Mathematical modeling of the Coronavirus (Covid-19) transmission dynamics using classical and fractional derivatives

This study focuses on formulating and analysing a COVID-19 transmission dynamics model using integer and fractional-order derivatives in the Caputo sense. The model considers two doses of vaccination, confinement, and treatment with limited resources. The control reproduction number is computed and the asymptotic stability analysis of the disease-free equilibrium is proved. We also prove the existence of at least one endemic equilibrium whenever Rc>1. Using real data from Germany, we calibrate our models by performing parameter estimations. We find that the control reproduction number is approximately equal to 1. 90, which implies that the disease remains endemic in Germany. We also perform global sensitivity analysis by computing partial rank correlation coefficients (PRCC) between Rc (respectively compartments of infected individuals) and each model parameter. By fixing vaccine coverage at 70%, we observe that it might be more effective to increase the vaccine efficacy than increasing the numbers of vaccinated people. After that, we formulate the corresponding fractional model in the Caputo sense, proving positivity, boundedness, existence, and uniqueness of solutions. We calculate the control reproduction number of the fractional model, and prove the asymptotic stability of the DFE, and existence of at least one endemic equilibrium point. We find from numerical simulations, that for a long-term forecasting, it seems better to use a fractional derivative in the range (0, 0. 87] than using just ordinary derivatives. Indeed, for this range of the fractional-order parameter, daily detected cases are closer to those the classical model predicts.