The diet problem, a mathematical approach

Background and aim: Differential equations have always been used to modelize physical phenomena from other branches of science: physics, biology, chemistry, engineering, computer science etc. The aim of this paper is to find a simple mathematical model that can describe the variation of weight depending on time and calories intake. The idea is simple and is based on the so-called Malthus mathematical model, an ordinary differential equation associated to an initial condition, which studies the growth of a population with respect to a certain phenomenon or under the influence of external/internal factors. Methods: The most basic and intuitive Malthus model is formalized as follows: given P=P(t) the function that describes the size of a population, the ordinary differential equation P’(t)=rP expresses the fact that the rate of change of the size of the population (i.e. the derivative P’(t) with respect to the time t) depends directly on the size of the population itself multiplied by a factor r that represents the population growth rate, sometimes called Malthusian parameter. The equation needs to be associated to an initial condition, say P0 = P(0), which represents the size of the population at the time t=0. The solution of this problem can be calculated explicitly and this allows to precisely link the weight loss (or gain) according to calories intake, expected time, gender, kind of physical activity etc. Results: Our model considers age, gender and physical activity and allows us to discuss how to calculate a reasonable diet plan depending on different variables. Morever, it can give an idea, by studying the asymptotic behaviour of the solution, why the so-called miracle-diets can’t work, why long diet plans usually fail and how to deal with severe obesity. Conclusions: The results obtained by means of this mathematical model shed new light on how to approach the creation of a reasonable diet plan. These results can be improved by introducing numerical simulations, which is the aim of a subsequent paper.