SIR模型的解析近似解
An analytic approximate solution of the SIR model
摘要：
SIR（D）流行病模型是通过超越方程组定义的，不能用初等函数求解。在本文中，这些方程被成功地用近似方程代替，近似方程的解以初等函数的形式明确给出，分段地，从广义逻辑函数出发：它们确保{ \em精确}（在数值意义上）渐近值，除了使用起来相当实用之外，例如与数据算法配合;此外，它们揭示了一个有用的特征，事实上，至少在非常严格的近似下，{ \em精确}方程的（数值）解也具有该特征。在工作中的新颖之处是：获得近似方程的方式，使用简单的，解析几何的考虑;容易和实用的制定最终的近似解决方案;所提到的有用的功能，从来没有透露过。这项工作的方法和结果证明是强大的范围内的值的众所周知的无量纲参数称为{ \em基本繁殖比}，至少涵盖了所有已知的流行病的情况下，从流感到麻疹：这是一个点，并没有出现在类似的作品中讨论。
英文摘要：
The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure {\em exact} (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms; moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the {\em exact} equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations; the easy and practical formulation of the final approximate solutions; the mentioned useful feature, never disclosed before. The work's method and result prove to be robust over a range of values of the well known non-dimensional parameter called {\em basic reproduction ratio}, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn't appear much discussed in analogous works.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
