一个修正的SIR模型的精确封闭解
Exact closed-form solution of a modified SIR model
摘要：
本文给出了一个修正的SIR系统的封闭形式的精确解析解，其中恢复的个体从种群中移除。在这个动力系统中，易感个体和恢复个体的种群$S(t)$和$R(t)$是广义Logistic函数，而感染个体的种群$I(t)$是广义Logistic函数乘以指数，它们都具有相同的特征时间.分析了这种改进的SIR系统的动力学，并对一些流行病学相关的量进行了精确计算。提出并解释了修正后的SIR模型与原SIR模型的主要区别，即它们各自守恒量的零点。此外，它表明，修正的SIR模型与时间相关的传输速率也可以解决某些现实的传输速率函数的封闭形式。
英文摘要：
The exact analytical solution in closed form of a modified SIR system where recovered individuals are removed from the population is presented. In this dynamical system the populations $S(t)$ and $R(t)$ of susceptible and recovered individuals are found to be generalized logistic functions, while infective ones $I(t)$ are given by a generalized logistic function times an exponential, all of them with the same characteristic time. The dynamics of this modified SIR system is analyzed and the exact computation of some epidemiologically relevant quantities is performed. The main differences between this modified SIR model and original SIR one are presented and explained in terms of the zeroes of their respective conserved quantities. Moreover, it is shown that the modified SIR model with time-dependent transmission rate can be also solved in closed form for certain realistic transmission rate functions.
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