Abstract
We study, in this paper, infection dynamics when an epidemic emerges to many regions which are connected with their neighbors by any kind of anthropological movement. For this, we devise a multi-regions discrete-time model with the three classical SIR compartments, describing the spatial-temporal behaviors of homogenous susceptible, infected and removed populations. We suppose a large geographical domain, presented by a grid of colored cells, to exhibit at each instant i the spatial propagation of an epidemic which affects its different parts or sub-domains that we call here cells or regions. In order to minimize the number of infected individuals in some regions, we suggest an optimal control approach based on a travel-blocking vicinity strategy which aims to control a group of cells, or a patch, by restricting movements of infected people coming from its neighboring cells. We apply a discrete version of Pontryagin’s maximum principle to state the necessary conditions and characterization of the travel-blocking optimal controls. We provide cellular simulations based on discrete progressive-regressive iterative schemes associated with the obtained multi-points boundary value problems. For illustrating the modeling and optimal control approaches, we consider an example of 100 regions.
Highlights
1.1 Main references and description of the problem In, Kermack and McKendrick devised the Susceptible-Infected-Removed (SIR) model which has presented an interesting contribution to the mathematical theory of epidemics [ ]
4.4 Discussions In Figures, and, we investigate the effectiveness of the travel-blocking vicinity optimal control approach on the SIR populations of when it is applied to two patches P and P = {C }
Not very far from the main goals of this kind of epidemic models treated in the mentioned references, which aim to highlight the nature of infection connections which participate in the rapid spread of an epidemic, in this paper we have devised a multi-regions discrete-time model which describes infection dynamics due to the presence of an epidemic in one region and its spreading to other regions via travel
Summary
1.1 Main references and description of the problem In , Kermack and McKendrick devised the Susceptible-Infected-Removed (SIR) model which has presented an interesting contribution to the mathematical theory of epidemics [ ]. In Figure (b), we can see the example of all nine regions presented in (a), how they can be converted to cells, assembled in one grid which represents a part of the earth as the global domain of interest Based on this new kind of representations, we can discuss the spread of the epidemic and the effectiveness of a control strategy in one region, with the possibility to analyze the SIR dynamics in this region without and with control, and exhibiting the importance of the direct influence between it and its vicinity. We suggest here a new modeling approach which is based on a multi-regions discrete-time epidemic model describing the spatial-temporal spread of an epidemic which emerges in a global domain of interest represented by a grid of colored cells which are uniform in size These cells are supposed to be connected by movements of their populations, and they represent sub-domains of or regions. In Section , we provide simulations of the numerical results for an example of hypothetical cities when an infection starts from one cell which has three neighboring cells (respectively, the case of a cell with eight neighboring cells is investigated), while aiming to control a patch of four cells, and in another example, two patches of one and four cells, respectively
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