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Long-time dynamics of a parabolic-ODE SIS epidemic model with saturated incidence mechanism

Rui Peng, Rachidi Salako, Yixiang Wu

TL;DR

This paper analyzes a parabolic-ODE SIS epidemic model with saturated incidence $\frac{SI}{m(x)+S+I}$ in a bounded, heterogenous environment under two movement-control regimes: $d_S=0$ (susceptible-imposed mobility restriction) and $d_I=0$ (infected-imposed mobility restriction). By partitioning the domain into high-, moderate-, and low-risk regions and introducing incidence-support sets, the authors derive threshold-type global dynamics, establishing when disease extinction or persistence occurs and how spatial patterns depend on the total population $N$ and the saturation function $m(x)$. They prove existence, positivity, and convergence results using semigroup theory, Lyapunov functionals, and eigenvalue arguments, and illustrate the findings with numerical simulations that corroborate the theoretical dichotomies and reveal spatial occupancy patterns. The study reveals a novel interplay between $N$, the transmission-recovery balance $\beta-\gamma$, and the saturated incidence, offering insights into targeted movement-control strategies for disease eradication in spatially heterogeneous settings.

Abstract

In this paper, we investigate a parabolic-ODE SIS epidemic model with no-flux boundary conditions in a heterogeneous environment. The model incorporates a saturated infection mechanism \({SI}/(m(x) + S + I)\) with \(m \geq,\,\not\equiv 0\). This study is motivated by disease control strategies, such as quarantine and lockdown, that limit population movement. We examine two scenarios: one where the movement of the susceptible population is restricted, and another where the movement of the infected population is neglected. We establish the long-term dynamics of the solutions in each scenario. Compared to previous studies that assume the absence of a saturated incidence function (i.e., $m\equiv 0$), our findings highlight the novel and significant interplay between total population size, transmission risk level, and the saturated incidence function in influencing disease persistence, extinction, and spatial distribution. Numerical simulations are performed to validate the theoretical results, and the implications of the results are discussed in the context of disease control and eradication strategies.

Long-time dynamics of a parabolic-ODE SIS epidemic model with saturated incidence mechanism

TL;DR

This paper analyzes a parabolic-ODE SIS epidemic model with saturated incidence in a bounded, heterogenous environment under two movement-control regimes: (susceptible-imposed mobility restriction) and (infected-imposed mobility restriction). By partitioning the domain into high-, moderate-, and low-risk regions and introducing incidence-support sets, the authors derive threshold-type global dynamics, establishing when disease extinction or persistence occurs and how spatial patterns depend on the total population and the saturation function . They prove existence, positivity, and convergence results using semigroup theory, Lyapunov functionals, and eigenvalue arguments, and illustrate the findings with numerical simulations that corroborate the theoretical dichotomies and reveal spatial occupancy patterns. The study reveals a novel interplay between , the transmission-recovery balance , and the saturated incidence, offering insights into targeted movement-control strategies for disease eradication in spatially heterogeneous settings.

Abstract

In this paper, we investigate a parabolic-ODE SIS epidemic model with no-flux boundary conditions in a heterogeneous environment. The model incorporates a saturated infection mechanism \({SI}/(m(x) + S + I)\) with . This study is motivated by disease control strategies, such as quarantine and lockdown, that limit population movement. We examine two scenarios: one where the movement of the susceptible population is restricted, and another where the movement of the infected population is neglected. We establish the long-term dynamics of the solutions in each scenario. Compared to previous studies that assume the absence of a saturated incidence function (i.e., ), our findings highlight the novel and significant interplay between total population size, transmission risk level, and the saturated incidence function in influencing disease persistence, extinction, and spatial distribution. Numerical simulations are performed to validate the theoretical results, and the implications of the results are discussed in the context of disease control and eradication strategies.

Paper Structure

This paper contains 14 sections, 22 theorems, 237 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Results for system \ref{['ds=0-model']}
  4. Results for system \ref{['dI=0-model']}
  5. Proof of main results for system \ref{['ds=0-model']}
  6. Proof of main results for system \ref{['dI=0-model']}
  7. Numerical simulations
  8. Simulations for system \ref{['ds=0-model']}
  9. Simulations for system \ref{['dI=0-model']}
  10. Discussion and Conclusion
  11. For system \ref{['ds=0-model']}: control on the susceptible's movement
  12. For system \ref{['dI=0-model']}: control on the infected's movement
  13. Control on both susceptible and infected's movements
  14. Summary

Key Result

Proposition 2.1

For any initial data $(S_0,I_0)\in [C^+(\bar{\Omega})]^2$, system ds=0-model admits a unique solution $(S(t,x),I(t,x))$ defined for all $t\ge 0$ and $x\in\bar{\Omega}$ with If $I_0\not\equiv 0$, then $S(t,x)>0$ and $I(t,x)>0$ for all $x\in\bar{\Omega}$ for all $t>0$.

Figures (5)

  • Figure 1: Simulation of model \ref{['ds=0-model']} with $\beta>\gamma$.
  • Figure 2: Simulation of model \ref{['ds=0-model']} with $\beta\ge\gamma$ and $H^0\neq\emptyset$.
  • Figure 3: Simulation of model \ref{['ds=0-model']} with $H^-\neq \emptyset$.
  • Figure 4: Simulation of model \ref{['dI=0-model']}. Left two figures: $\beta=1+x$, $\gamma=0.8$, $m=1$. Right figure: $\beta=1+x$, $\gamma=2.5$, $m=1$.
  • Figure 5: Simulation of model \ref{['dI=0-model']} with $\gamma=1$, $m=m_0$, and initial data $(aS_0, aI_0)$. Fig.(A)-(B) show $I(x, 600)$; Fig.(C) shows $\int_0^1 I(x, 600)dx$ as a function of $a$.

Theorems & Definitions (44)

  • Proposition 2.1
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.3
  • Corollary 2.1
  • Remark 2.2
  • Proposition 2.4
  • ...and 34 more