In this paper, we deal with the sharp threshold results for a host-pathogen model with general incidence rates. We formulate the model by a system of degenerated reaction-diffusion equations with heterogeneous parameters, where the movement of pathogens are ignored. The basic reproduction number, ${\Re }_0$, is defined to govern and it is shown to be a threshold determining whether or not the disease will be extinct or be persistent. We also confirm that disease will be extinct in the critical case ${\Re }_0=1$. By three examples of homogeneous cases in the sense that parameters are all constants, we obtain the specific formula for ${\Re }_0$, and explore the stability problems of the unique constant positive equilibrium by using the technique of Lyapunov function. Our theoretical results can also be potentially applied to explore the effect of the spatial heterogeneity on disease dynamics, and also evaluate the risk of disease transmission.

在本文中，我们处理了具有一般发生率的宿主-病原体模型的尖锐阈值结果。我们通过具有异质性参数的简并反应扩散方程系统来建立模型，其中忽略了病原体的运动。基本复制号，${\Re }_0$被定义为可治病，并且显示为确定该疾病是否将灭绝或持续的阈值。我们还确认，在危急情况下疾病将灭绝${\Re }_0=\mathrm{1个}$。在参数均为常数的意义上，通过三个同类情况的例子，我们获得了以下公式的特定公式：${\Re }_0$，并利用Lyapunov函数技术探讨唯一常数正平衡的稳定性问题。我们的理论结果也可以潜在地用于探索空间异质性对疾病动力学的影响，并评估疾病传播的风险。