Epidemic spreading on networks with vaccination

In this paper, a new susceptible-infected-susceptible (SIS) model on complex networks with imperfect vaccination is proposed. Two types of epidemic spreading patterns (the recovered individuals have or have not immunity) on scale-free networks are discussed. Both theoretical and numerical analyses are presented. The epidemic thresholds related to the vaccination rate, the vaccination-invalid rate and the vaccination success rate on scale-free networks are demonstrated, showing different results from the reported observations. This reveals that whether or not the epidemic can spread over a network under vaccination control is determined not only by the network structure but also by the medicine's effective duration. Moreover, for a given infective rate, the proportion of individuals to vaccinate can be calculated theoretically for the case that the recovered nodes have immunity. Finally, simulated results are presented to show how to control the disease prevalence.     

从整体上探索发现和提出数学问题之途——以高中复习课“导数应用”为例

从数学角度发现和提出问题的能力、分析和解决问题的能力(简称“四能”),[1]已写入普通高中数学课程标准2017年版的课程目标中.不争的事实是,中学数学课堂教学中,分析和解决问题的教学活动易寻,而学生在教师引导下发现和提出问题的活动难觅.我们认为,把研究或处理的对象当作一个系统,研究系统、要素及环境三者的相互关系和变动的规律性,强调整体、联系、动态和最佳等观点.[2]不仅有利于分析和解决问题,更有助于发现和提出问题.下面以高中复习课“导数应用”为例,呈现我们的教学思考、实践与反思.