A dynamical model of SARS-CoV-2 in fractional derivative using the cases of coronavirus of the fourth wave is presented. We construct basically the model in an integer case, and later it is extended to a fractional-order system by applying the Atangana–Baleanu operator definition. We give some background definitions and results for the fractional-order model. We present for the disease-free case that the model is locally asymptotically stable when ${{ℛ}}_0<1$. The global dynamics of the fractional model are given when ${{ℛ}}_0\le 1$ for the disease-free case. The model is further extended to fractional stochastic piecewise equations in the Atangana–Baleanu case. The reported cases from the fourth wave in Pakistan starting from July 1 up to November 16, 2021 are considered for the estimation of the parameters. We fitted our model to the suggested data and obtained the numerical value of the basic reproduction number ${{ℛ}}_0\approx 0.9775$ for fractional order. We give the data fitting to both the fractional and piecewise stochastic differential equations, and show them both as having a good fitting to the data. We use further the numerical values of the model parameters and present its numerical results graphically using the effective numerical approaches. Some sensitive parameters that are reasonable for disease eliminations are used to obtain the graphical results.

提出了使用第四波冠状病毒病例的分数阶 SARS-CoV-2 动力学模型。我们基本上在整数情况下构建模型，然后通过应用 Atangana–Baleanu 算子定义将其扩展到分数阶系统。我们给出了分数阶模型的一些背景定义和结果。对于无病病例，我们提出该模型在以下情况下局部渐近稳定${{ℛ}}_0<\mathrm{1个}$. 分数模型的全局动力学在以下时间给出${{ℛ}}_0\le \mathrm{1个}$对于无病病例。该模型进一步扩展到 Atangana–Baleanu 案例中的分数阶随机分段方程。从 2021 年 7 月 1 日到 2021 年 11 月 16 日，巴基斯坦第四波报告的病例被考虑用于参数的估计。我们将我们的模型拟合到建议的数据并获得基本再生数的数值${{ℛ}}_0\approx 0.977\mathrm{5个}$对于分数阶。我们将数据拟合到分数阶和分段随机微分方程，并显示它们都与数据具有良好的拟合。我们进一步使用模型参数的数值，并使用有效的数值方法以图形方式显示其数值结果。一些对于疾病消除是合理的敏感参数用于获得图形结果。