In this paper, we devise a discrete time SIR model depicting the spread of infectious diseases in various geographical regions that are connected by any kind of anthropological movement, which suggests disease-affected people can propagate the disease from one region to another via travel. In fact, health policy-makers could manage the problem of the regional spread of an epidemic, by organizing many vaccination campaigns, or by suggesting other defensive strategies such as blocking movement of people coming from borders of regions at high-risk of infection and entering very controlled regions or with insignificant infection rate. Further, we introduce in the discrete SIR systems, two control variables which represent the effectiveness rates of vaccination and travel-blocking operation. We focus in our study to control the outbreaks of an epidemic that affects a hypothetical population belonging to a specific region. Firstly, we analyze the epidemic model when the control strategy is based on the vaccination control only, and secondly, when the travel-blocking control is added. The multi-points boundary value problems, associated to the opti-This work is supported by the Systems Theory Network (Réseau Théorie des Systèmes), and Hassan II Academy of Sciences and Technologies-Morocco.
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.
In this work, a mathematical model for studying the impact of awareness programs on HIV/AIDS outbreak is proposed. The main idea is that people who are susceptible to infection can prevent it, if they are aware how the disease spreads and its consequences, and also the measures to control it. Various forms of communication media, educational, heath institutions and non-governmental organizations play a significant role to promote HIV/AIDS awareness amongst the most concerned people, namely couples and senior secondary school children. The developed HIV model is inspired from the classical SIR epidemic model where a control function is introduced to represent the effectiveness of an awareness program. The obtained optimal control, is characterized in terms of the optimality system, based on Pontryagin's maximum principle, and it is simulated using the Forward-Backward Sweep Method with a progressive-regressive Runge-Kutta fourth order scheme, which is adapted to solve a two-point boundary value problem.
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