具有非局部扩散和非局部反应的传染病模型自由边界问题：传播-消亡二择一
The free boundary problem of an epidemic model with nonlocal diffusions and nonlocal reactions: spreading-vanishing dichotomy
AI总结：
本文研究具有非局部扩散和非局部反应的传染病模型自由边界问题，通过初等分析得到主特征值性质，证明传播-消亡二择一并给出判别准则，发现非局部反应项越多传播越难发生。
摘要：
本文研究一个传染病模型的自由边界问题。传染源和感染人群的空间移动由非局部扩散算子近似描述。特别地，传染源的增长率和人群的感染率均由非局部反应项表示。因此，该模型包含四个积分项，这给相应主特征值问题的研究带来了一些困难。首先，利用一些初等分析代替Krein-Rutman定理和变分特征，我们得到了主特征值的存在性和渐近行为。然后证明了传播-消亡二择一成立，并导出了传播和消亡的判别准则。最后，将我们的结果与已有文献中的结果进行比较，讨论了非局部反应项对传播和消亡的影响，发现模型中的非局部反应项越多，传播越难发生。
英文摘要：
This paper concerns the free boundary problem of an epidemic model. The spatial movements of the infectious agents and the infective humans are approximated by nonlocal diffusion operators. Especially, both the growth rate of the agents and the infective rate of humans are represented by nonlocal reaction terms. Thus our model has four integral terms which bring some diffculties for the study of the corresponding principal eigenvalue problem. Firstly, using some elementray analysis instead of Krein-Rutman theorem and the variational characteristic, we obtain the existence and asymptotic behaviors of principal eigenvalue. Then a spreading-vanishing dichotomy is proved to hold, and the criteria for spreading and vanishing are derived. Lastly, comparing our results with those in the existing works, we discuss the effect of nonlocal reaction term on spreading and vanishing, finding that the more nonlocal reaction terms a model has, the harder spreading happens.
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