Thisarticleisdevotedtotheproblemofoptimalcontrolofareaction-diffusionsystemforan
SEIRDS-typeepidemiologicalmodel,wherethedynamicsevolveinaspatiallyheterogeneousenvironment.
Thecontrolvariablesarethetransmissionratesβe, β1,andβ2,correspondingrespectively tothe contagion
resultingfromcontactwithasymptomaticandsymptomaticindividuals. Theaimistooptimizethenumber
of exposed and infected individuals at a final time T within the frameworkof the controlled evolution of
thesystem. Moreprecisely,theaimistodeterminetheoptimalrates¯
βe,¯
β1,and¯
β2 sothatthenumbers
ofexposedEandinfectedI1 andI2 donotexceed,atthefinaltimeT,thepre-establishedthresholdse,
i1,andi2. Inthisarticle,wedemonstratetheexistenceoftheseoptimalcontrolsinasuitablefunctional
framework, and wederive the necessary first-order optimality conditions basedon the adjoint variables.

reaction-diffusion;SEIRDSmodel;optimalcontrol;first-ordernecessaryoptimality conditions;adjoint system.