Modeling and multi-objective optimal control of reaction-diffusion COVID-19 system due to vaccination and patient isolation

Yunbo Tu; Tasawar Hayat; Aatef Hobiny; Xinzhu Meng

doi:10.1016/j.apm.2023.02.002

Modeling and multi-objective optimal control of reaction-diffusion COVID-19 system due to vaccination and patient isolation

Appl Math Model. 2023 Jun:118:556-591. doi: 10.1016/j.apm.2023.02.002. Epub 2023 Feb 9.

Authors

Yunbo Tu¹, Tasawar Hayat^{2

3}, Aatef Hobiny², Xinzhu Meng^{1

2}

Affiliations

¹ College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, PR China.
² Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.
³ Department of Mathematics, Quaid-i-Azam University 45320, Isamabad 44000, Pakistan.

Abstract

In this paper, a reaction-diffusion COVID-19 model is proposed to explore how vaccination-isolation strategies affect the development of the epidemic. First, the basic dynamical properties of the system are explored. Then, the system's asymptotic distributions of endemic equilibrium under different conditions are studied. Further, the global sensitivity analysis of $R_{0}$ is implemented with the aim of determining the sensitivity for these parameters. In addition, the optimal vaccination-isolation strategy based on the optimal path is proposed. Meantime, social cost $C (m, σ)$ , social benefit $B (m, σ)$ , threshold $R_{0} (m, σ)$ three objective optimization problem based on vaccination-isolation strategy is explored, and the maximum social cost $(M S C)$ and maximum social benefit $(M S B)$ are obtained. Finally, the instance prediction of the Lhasa epidemic in China on August 7, 2022, is made by using the piecewise infection rates $β_{1} (t)$ , $β_{2} (t)$ , and some key indicators are obtained as follows: (1) The basic reproduction numbers of each stage in Lhasa, China are $R_{0} (1 : 8) = 0.4678, R_{0} (9 : 20) = 2.7655, R_{0} (21 : 30) = 0.3810$ and $R_{0} (31 : 100) = 0.7819$ ; (2) The daily new cases of this epidemic will peak at 43 on the 20th day (August 26, 2022); (3) The cumulative cases in Lhasa, China will reach about 640 and be cleared about the 80th day (October 28, 2022). Our research will contribute to winning the war on epidemic prevention and control.

Keywords: Instance prediction; Multi-objective optimization; Piecewise infection rate; Reaction-diffusion COVID-19 system; Vaccination-isolation strategy.