Ponente
Descripción
Understanding demographic dynamics is essential for socioeconomic planning and for analyzing demographic transition processes observed in countries such as Brazil. In this work, we propose an age-structured model based on the McKendrick-von Foerster equation to describe the evolution of the Brazilian population from 2000 to 2022, using demographic data provided by the United Nations (United Nations, World Population Prospects 2024, UN, 2024).
Annual mortality data for single ages from 0 to 100 years and fertility data for ages between 15 and 49 years were considered. Mortality rates were adjusted using the Gompertz function, whose parameters describe baseline mortality and its growth with age, while fertility was modeled through a shifted Gamma distribution, allowing the asymmetric behavior and temporal displacement of age-specific fertility curves to be represented.
The model parameters were estimated using the least squares method considering the period from 2000 to 2015 as the calibration window. The resulting coefficients of determination $R^2$ indicated high agreement between the parametric models and the observed data, with values above 0.90.
Time series were then constructed for each mortality and fertility parameter, allowing the model coefficients to evolve continuously over time. This approach enabled the incorporation of demographic transition trends directly into the age-structured population dynamics, including changes in fertility intensity and the progressive shift of fertility profiles toward older ages. Population dynamics were described by the McKendrick-von Foerster equation (Martcheva, An Introduction to Mathematical Epidemiology, Springer, 2015),
$$\frac{\partial n(a,t)}{\partial t}+\frac{\partial n(a,t)}{\partial a} = -\mu(a,t) n(a,t)$$
with a birth boundary condition containing the fertility rates $\beta(a,t)$, a correction factor based on the female population proportion and historical calibration data.
The model was solved numerically and extrapolated to the period from 2015 to 2022. Validation was performed by comparison with observed United Nations data for 2022, resulting in a final relative error for the 2022 population estimate of approximately 1.13\%. In addition to consistently reproducing the Brazilian population aging process, the temporal evolution of fertility parameters indicated a progressive shift of fertility curves toward older ages, accompanied by a reduction in fertility peak intensity over the analyzed period.
The results indicate that incorporating time-dependent demographic parameters into the McKendrick-von Foerster framework provides a consistent approach for representing demographic transition processes and extrapolating Brazilian population dynamics in medium-term scenarios.