On a two-season faecal-oral model with impulsive intervention

TL;DR

This paper develops a two-season faecal-oral transmission model with impulsive intervention and free boundaries to capture rainfall-driven seasonality in disease spread. It combines a fixed-dry-season boundary with a moving-wet-season boundary and impulsive environmental control applied at the end of each wet season, establishing a rigorous spectral framework via a principal eigenvalue $oldsymbol{\lambda_1}$ to characterize long-term dynamics. The authors prove a spreading-vanishing dichotomy and derive sharp criteria linking outcomes to domain length, impulse strength, and dry-season duration, complemented by numerical simulations that illustrate how stronger impulses and longer dry seasons improve control. The work provides theoretical and computational insights into intervention strategies under seasonal switching, with implications for environmental disinfection timing and public health planning.

Abstract

Rainfall is associated with the outbreak of certain waterborne faecal-oral diseases, driving the implementation of various human interventions for their control and prevention. Taking into account human intervention and temporal variation in rainfall, this paper develops a two-season switching faecal-oral model with impulsive intervention and free boundaries. In this model, the infection fronts are represented by fixed boundaries during the dry season and by moving boundaries during the wet season, with impulsive intervention occurring at the end of each wet season. The simultaneous introduction of impulsive intervention and seasonal switching creates new difficulties for mathematical analysis. We overcome these challenges through novel analytical techniques, resulting in a spreading-vanishing dichotomy and a sharp criteria governing this dichotomy. Finally, numerical simulations are presented to validate the theoretical results and to visually illustrate the influence of seasonal switching and impulsive intervention. Our results mathematically explain that two factors, the duration of the dry season and the intensity of impulsive intervention are both positively correlated with effective disease control.

Figures (4)

• Figure 1: Distribution graph of the unique solution of \ref{['3-20']}
• Figure 2: Monotonicity of the principal eigenvalue $\lambda_{1}$ with respect to $l_{2}-l_{1}$, $H'(0)$, and $\tau$, respectively. (a) decreasing of $\lambda_{1}$ with respect to $l_{2}-l_{1}$ under $H'(0)=0.9$ and $\tau=5$. (b) decreasing of $\lambda_{1}$ with respect to $H'(0)$ under $\tau=5$ and $l_{2}-l_{1}=20$. (c) increasing of $\lambda_{1}$ with respect to $\tau$ under $H'(0)=0.9$ and $l_{2}-l_{1}=20$.
• Figure 3: The left four images (a-d), in the absence of pulse intervention, show that the infectious agents $u$ is spreading. In contrast, the right four images (e-h), with the application of the impulsive function $H(u)=\frac{4u}{10+u}$, demonstrate that $u$ is vanishing.
• Figure 4: The left four images (a-d), with short dry season duration ($\tau=3$), show that the infectious agents $u$ is spreading. In contrast, the right four images (e-h), with long dry season duration ($\tau=4.7$), demonstrate that $u$ is vanishing.

Theorems & Definitions (47)

• Remark 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.2
• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• Remark 3.1