Sparse Optimal Control of Pattern Formations for an SIR Reaction-Diffusion Epidemic Model
Abstract.
Pattern formation arising from the reaction-diffusion epidemic model is a space-time depiction of the distribution and transmission of infectious diseases. Disease control can be achieved by controlling the associated pattern formations. For an SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates under the case of a constant recovery rate. To control pattern formations of the SIR epidemic model, we introduce a regulator as a control function of the recovery rate. For a perfect control strategy, it not only leads to a desired pattern formation, but also has a small support in space-time domains. In order to obtain such a control strategy, we propose a sparse optimal control problem governed by the SIR epidemic model. We study the existence of optimal solutions, derive the first order necessary optimality system, obtain the sparsity structure of the control function, and numerically solve the control problem. Numerical results demonstrate the feasibility and effectivity of our method.
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Information & Authors
Information
Published In
SIAM Journal on Applied Mathematics
Pages: 1764 - 1790
DOI: 10.1137/22M1472127
ISSN (online): 1095-712X
Copyright
© 2022 Society for Industrial and Applied Mathematics.
History
Submitted: 18 January 2022
Accepted: 7 July 2022
Published online: 18 October 2022
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Funding Information
Basic Applied Study Program of Shanxi Province: 20210302123453
Strategic Priority Research Program of Chinese Academy of Sciences: XDB 41000000
National Key Basic Research Program: 2018YFB0704304
Health Commission of Shanxi Province: 2020XM18
The work of the first author was supported by the Basic Applied Study Program of Shanxi Province under grant 20210302123453. The work of the second author was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant XDB 41000000), the National Key Basic Research Program (Grant 2018YFB0704304), and the National Natural Science Foundation of China under grant 12071468. The work of the third author was supported by the National Natural Science Foundation of China under grant 61873154 and the Health Commission of Shanxi Province under grant 2020XM18. The work of the fourth author was supported by the National Natural Science Foundation of China under grant 12022113.
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