In this paper the mathematical SIRC epidemic model is considered. It efficiently describes diseases in which a cross immune class (C) is present, along with the susceptible (S), the infected (I) and the removed (R) ones. Controlling epidemic diseases corresponds to the introduction of vaccination, quarantine and treatment strategies; generally only one of these actions is considered. In this paper the possibility of optimal controls both over the susceptible and the infected subjects is assumed, taking into account also limitations of resources. A suitable cost index is introduced and via the Pontryagin's Minimum Principle the optimal control strategy is determined and the existence of the optimal solution is assessed. Numerical results are developed analyzing the effects of different control strategies.

Endoscopic imaging through a multicore fiber (MCF) is widely used but is affected by pixelated images, which limits its resolution to a few micrometers. This is due to the spacing between the cores in the MCF, which is large enough to avoid core-to-core light coupling. Wavefront shaping techniques have been shown to focus light to a resolution finer than the inter core spacing, however a long calibration procedure is needed. Moreover the calibration depends on the optical fiber conformation. Here, we show a calibration method using only one digital hologram. The method is based on digital phase conjugation and the memory effect of the MCF to focus and scan a spot. In addition, we show how simple patterns can be projected using the same multicore fiber.

The images produced by multicore endoscopes are pixelated, and their resolution is limited by the core-to-core spacing. Lenses can be used to improve the resolution, but this reduces the field of view proportionally. Lensless endoscopy through multicore fibers can be achieved by using wavefront shaping techniques. This requires a calibration step, and the conformation of the fiber must remain constant over time. Here we demonstrate that, without a calibration step and in the presence of core-to-core coupling, we can obtain fluorescence images with a resolution better than the core-to-core spacing. This is accomplished by taking advantage of the memory effect present in these kinds of fibers.

, "Lensless two-photon imaging through a multicore fiber with coherence-gated digital phase conjugation," J. Abstract. We performed near-diffraction limited two-photon fluorescence (TPF) imaging through a lensless, multicore-fiber (MCF) endoscope utilizing digital phase conjugation. The phase conjugation technique is compatible with commercially available MCFs with high core density. We demonstrate focusing of ultrashort pulses through an MCF and show that the method allows for resolution that is not limited by the MCF core spacing. We constructed TPF images of fluorescent beads and cells by digital scanning of the phase-conjugated focus on the target object and collection of the emitted fluorescence through the MCF.

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