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Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations. [ABSTRACT FROM AUTHOR]

The nonlinear fractional stochastic differential equation approach with Hurst parameter H within interval H ∈ (0 , 1) to study the time evolution of the number of those infected by the coronavirus in countries where the number of cases is large as Brazil is studied. The rises and falls of novel cases daily or the fluctuations in the official data are treated as a random term in the stochastic differential equation for the fractional Brownian motion. The projection of novel cases in the future is treated as quadratic mean deviation in the official data of novel cases daily since the beginning of the pandemic up to the present. Moreover, the rescaled range analysis (RS) is employed to determine the Hurst index for the time series of novel cases and some statistical tests are performed with the aim to determine the shape of the probability density of novel cases in the future. [ABSTRACT FROM AUTHOR]