Mathematical Analyses of Spatial Spread of Pine Wilt Disease

松材线虫病是由松材线虫松材线虫引起的，松材线虫通过松墨天牛传播。日本本土松树黑松(Pinus Thunbergii)和红松(P.densiflora)对线虫感染极其敏感，尽管采取了密集的控制措施，但在过去几十年里，这种疾病一直在全国范围内蔓延。甲虫的扩散通常有两种方式，即近距离扩散和远距离扩散。短距离的扩散通常通过它们自己的主动飞行进行，而远程的扩散是通过风或人工交通设施的被动运输来调节的。为了评估疾病的扩散速度，我们提出了一个将松材线虫病的局部动态与甲虫的扩散相结合的模型。我们的模型预测了几个结果：(1)甲虫传播距离的分布严重影响了疾病的传播速度。随着经历远距离传播的甲虫比例从零增加，扩展速度急剧放大。(2)然而，如果远距离传播变得过于频繁，由于Allee效应，疾病停止扩展。因此，正确评估甲虫的活动能力对预测疾病的扩展速度具有重要意义。(3)随着疾病根治率的提高，扩展速度先是逐渐降低，在一定的根治率值下突然降为零。

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.