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Nonlocal diffusion and pulse intervention in a faecal-oral model with moving infected fronts

Qi Zhou, Michael Pedersen, Zhigui Lin

TL;DR

This work develops a pulsed, nonlocal faecal-oral epidemic model with moving infected fronts and free boundaries, capturing periodic disinfection and long-range dispersal. It establishes global well-posedness and analyzes a time-periodic principal eigenvalue via resolvent-positive operators to derive a spreading-vanishing dichotomy with explicit criteria. The results show that impulsive interventions can effectively curb disease spread, with kernel choice and impulse strength playing crucial roles. Numerical simulations corroborate the theory and illuminate how intervention timing and diffusion structure influence outcomes, offering guidance for policy design in faecal-oral disease control.

Abstract

How individual dispersal patterns and human intervention behaviours affect the spread of infectious diseases constitutes a central problem in epidemiological research. This paper develops an impulsive nonlocal faecal-oral model with free boundaries, where pulses are introduced to capture a periodic spraying of disinfectant, and nonlocal diffusion describes the long-range dispersal of individuals, and free boundaries represent moving infected fronts. We first check that the model has a unique nonnegative global classical solution. Then, the principal eigenvalue, which depends on the infected region, the impulse intensity, and the kernel functions for nonlocal diffusion, is examined by using the theory of resolvent positive operators and their perturbations. Based on this value, this paper obtains that the diseases are either vanishing or spreading, and provides criteria for determining when vanishing and spreading occur. At the end, a numerical example is presented in order to corroborate the theoretical findings and to gain further understanding of the effect of the pulse intervention. This work shows that the pulsed intervention is beneficial in combating the diseases, but the effect of the nonlocal diffusion depends on the choice of the kernel functions.

Nonlocal diffusion and pulse intervention in a faecal-oral model with moving infected fronts

TL;DR

This work develops a pulsed, nonlocal faecal-oral epidemic model with moving infected fronts and free boundaries, capturing periodic disinfection and long-range dispersal. It establishes global well-posedness and analyzes a time-periodic principal eigenvalue via resolvent-positive operators to derive a spreading-vanishing dichotomy with explicit criteria. The results show that impulsive interventions can effectively curb disease spread, with kernel choice and impulse strength playing crucial roles. Numerical simulations corroborate the theory and illuminate how intervention timing and diffusion structure influence outcomes, offering guidance for policy design in faecal-oral disease control.

Abstract

How individual dispersal patterns and human intervention behaviours affect the spread of infectious diseases constitutes a central problem in epidemiological research. This paper develops an impulsive nonlocal faecal-oral model with free boundaries, where pulses are introduced to capture a periodic spraying of disinfectant, and nonlocal diffusion describes the long-range dispersal of individuals, and free boundaries represent moving infected fronts. We first check that the model has a unique nonnegative global classical solution. Then, the principal eigenvalue, which depends on the infected region, the impulse intensity, and the kernel functions for nonlocal diffusion, is examined by using the theory of resolvent positive operators and their perturbations. Based on this value, this paper obtains that the diseases are either vanishing or spreading, and provides criteria for determining when vanishing and spreading occur. At the end, a numerical example is presented in order to corroborate the theoretical findings and to gain further understanding of the effect of the pulse intervention. This work shows that the pulsed intervention is beneficial in combating the diseases, but the effect of the nonlocal diffusion depends on the choice of the kernel functions.

Paper Structure

This paper contains 13 sections, 27 theorems, 210 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model formulation and preliminaries
  3. Model formulation
  4. Preliminaries
  5. The associated problem with fixed boundaries
  6. The corresponding eigenvalue problem
  7. The asymptotic profile in a fixed domain
  8. The asymptotic profile in a moving environment
  9. Some introductory results
  10. Spreading-vanishing dichotomy
  11. Spreading-vanishing criteria
  12. Numerical simulation
  13. Conclusion and future work

Key Result

Lemma 2.1

Assume that $h_{0}$, $\tau>0$, and that $(u_{0}, v_{0})$ satisfies initial value. Then, for any $T>0$ and $(g,h)\in\mathbb{G}^{h_{0}}_{T, \tau}\times\mathbb{H}^{h_{0}}_{T, \tau}$, the following problem has a unique solution $(u,v)\in \mathbb{X}^{T}=\mathbb{X}^{T}(g,h,u_{0}, v_{0})$. Moreover, $(u,v)$ satisfies

Figures (3)

  • Figure 1: Dispersal kernel function $J_{i}(x)(i=1,2)$ defined by \ref{['5-1']}.
  • Figure 2: When $H(u)=u$, which implies no pulse, $u$ converges to a steady state and $\lambda_{1}<0$.
  • Figure 3: When $H(u)=\frac{0.1u}{10+u}$, $u$ decays to $0$ and $\lambda_{1}>0$.

Theorems & Definitions (60)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 50 more