Zeraick Monteiro Noemi, Weber Dos Santos Rodrigo, Rodrigues Mazorche Sandro
Graduate Program in Computational Modeling, Federal University of Juiz de Fora, Juiz de Fora, Minas Gerais 36036-900, Brazil.
Department of Mathematics, Federal University of Juiz de Fora, Juiz de Fora, Minas Gerais 36036-900, Brazil.
Proc Natl Acad Sci U S A. 2024 May 7;121(19):e2322424121. doi: 10.1073/pnas.2322424121. Epub 2024 May 2.
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.
具有卷积型积分算子的演化方程已有研究历史，但在文献中，关于某些卷积核与新模型（包括延迟微分方程和分数阶微分方程）之间的联系仍存在空白。我们从逻辑模型结构出发证明，经典模型、延迟模型和分数阶模型都是使用伽马 Mittag-Leffler 记忆核的框架的特殊情况。我们讨论并分类了这种一般核的不同类型，分析了一般模型的渐近行为，并提供了数值模拟。通过参数分析给出了记忆核的详细分类。我们构建的分数阶模型具有独特的特征，因为它们保持了维度平衡，并明确地将分数阶与过去的数据点联系起来。此外，我们说明了我们的模型如何能够再现澳大利亚、巴西和秘鲁的 COVID-19 感染动态。我们的研究通过提出一个统一的框架扩展了数学建模，该框架通过利用积分 - 微分方程、分数阶或延迟微分方程以及经典常微分方程组来促进历史数据的纳入。
