Abstract
Fractional programming problems have attracted the attention of many specialists due to their applications in many fields as Economics and Finance, Engineering, Biology, Epidemiology, Medicine, etc. In this article, we present examples of the problems of the fractional programming, and we discussed a parallel between the SIR models built using classical differential equations and using fractional differential equations. Epidemiological models (i.e. SIR model and it’ s generalizations) or pollution models, can provide particular cases of numerators and denominators for objective functions in fractional programming problems. For the SIR epidemiological model, the article discusses the existence of the solution for its generalization and brings a novelty regarding its approach as a model with partial derivatives. In the classical models, the rate of change is more abrupt, which suggests a more deterministic and predictable behavior in the spread of the disease. The fractional derivative brings a memory effect, which makes the spread and recovery slower and more extended over time. Simulations of many optimization problems were also realized in GeoGebra, Maple and MATLAB software. A case study related to optimizing vaccination rates between two regions, applied the Frank-Wolfe method for a fractional programming problem and an implementation in Maple. In the case study, in a context of a city divided into two regions, we supposed that the authorities aim to optimize the vaccination rate to minimize the combined infection rate. The aim was to maximize the ratio between vaccination effectiveness and total costs. A principal conclusion is that the use of simulations, for example, in GeoGebra, Maple or MATLAB software, can increase the quality of the teaching-learning process for students, in subjects such as Operational Research or Optimization Techniques.
Published Version
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