Supervision of the Infection in an SI (SI-RC) Epidemic Model by Using a Test Loss Function to Update the Vaccination and Treatment Controls

This paper studies and proposes some supervisory techniques to update the vaccination and control gains through time in a modified SI (susceptible-infectious) epidemic model involving the susceptible and subpopulations. Since the presence of linear feedback controls are admitted, a compensatory recovered (or immune) extra subpopulation is added to the model under zero initial conditions to deal with the recovered subpopulations transferred from the vaccination and antiviral/antibiotic treatment on the susceptible and the infectious, respectively. Therefore, the modified model is referred to as an SI(RC) epidemic model since it integrates the susceptible, infectious and compensatory recovered subpopulations. The defined time-integral supervisory loss function can evaluate weighted losses involving, in general, both the susceptible and the infectious subpopulations. It is admitted, as a valid supervisory loss function, that which involves only either the infectious or the susceptible subpopulations. Its concrete definition involving only the infectious is related to the Shannon information entropy. The supervision problem is basically based on the implementation of a parallel control structure with different potential control gains to be judiciously selected and updated through time. A higher decision level structure of the supervisory scheme updates the appropriate active controller (i.e., that with the control gain values to be used along the next time window), as well as the switching time instants. In this way, the active controller is that which provides the best associated supervisory loss function along the next inter-switching time interval. Basically, a switching action from one active controller to another one is decided as a better value of the supervisory loss function is detected for distinct controller gain values to the current ones.The authors are grateful to the Spanish Government for Grants RTI2018-094336-B-I00 and RTI2018-094902-B-C22 (MCIU/AEI/FEDER, UE), to the Institute of Health Carlos III for Grant COV20/01213 and to the Basque Government for Grant IT1207-19. They also thank the referees for their useful suggestions and corrections