A mathematical model with fractional order for obesity with positive and negative interactions and its impact on the diagnosis of diabetes

Abstract

Overweight and obesity are current problems humankind faces and have serious health consequences because they contribute to diseases such as heart diseases and diabetes. In this paper, we present a mathematical model for the study of overweight and obesity in a population and its impact on the growth of the number of diabetics. For the construction of the model, we take into account social factors and the interactions between different elements of society. We use fractional-order derivatives in the Caputo sense because of the advantages of this type of technique with respect to the memory effect, and it shows different behaviors depending on the fractional order. We find the basic reproduction number and prove the local and global stability of the disease-free equilibrium points. We study the sensitivity index with respect to the basic reproduction number for parameters associated with weight gain due to social pressure and the rate of diagnosis of diabetes not associated with body weight. To validate the model, we perform computational simulations with data extracted from the literature. We conclude that for higher fractional orders a higher value of the basic reproduction number was reached. We show that at the end of the study for different fractional orders that normal-weight individuals are decreasing, and overweight, obese, and diabetic people are increasing.