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**Cell and patch vicinity travel restrictions in a multi-regions SI discrete epidemic control model.**
*(English)*
Zbl 1467.92236

Summary: Some epidemics can spread in geographical large scales and in unprecedented durations if the authorities concerned do not hurry to save susceptible populations. In such cases, mathematical modelers are required to take into account the spatial aspect of the propagation of an epidemic. Infected travelers either exiting the main source of epidemic or coming from other locations that have been partially affected later, and aiming to enter safer regions, represent the population category that should be hastily controlled. Based on these assumptions, and with the consideration of the factor of their mobility, we devise a discrete-time susceptible-infected (SI) model which describes the spatial spread of an epidemic emerging in regions that are connected by any kind of movement. Furthermore, we introduce into SI discrete equations, controls variables which restrict movements of infected people for avoiding any contact with the susceptible class. Explicitly, we present regions by \(M^2\) cells assembled in one grid, and we seek the optimal values of controls using a discrete version of Pontryagin’s maximum principle. In a first case, we aim to control only one cell by restricting movements of infected people coming from its neighbors, and in a second case, we aim to control a group of cells or patches, based on the same logic of inter-interventions and which is related to the vicinity travel restrictions optimal control strategy. In a first time, the patch is supposed to be located in the opposite side of the epidemic source, and in a second time, in the interior of the domain of study in order to verify the effectiveness of such control policies when the number of regions in the vicinity set, is either small or important.

### MSC:

92D30 | Epidemiology |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

### Keywords:

spatial spread of epidemic; discrete-time model; SI epidemic model; multi-regions; optimal control; travel restrictions; vicinity; cell; patch
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\textit{O. Zakary} et al., Int. J. Adv. Appl. Math. Mech. 6, No. 2, 30--41 (2018; Zbl 1467.92236)

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### References:

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