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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. [ABSTRACT FROM AUTHOR]

The main goal of this article is to exhibit the importance of awareness programs and travel-blocking operations, in the prevention of HIV/AIDS outbreaks, based on a multi-domains SIR epidemic model. The model devised describes the spatial-temporal spread of HIV/AIDS, in p neighboring domains, taking into account the epidemiological diversity of their populations. For a first case of the study, we focus on discussing the benefits of awareness campaigns that aim to raise public consciousness among susceptible people, about the danger of HIV/AIDS. Thus, for the reason that intra-domains interventions are not always sufficient to prevent people belonging to a domain (neighborhood, town or city for example), from a rapid expansion of HIV/AIDS, we propose the travel-blocking strategy, as a second approach presenting inter-domains interventions that health-policy makers could follow, by restricting the number of suspicious people that can participate in spreading HIV/AIDS via travel, coming from domains at high-risk of infection to enter domains with lower risk. The optimal control theory, based on Pontryagin's maximum principle, is applied twice in this paper, for the characterizations of the awareness and travel-blocking controls. The numerical results associated to the multi-points boundary value problems are obtained based on the Forward-Backward Sweep Method combined with progressive-regressive Runge-Kutta fourth-order schemes. [ABSTRACT FROM AUTHOR]