Introduction

Mathematical modelling is a rapidly expanding field that offers new and interesting opportunities for both mathematicians and biologists. Concerning COVID-19, this powerful tool may help humans to prevent the spread of this disease, which has affected the livelihood of all people badly.

Objectives

The main objective of this research is to explore an efficient mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework.

Methods

The new model in this paper is formulated in the Caputo sense, employs a nonlinear time-varying transmission rate, and consists of ten population classes including susceptible, infected, diagnosed, ailing, recognized, infected real, threatened, diagnosed recovered, healed, and extinct people. The existence of a unique solution is explored for the new model, and the associated dynamical behaviours are discussed in terms of equilibrium points, invariant region, local and global stability, and basic reproduction number. To implement the proposed model numerically, an efficient approximation scheme is employed by the combination of Laplace transform and a successive substitution approach; besides, the corresponding convergence analysis is also investigated.

Results

Numerical simulations are reported for various fractional orders, and simulation results are compared with a real case of COVID-19 pandemic in Italy. By using these comparisons between the simulated and measured data, we find the best value of the fractional order with minimum absolute and relative errors. Also, the impact of different parameters on the spread of viral infection is analyzed and studied.

Conclusion

According to the comparative results with real data, we justify the use of fractional concepts in the mathematical modelling, for the new non-integer formalism simulates the reality more precisely than the classical framework.

中文翻译：

---

具有实际案例研究的广义分数模型的动力学行为和稳定性分析

介绍

数学建模是一个迅速发展的领域，它为数学家和生物学家提供了新的有趣的机会。关于 COVID-19，这种强大的工具可以帮助人类预防这种严重影响所有人生计的疾病的传播。

目标

本研究的主要目的是探索一种有效的数学模型，用于在广义分数框架中研究 COVID-19 动力学。

方法

本文中的新模型是在 Caputo 意义上制定的，采用非线性时变传输率，并由十个人口类别组成，包括易感、感染、诊断、患病、识别、感染真实、威胁、诊断恢复、治愈和灭绝的人。探索了新模型唯一解的存在性，并从平衡点、不变区域、局部和全局稳定性以及基本再生数等方面讨论了相关的动力学行为。为了在数值上实现所提出的模型，结合拉普拉斯变换和逐次替换方法采用了一种有效的近似方案；此外，还研究了相应的收敛性分析。

结果

报告了各种分数阶的数值模拟，并将模拟结果与意大利 COVID-19 大流行的真实案例进行了比较。通过使用模拟数据和测量数据之间的这些比较，我们找到了具有最小绝对和相对误差的分数阶的最佳值。此外，分析和研究了不同参数对病毒感染传播的影响。

结论

根据与真实数据的比较结果，我们证明在数学建模中使用分数概念是合理的，因为新的非整数形式比经典框架更精确地模拟现实。