SIR类型新型冠状病毒肺炎多阶段最优控制模型

doi:10.15960/j.cnki.issn.1007-6093.2023.01.003

Abstract

Abstract:

The coronavirus disease 2019 (COVID-19) spreads all over the world, and it makes serious threat to people's health. Being faced with the data of anticipated development of COVID-19, we need to determine the epidemic spreading parameters under limited medical resources to give guidance for implementation intensities of the main epidemic prevention and control measures. In this paper, we describe the development of COVID-19 based on the SIR model. What's more, we propose a multi-stage optimal control model to determine the epidemic spreading parameters. In order to determine the values of parameters efficiently, we construct an SDP approximation model which is a polynomial-time computable problem. Based on the data of COVID-19 published by WHO, we apply our approximation model to obtain the epidemic spreading parameters which describe the development of COVID-19 in the USA within a given period of time, and analyze the epidemic prevention and control strategies.

Key words: coronavirus disease 2019, multi-stage optimal control model, SDP problem

CLC Number:

O221

Jintao XU, Wenxun XING. SIR type COVID-19 multi-stage optimal control model[J]. Operations Research Transactions, 2023, 27(1): 43-52.

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URL: https://www.ort.shu.edu.cn/EN/10.15960/j.cnki.issn.1007-6093.2023.01.003

https://www.ort.shu.edu.cn/EN/Y2023/V27/I1/43

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$\lambda_{a}$	相邻两阶段易感者($\rm S$)被新冠病毒感染, 转变为无症状感染者($\rm I_{a}$) 的比例
$\lambda_{s}$	相邻两阶段易感者($\rm S$)被新冠病毒感染, 转变为有症状感染者($\rm I_{s}$) 的比例
$\mu_{a}$	相邻两阶段无症状感染者($\rm I_{a}$)痊愈的比例
$\mu_{s}$	相邻两阶段有症状感染者($\rm I_{s}$)痊愈的比例
$\gamma $	相邻两阶段易感者($\rm S$)通过接种疫苗获得免疫的比例
$\delta_{S}$	相邻两阶段易感者($\rm S$)保持不变的比例
$\delta_{I_{a}}$	相邻两阶段无症状感染者($\rm I_{a}$)保持不变的比例
$\delta_{I_{s}}$	相邻两阶段有症状感染者($\rm I_{s}$)保持不变的比例
$\delta_{R}$	相邻两阶段免疫者($\rm R$)保持不变的比例
$\beta_{S1}$	相邻两阶段由于输入输出导致的易感者($\rm S$)人数变化量
$\beta_{S2}$	相邻两阶段内新生儿人数
$\beta_{I_{a}}$	相邻两阶段由于输入输出导致的无症状感染者($\rm I_{a}$)人数变化量
$\beta_{I_{s}}$	相邻两阶段由于输入输出导致的有症状感染者($\rm I_{s}$)人数变化量
$\beta_{R}$	相邻两阶段由于输入输出导致的免疫者($\rm R$)人数变化量

方程组(1)疫情传播参数需满足的关系式	原因解释
$L_{1}\leq \delta_{S}+\lambda_{a}+\lambda_{s}+\gamma\leq U_{1}$	易感者($\rm S$)存在死亡率
$L_{2}\leq \delta_{I_{a}}+\mu_{a}\leq U_{2}$	无症状感染者($\rm I_{a}$)存在死亡率
$L_{3}\leq \delta_{I_{s}}+\mu_{s}\leq U_{3}$	有症状感染者($\rm I_{s}$)存在死亡率
$L_{4}\leq \delta_{R}\leq U_{4}$	免疫者($\rm R$)存在死亡率
$9\lambda_{a}\leq \lambda_{s}$	依据文献[11], 本文的合理假设
$L_{5}\leq \gamma\leq U_{5}$	本文的合理假设

$\tilde{U}$的取值	0.6	1	2
AMOCM的最优值	0.363 729	0.110 419	0.003 306 57

矩阵及向量	$\lambda_{a}$	$\lambda_{s}$	$\gamma$	$\delta_{I_{a}}$	$\delta_{I_{s}}$	$\delta_{S}$	$\mu_{a}$	$\mu_{s}$	$\beta_{I_{a}}$
$A_{0}$, $b_{0}$	0.000 46	0.004 14	0	0.325 18	0.315 17	0.995 4	0.674 82	0.677 55	-0.000 06
$A_{1}$, $b_{1}$	0.000 09	0.000 82	0.000 32	0.441 04	0.555 47	0.998 77	0.558 96	0.420 39	0.000 04

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SIR type COVID-19 multi-stage optimal control model

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