes equations describing the motion of fluids, which enhances the existing literature by simultaneously taking into account two aspects: the pathogen being transported by the water currents and the dependence of the effective viscosity of the fluid on the pathogen concentration. We apply the Faedo-Galerkin method and compactness arguments to prove the existence of a global, biologically feasible solution to the coupled SIR--Pathogen--Navier-Stokes (SIRPNS) system. Additionally, we investigate the uniqueness of such solutions in the two-dimensional case. Finally, by constructing a numerical scheme based on the semi-implicit scheme in time and the finite element method in space, we run several numerical simulations to show how infection dispersal, environmental contamination, and hydrodynamic feedback together govern the spatial dynamics, persistence, and eventual decline of waterborne epidemics.

Paper Structure

This paper contains 21 sections, 5 theorems, 75 equations, 24 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and motivation
  2. Model formulation
  3. Fluid dynamics
  4. Pathogen concentration dynamics
  5. Host dynamics
  6. The resulting coupled system
  7. Mathematical analysis of the SIRPNS model
  8. Functional framework and assumptions
  9. Existence of weak solutions
  10. Construction of a sequence of Faedo-Galerkin solutions
  11. A priori estimates of the sequence of Faedo-Galerkin solutions
  12. Passage to the limit in Faedo-Galerkin solutions
  13. Uniqueness of weak solutions in the two-dimensional case $(d=2)$
  14. Numerical simulations of the SIRPNS model
  15. Numerical scheme
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $X_0, X$, and $X_1$ be three Banach spaces with $X_0 \subset X \subset X_1$. Suppose that $X_0$ is compactly embedded in $X$ and that $X$ is continuously embedded in $X_1$. Then:

Figures (24)

  • Figure 1: The cycle of waterborne epidemics transmission.
  • Figure 2: Schematic diagram of the SIR–Pathogen–Navier–Stokes (SIRPNS) model couplings. The diagram illustrates the two-way coupling between epidemiological dynamics (blue), pathogen transport (red), and fluid mechanics (green).
  • Figure 4: Evolution of the pathogen concentration field $C$ in the absence of epidemic feedback. The contaminant spreads outward and gradually decays in amplitude, producing smooth, symmetric profiles. This preliminary behaviour represents possible contamination zones that may act as sources for subsequent infection.
  • Figure : (a) Spatial distribution of $S$ at $t=0$, $t=20$, and $t=40$.
  • Figure : (a) $S$ at $t=0$, $t=20$, and $t=40$.
  • ...and 19 more figures

Theorems & Definitions (10)

  • Lemma 3.1
  • Definition 3.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1