仓室型流行病中传播率的最优控制
Optimal control of the transmission rate in compartmental epidemics
AI总结：
该研究提出了通用的仓室型流行病传播向量常微分方程模型，证明了其初值问题的适定性、正性及最优控制解的存在性与小时间唯一性，还研究了线性代价下的奇异弧并给出n=1情形的细节与数值模拟。
摘要：
我们提出了一个通用常微分方程系统，它涵盖了若干经典及近期的模型，用于描述有限时间范围内无生命动力学的封闭种群中的流行病传播。该模型是向量型的，即它采用向量值状态函数，其分量代表各类暴露/感染亚种群，同时配有对应的控制函数向量，不同亚种群的控制函数可能不同。在通用设定下，我们证明了状态方程系统初值问题的适定性与正性，且在非常通用的代价泛函下，系统非线性部分系数的最优控制问题存在解。我们还证明了：当代价在所有控制变量上为超线性、且各变量指数可不同并处于区间(1,2]内时，小时间范围内最优解具有唯一性。随后我们考虑控制变量的线性代价，研究了奇异弧。我们给出了n=1情形下的全部细节，并通过若干数值模拟对结果进行了说明。
英文摘要：
We introduce a general system of ordinary differential equations that includes some classical and recent models for the epidemic spread in a closed population without vital dynamic in a finite time horizon. The model is vectorial, in the sense that it accounts for a vector valued state function whose components represent various kinds of exposed/infected subpopulations, with a corresponding vector of control functions possibly different for any subpopulation. In the general setting, we prove well-posedness and positivity of the initial value problem for the system of state equations and the existence of solutions to the optimal control problem of the coefficients of the nonlinear part of the system, under a very general cost functional. We also prove the uniqueness of the optimal solution for a small time horizon when the cost is superlinear in all control variables with possibly different exponents in the interval (1,2]. We consider then a linear cost in the control variables and study the singular arcs. Full details are given in the case n=1 and the results are illustrated by the aid of some numerical simulations.
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