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In this paper, we provide a unique, cost-effective numerical method for solving the SIR model of a COVID-19 disease using the method of Taylor wavelets and collocation technique. The model’s mathematical structure is represented by an ordered collection of ordinary differential equations that are nonlinear. We use the functional matrix of integration of Fibonacci wavelets to translate the given model into a set of computational equations, which we next simplify using the Newton–Raphson method. The given model is solved using the proposed method under various conditions to analyze the outcomes graphically and numerically as well. To demonstrate the efficiency and easy applicability of the method, we have also made a numerical comparison of the solution with some other methods in tables. The behavior of the approximate solutions of susceptible, infectible, and recovery curves under the proposed method is analyzed. MATLAB software is used to perform all the related calculations.