This work is devoted to present temporal-only and spatio-temporal COVID-19 epidemic models when virus mutations and vaccination influences are considered. Firstly, the proposed non-diffusive COVID-19 model is introduced. The nonlinear incidence rate is employed to better model the strict measures forced by governmental authorities to control pandemic spread. The immunity acquired by vaccinations are assumed to be incomplete for realistic considerations. The existence, uniqueness and continuous dependence on initial conditions are studied for the solution. The study of stability along with bifurcation analysis are carried out to investigate the influences of variations in model's parameters. Moreover, the basic reproduction number is obtained for the proposed model. The stability regions for equilibrium points are depicted in space of parameters to explore their effects. Secondly, the diffusive version of the model is considered where possible occurrence of Turing instability is investigated. Finally, numerical simulations are employed to verify theoretical results of the work.