## 3 Supervised Selection of the Control Gains
### Updating Algorithms
This paper has been devoted to the investigation of supervisory techniques to update through time the feedback vaccination and control gains in a modified SI-epidemic model involving the susceptible and infectious subpopulations which is built from a basic uncontrolled SIS epidemic model. The presence of linear feedback controls motivates the incorporation of a new subpopulation to the basic model. Such an incorporated subpopulation is referred to as a compensatory recovered (or immune) one and it is subject to zero initial conditions. Therefore, the modified model is referred to as an SI (RC) epidemic model since it integrates susceptible, infectious and compensatory recovered subpopulations. The supervisory loss function can evaluate weighted losses involving the susceptible and the infectious subpopulations or, in particular, any of them only. The supervision problem is basically based on the implementation of a parallel control structure with different potential control gains to be judiciously selected and switched to be operated on the controlled model through time. A higher decision level structure selects the appropriate gains along a time interval and the time instants to switch between the successive active controllers in operation. Basically, switching to another controller on a time interval between two consecutive switching time instants is decided when a better supervisory loss value is detected associated with an alternative controller structure to that currently

Figure 2:2 Comparison of the performances obtained by Algorithms 2–4. (**a**) Vaccination functions. (**b**) Treatment functions. (**c**) Comparison of the evolution of Infectious. in operation within the parallel controllers' disposal structure. Several algorithms for the design of the controller gains have been described and some numerical testing examples have also been given and discussed. The various proposed algorithms are able to: a) involve predefined set of gains; b) they can incorporate eventual variations of the transmission and recovery rates at the expense of a larger whole model complexity; c) they can update the control gains from a dynamic map of eventual candidates built from the current active ones; or d) the tests about switching the active parameterization decision might be performed at constant sampling periods. A related future work is foreseen on the extension of the above method to other types of epidemic models which involve more subpopulations, e.g., several types of infected individuals, such as those exposed without symptoms which can later on have transitions to the infectious symptomatic and the infectious asymptomatic subpopulations. It is also intended to extend the above results to patchy environments where the disease is spreading in different regions, towns, etc., with mutual populations interchanges. Conceptualization, M.D.I.S., A.I. and S.A.-Q. ; methodology, M.D.I.S. and A.I. ; software, A.I. and R.N. ; validation, A.I. and S.A.-Q., Y.Y. ; formal analysis, M.D.I.S. ; investigation, M.D.I.S. and A.G.; resources, M.D.I.S., R.N. and A.I. ; data curation, R.N. ; A.I. and S.A.-Q. ; writing--original draft preparation, M.D.I.S., A.I. and A.G.; writing--review and editing, A.I., R.N. and A,G. ; visualization, A.I. ; supervision, A.I. and M.D.I.S. ; project administration, M.D.I.S. ; funding acquisition, A.G. and M.D.I.S. All authors have read and agreed to the published version of the manuscript. The authors are grateful to the Spanish Government for Grants RTI2018-094336-B-100 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. The authors declare that they have no competing interest regarding the publication of this manuscript. ## References

* (1) [PERSON] _Mathematical Foundations of Information Theory_; Dover Publications Inc.: New York, NY, USA, 1957.