Generalized Fractional Calculus for Gompertz-Type Models

This paper focuses on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Precisely, we first introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of its solutions. This is done by proving the existence and uniqueness of Gaussian solutions of such equations via a fixed point argument and then by showing that, under suitable conditions, the expected value of the solution solves a generalized fractional linear equation. Regularity of the absolute p-moment functions is proved by using generalized Grönwall inequalities. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. Their stochastic counterparts are then constructed by using the previously considered integral equations to define a rate process and a generalization of lognormal distributions to ensure that the median of the newly constructed process coincides with the deterministic curve.

A Bayesian Model of COVID-19 Cases Based on the Gompertz Curve

The COVID-19 pandemic has highlighted the need for finding mathematical models to forecast the evolution of the contagious disease and evaluate the success of particular policies in reducing infections. In this work, we perform Bayesian inference for a non-homogeneous Poisson process with an intensity function based on the Gompertz curve. We discuss the prior distribution of the parameter and we generate samples from the posterior distribution by using Markov Chain Monte Carlo (MCMC) methods. Finally, we illustrate our method analyzing real data associated with COVID-19 in a specific region located at the south of Spain.

Body growth description of male and female from three broilers strains in the hot climate

To evaluate the growth of males and females from three broiler strains (Cobb 500, Ross 308, and Hubbard Flex) we used a completely randomized experimental design with a factorial arrangement (3 strains and 2 sexes) and six replicates of 60 birds each, 2160 chicks of both sexes in total followed for 56 days. Once a week, the birds from each pen were weighed to estimate the average weight. The sample birds were fasted for 24 hours, re-weighed, slaughtered, eviscerated, and dissected to obtain the breast, thigh and drumstick cuts. The fitted Gompertz equations were used for each strain, and sex, for the fasted weight, and the weight of breasts, thighs, and drumsticks. To verify the effect of strain on the Gompertz curve parameters we assessed the weight at maturity (Wm), rate of maturing (b), and the time at which maximum growth rate was attained (t*) for both sexes and verified the equivalence of parameters and identity of non-linear models. For the fasted weight the rate of maturing was very similar for all strains and sexes, but the Hubbard strain showed the highest Wm and t*. The Wm parameter was common for most cuts among the strains, and the Hubbard was the dominant strain with the highest b parameter. All three parameters were similar for the male breast weight, and only one equation required adjustment to describe the breast weight growth for the three strains. Individually adjusted equations were necessary to describe the weight of drumsticks because females showed no similarity in any of the parameters for any strain.