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In this paper, we present a definition of the generalized fractional local derivative, which contains as a particular case several of those reported in the literature. In this framework, we apply this operator to the study of the solutions of some fractional differential equations and we focus mainly on a Gompertz model applied to the study of tuberculosis in Mexico. We also show the effectiveness of this model by fitting it to real data and by comparing it with other models using other well‐known definitions of fractional derivative. [ABSTRACT FROM AUTHOR]

Sustainable water management is important to ensure its availability for future generations. The study of water quality is fundamental for this purpose. Assessing the health of aquatic ecosystems through bioindicators has been shown to be reliable and inexpensive. The objective of this work was to evaluate water quality through a biomathematical model that involves environmental stress indicator organisms and their close relationship with dissolved oxygen. In this direction, a system of differential equations describing the population dynamics of aquatic macroinvertebrates under the influence of dissolved oxygen is proposed. The model is validated by its application in the Coyuca Lagoon, Mexico. Likewise, population changes over time were represented, which allowed us to deduce that the increase or decrease in the aeration/oxygenation rate significantly affects the population dynamics of the bioindicator organisms. In addition, to classify water quality, a one-to-one correspondence was established between water quality and the equilibrium points of the system of differential equations. The results obtained allow inferring that the proposed techniques are useful for the study of water quality since they can predict significant changes in the ecosystem and provide researchers and water managers with tools for decision making. [ABSTRACT FROM AUTHOR]

Objective: To identify specific fractures risks of the following bones: distal femur, patella and tibia plateau, according to sex and age, with special interest in women >or= 50 y.o., Material and Methods: An epidemiologic case-control study was done between January 2002 and December 2005. The incidence rate for fractures were got by year, sex and age. The group exposed to the risk factors, was represented by female patients >or=50 y.o. The cases with fracture's worst prognosis were: distal femur (supracondylar, supraintercondylar or condylar), tibial plateau (Schatzker IV-VI); the control group was made with patients which fractures were in: proximal femur (Subtrocantheric, diaphysis), tibial plateau (Schatzker I-III). Descriptive and inferential analysis were done separately and together., Results: 1578 Patients were studied. Incidence of fractures in the studied regions was of 5.9 per 10 000 persons/year (py), being of 5.7 and 4.9 per 10,000 py for men and women, respectively (p < 0.001). In subjects with age >or=50 y.o. the incidence rate was of 10.5 and 6.6 per 10 000 py for men and women, respectively. In the study patients less than 50 y.o. the incidence rate was of 1.6 per 10 000 py for both sexes (p < 0.001). The women >or=50 y.o. presented an OR of 5.1 (95% CI: 2.7-9.8, p < 0.001)., Conclusions: In this study sample, the risk of fracture of the femur and the knee was greater in men than in women. In the >or= 50 y.o. age group, the ratio was inverted, with special interest in distal femur fractures. According to sex and age, no differences were identified in tibia plateau fractures.