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In this paper, new model on novel coronavirus disease (COVID-19) with four compartments including susceptible, exposed, infected, and recovered class with fractal-fractional derivative is proposed. Here, Banach and Leray–Schauder alternative type theorems are used to establish some appropriate conditions for the existence and uniqueness of the solution. Also, stability is needed in respect of the numerical solution. Therefore, Ulam–Hyers stability using nonlinear functional analysis is used for the proposed model. Moreover, the numerical simulation using the technique of fundamental theorem of fractional calculus and the two-step Lagrange polynomial known as fractional Adams-Bashforth (AB) method is proposed. The obtained results are tested on real data of COVID-19 outbreak in Malaysia from 25 January till 10 May 2020. The numerical simulation of the proposed model has performed in terms of graphs for different fractional-order q and fractal dimensions p via number of considered days of disease spread in Malaysia. Since COVID-19 transmits rapidly, perhaps, the clear understanding of transmission dynamics of COVID-19 is important for countries to implement suitable strategies and restrictions such as Movement Control Order (MCO) by the Malaysian government, against the disease spread. The simulated results of the presented model demonstrate that movement control order has a great impact on the transmission dynamics of disease outbreak in Malaysia. It can be concluded that by adopting precautionary measures as restrictions on individual movement the transmission of the disease in society is reduced. In addition, for such type of dynamical study, fractal-fractional calculus tools may be used as powerful tools to understand and predict the global dynamics of the mentioned disease in other countries as well. [ABSTRACT FROM AUTHOR]