A COVID-19 deterministic compartmental mathematical model with different types of quarantine and isolation is proposed to investigate their role in the disease transmission dynamics. The quarantine compartment is subdivided into short and long quarantine classes, and the isolation compartment is subdivided into tested and non-tested home-isolated individuals and institutionally isolated individuals. The proposed model has been fully analyzed. The analysis includes the positivity and boundedness of solutions, calculation of the control reproduction number and its relation to all transmission routes, existence and stability analysis of disease-free and endemic equilibrium points and bifurcation analysis. The model parameters have been estimated using a dataset for Oman. Using the fitted parameters, the estimated values of the control reproduction number and the contribution of all transmission routes to the reproduction number have been calculated. Sensitivity analysis of the control reproduction number to model parameters has also been performed. Finally, numerical simulations to demonstrate the effect of some model parameters related to the different types of quarantine and isolation on the disease transmission dynamics have been carried out, and the results have been demonstrated graphically.