This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang‐Gao‐Tenreiro Machado‐Baleanu and Yang‐Abdel‐Aty‐Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional‐exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang‐Abdel‐Aty‐Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang‐Abdel‐Aty‐Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.

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Recently, novel coronavirus is a serious global issue and having a negative impact on the economy of the whole world. Like other countries, it also effected the economy and people of Pakistan. According to the publicly reported data, the first case of novel corona virus in Pakistan was reported on 27th February 2020. The aim of the present study is to describe the mathematical model and dynamics of COVID-19 in Pakistan. To investigate the spread of coronavirus in Pakistan, we develop the SEIR time fractional model with newly, developed fractional operator of Atangana–Baleanu. We present briefly the analysis of the given model and discuss its applications using world health organization (WHO) reported data for Pakistan. We consider the available infection cases from 19th March 2020, till 31st March 2020 and accordingly, various parameters are fitted or estimated. It is worth noting that we have calculated the basic reproduction number $${\mathfrak{R}}_{0} \approx 2.30748$$ R 0 ≈ 2.30748 which shows that virus is spreading rapidly. Furthermore, stability analysis of the model at disease free equilibrium DFE and endemic equilibriums EE is performed to observe the dynamics and transmission of the model. Finally, the AB fractional model is solved numerically. To show the effect of the various embedded parameters like fractional parameter $$\alpha$$ α on the model, various graphs are plotted. It is worth noting that the base of our investigation, we have predicted the spread of disease for next 200 days.

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