Optimal control of a reaction–diffusion model related to the spread of COVID-19

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Abstract
This paper is concerned with the well-posedness and optimal control problem of a reaction–diffusion system for an epidemic susceptible–exposed–infected–recovered–susceptible mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates [ ] and [ ] of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time [ ] of the controlled evolution of the system. More precisely, we search for the optimal [ ] and [ ] such that the number of infected plus exposed does not exceed at the final time a threshold value [ ], fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.
Funding Information
  • Italian CNR and the Romanian Academy (2020F3NCPX)
  • Ministry of Research Innovation and Digitization, CNCS-UEFISCDI (PN-III-P4-PCE-2021-0006)

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