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On a reaction-diffusion virus model with general boundary conditions in heterogeneous environments

Mingxin Wang, Lei Zhang

TL;DR

The paper analyzes a time-periodic reaction-diffusion host–vector model in heterogeneous environments with four populations $(H_u,H_i,V_u,V_i)$. It establishes a threshold framework via reproduction numbers $\mathcal{R}_{0,1}$, $\mathcal{R}_{0,2}$, and the coupled $\mathcal{R}_0$, proving that a positive $T$-periodic solution exists and is unique and globally asymptotically stable if $\mathcal{R}_0>1$, while disease extinction occurs when $\mathcal{R}_0\le1$. Section 3 provides the global well-posedness and characterizes long-time behavior across all threshold regimes, showing convergence to positive periodic states when the threshold is exceeded and to disease-free states otherwise. Section 4 analyzes autonomous and constant-coefficient Neumann cases, offering explicit equilibria and clear threshold criteria. Section 5 uses numerical simulations to illustrate how spatiotemporal heterogeneity shapes $\mathcal{R}_0$ and disease persistence, highlighting the substantial role of vector control. Overall, the work delivers a rigorous, threshold-driven understanding of periodic virus spread in heterogeneous environments, with practical implications for WNV/Zika-like dynamics.

Abstract

To describe the propagation of West Nile virus and/or Zika virus, in this paper, we propose and study a time-periodic reaction-diffusion model with general boundary conditions in heterogeneous environments and with four unknowns: susceptible host, infectious host, susceptible vector and infectious vector. We can prove that such problem has a positive time periodic solution if and only if host and vector persist and the basic reproduction ratio is greater than one, and moreover the positive time periodic solution is unique and globally asymptotically stable when it exists.

On a reaction-diffusion virus model with general boundary conditions in heterogeneous environments

TL;DR

The paper analyzes a time-periodic reaction-diffusion host–vector model in heterogeneous environments with four populations . It establishes a threshold framework via reproduction numbers , , and the coupled , proving that a positive -periodic solution exists and is unique and globally asymptotically stable if , while disease extinction occurs when . Section 3 provides the global well-posedness and characterizes long-time behavior across all threshold regimes, showing convergence to positive periodic states when the threshold is exceeded and to disease-free states otherwise. Section 4 analyzes autonomous and constant-coefficient Neumann cases, offering explicit equilibria and clear threshold criteria. Section 5 uses numerical simulations to illustrate how spatiotemporal heterogeneity shapes and disease persistence, highlighting the substantial role of vector control. Overall, the work delivers a rigorous, threshold-driven understanding of periodic virus spread in heterogeneous environments, with practical implications for WNV/Zika-like dynamics.

Abstract

To describe the propagation of West Nile virus and/or Zika virus, in this paper, we propose and study a time-periodic reaction-diffusion model with general boundary conditions in heterogeneous environments and with four unknowns: susceptible host, infectious host, susceptible vector and infectious vector. We can prove that such problem has a positive time periodic solution if and only if host and vector persist and the basic reproduction ratio is greater than one, and moreover the positive time periodic solution is unique and globally asymptotically stable when it exists.
Paper Structure (6 sections, 7 theorems, 104 equations, 7 figures)

This paper contains 6 sections, 7 theorems, 104 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Positive time periodic solutions
  3. Dynamical properties of \ref{['1.1']}
  4. Special cases: autonomous case and constant case
  5. Numerical simulation
  6. Discussion

Key Result

Theorem 2.1

Assume that $\mathcal{R}_{0,j}>1$, $j=1,2$. Then problem 2.4 has a positive solution $({\mathsf H}_i, {\mathsf V}_i)$ if and only if $\mathcal{R}_0>1$. Moreover, the positive solution $({\mathsf H}_i, {\mathsf V}_i)$ of 2.4 is unique and satisfies when it exists. Therefore, if $\mathcal{R}_0>1$, then 2.1 has a unique positive solution

Figures (7)

  • Figure 2: Positive periodic solution when $p_1=p_2=p_3=p_4=0.5$ and $q_1=q_2=q_3=q_4=0$.
  • Figure 3: Level sets of $\mathcal{R}_0$ with $p_1=p_2=q_1=q_2=0$. The two figures in the first row show that the effect of $p_3$ and $p_4$, as well as, $q_3$ and $q_4$ on $\mathcal{R}_0$ are very weak. However, the superposition of weak effects may lead to significant influences, as shown in the two figures in the second row.
  • Figure 4: Level sets of $\mathcal{R}_0$ with $p_3=p_4=q_3=q_4=0$. Simultaneous changes of multiple parameters, compared to changes of individual parameters, can either increase or decrease the impact on $\mathcal{R}_0$.
  • Figure 5: Level sets of $\mathcal{R}_0$ with $q_1=q_2=q_3=q_4=0$. The spatial heterogeneity may have greater influence on $\mathcal{R}_0$ than the temporal one in general.
  • Figure : (a) $p_1=p_2=p_3=p_4=q_1=q_2=q_3=q_4=0$
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4