Mathematical Analyses of Spatial Spread of Pine Wilt Disease

松材线虫病是由松墨天牛传播的松材线虫引起的。日本本土松树黑松和赤松对线虫感染极为敏感，尽管进行了密集的防治工作，但在过去几十年中，该疾病仍在全国范围内蔓延。金龟子的扩散通常有近距离扩散和远距离扩散两种方式。短距离扩散通常通过其自身的主动飞行进行，而远距离扩散则通过风或人工运输设施的被动运输进行。为了评估疾病的扩展速度，我们提出了一个模型，结合当地的动态松树枯萎病与甲虫的扩散。我们的模型预测了几个结果。(1)甲虫扩散距离的分布决定了病害的扩散速度。随着甲虫远距离扩散的比例从零开始增加，扩张速度急剧放大。(2)然而，如果远距离传播者变得过于频繁，则由于Allee效应，疾病停止扩展。因此，正确评估甲虫的移动性在预测疾病扩展速度方面的重要性。(3)随着根除率的增加，扩张速度先是逐渐减小，到某一根除率值时，扩张速度突然降为零。

Pine wilt disease is caused by the pinewood nematode, Bursaphelenchus xylophilus, that is vectored by the pine sawyer beetle, Monochamus alternatus. Native Japanese pines, black pine (Pinus thunbergii) and red pine (P.densiflora), are extremely sensitive to the nematode infection and the disease has been expanding nationwide in the last several decades despite intensive control efforts. The dispersal of the beetle often involves two modes, short-range dispersal and long-range dispersal. Short-range dispersal generally occurs through their own active flying, while long-range dispersal is mediated by passive transport on wind or artificial transportation facilities. To evaluate the speed of disease's expansion, we present a model that combines the local dynamics of the pine wilt disease with the dispersal of beetles. Our model predicts several results.(1) The distribution of the dispersal distance of beetles critically affects the expansion speed of the disease. The expansion speed is sharply amplified, as the fraction of beetles undergoing long distance dispersal increases from zero.(2) However, if long range dispersers become too frequent, the disease ceases to expand due to the Allee effect. Thus suggests the importance of correct assessment of the beetle's mobility in predicting the expansion speed of the disease.(3) As the eradication rate is increased, the expansion speed decreases gradually at first and suddenly drops to zero at a certain value of the eradication rate.