Hwang Yoon-Gu, Kwon Hee-Dae, Lee Jeehyun
Department of Computational Science and Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea.
Department of Mathematics, Inha University, 100 Inha-ro, Michuhol-gu, Incheon 22212, Republic of Korea.
Math Biosci Eng. 2020 Jan 16;17(3):2284-2301. doi: 10.3934/mbe.2020121.
We consider a feedback control problem of a susceptible-infective-recovered (SIR) model to design an efficient vaccination strategy for influenza outbreaks. We formulate an optimal control problem that minimizes the number of people who become infected, as well as the costs of vaccination. A feedback methodology based on the Hamilton-Jacobi-Bellman (HJB) equation is introduced to derive the control function. We describe the viscosity solution, which is an approximation solution of the HJB equation. A successive approximation method combined with the upwind finite difference method is discussed to find the viscosity solution. The numerical simulations show that feedback control can help determine the vaccine policy for any combination of susceptible individuals and infectious individuals. We also verify that feedback control can immediately reflect changes in the number of susceptible and infectious individuals.
我们考虑一个易感-感染-康复（SIR）模型的反馈控制问题，以设计针对流感爆发的有效疫苗接种策略。我们制定了一个最优控制问题，该问题使感染人数以及疫苗接种成本最小化。引入了一种基于哈密顿-雅可比-贝尔曼（HJB）方程的反馈方法来推导控制函数。我们描述了粘性解，它是HJB方程的一种近似解。讨论了一种结合迎风有限差分法的逐次逼近方法来找到粘性解。数值模拟表明，反馈控制有助于为易感个体和感染个体的任何组合确定疫苗接种策略。我们还验证了反馈控制可以立即反映易感个体和感染个体数量的变化。
