背景:数学建模是一个迅速发展的领域,为数学家和生物学家提供了新的有趣的机会。关于COVID-19,这个强大的工具可能有助于人类防止这种疾病的传播,这严重影响了所有人的生活。
目的:这项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:本文的新模型是在卡普托意义上制定的,采用非线性时变传输速率,由十个人口类别组成,包括易感人群,感染,诊断,Ailing,认可,受感染的真实,威胁,确诊康复,治愈了,和灭绝的人。探索了新模型的唯一解的存在性,并根据平衡点讨论了相关的动力学行为,不变区域,本地和全球稳定,和基本繁殖数。要在数值上实现所提出的模型,通过结合拉普拉斯变换和连续替换方法,采用了一种有效的近似方案;此外,并进行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大流行的真实病例进行比较。通过使用模拟和测量数据之间的这些比较,我们发现分数阶的最佳值具有最小的绝对和相对误差。此外,分析研究了不同参数对病毒感染传播的影响。
结论:根据与实际数据的比较结果,我们证明了在数学建模中使用分数概念的合理性,因为新的非整数形式主义比经典框架更精确地模拟了现实。
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.
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.
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.
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.
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.