In this paper, we investigate a logistic SI epidemic model with partially degenerate diffusion and double free boundaries to describe the spatial spreading of disease. The existence, uniqueness, and estimates of the global solution are discussed firstly. Then, we prove a spreading–vanishing dichotomy. Namely the infective class either successfully spreads to infinity as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, or vanishes in a finite area. Besides, the long-time behavior of the solution and criteria for spreading and vanishing are also obtained. Especially, we find that the Basic Reproduction Number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathcal {R}}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is not the unique factor which determines whether or not an infectious disease can spread through a population: when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {R}}_0\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, vanishing always happens and the disease will die out; when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathcal {R}}_0>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, whether or not to vanish depends on the size of the initial habitat and the rate of expansion. This phenomenon reveals the role of free boundaries in the epidemic. In the end, we give some numerical results as supplements to the theoretical results.

Spatial spreading of a logistic SI epidemic model with partially degenerate diffusion and double free boundaries

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发表日期：

2026年2月4日

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Siyu Liu,
Haomin Huang

摘要

In this paper, we investigate a logistic SI epidemic model with partially degenerate diffusion and double free boundaries to describe the spatial spreading of disease. The existence, uniqueness, and estimates of the global solution are discussed firstly. Then, we prove a spreading–vanishing dichotomy. Namely the infective class either successfully spreads to infinity as $t\rightarrow \infty $ $t \to \infty$ , or vanishes in a finite area. Besides, the long-time behavior of the solution and criteria for spreading and vanishing are also obtained. Especially, we find that the Basic Reproduction Number ${\mathcal {R}}_0$ $R_{0}$ is not the unique factor which determines whether or not an infectious disease can spread through a population: when ${\mathcal {R}}_0\le 1$ $R_{0} \leq 1$ , vanishing always happens and the disease will die out; when ${\mathcal {R}}_0>1$ $R_{0} > 1$ , whether or not to vanish depends on the size of the initial habitat and the rate of expansion. This phenomenon reveals the role of free boundaries in the epidemic. In the end, we give some numerical results as supplements to the theoretical results.