Multi-region optimal impulse control through example of immunization policy at large geographical scale with discussion of the closure case

Abstract

Most epidemiological studies prefer to rely on differential systems because of their mathematical tractability, however, in addition to the need to control modeling framework that takes into account factors of interconnections, there is also a need to rely on the practical pulsed controls. For that, we define first a susceptible-immunized-infected-removed (SVIR) multi-region control differential system, then we explain how this problem is reformulated after the introduction of discrete impulse controls via the compartment of the V variable and that is associated to the example of the immunization policy whose goals can be reached either by following those who recommend the application of discrete-time awareness seasonally or those who advise to use some potentially effective antiviral medications. Then, we characterize our sought optimal controls in the multi-region impulse case. Finally, we take the example of three interconnected regions and conclude that prompt control measures are not always necessary as we observe that even when their values are not that significant in the first six weeks, there is a decrease in infection and which may be also due to natural immunity and that could lead to some herd immunity later enhanced by impulse controls for other months. We also discuss in the end, the pandemic crisis scenario when impulse closure policies are needed to enhance the effectiveness of the impulse immunization strategy.