This study deals with a theoretical analysis of the integrability properties and analytical solutions of an initial-value problem for a SIRD model with a constant amount of alive population (SIRD-CAAP), which is in the form of a fourth-dimensional and first-order coupled system of nonlinear ordinary differential equations, by using the partial Hamiltonian method. This research represents a COVID-19 study as a real-world problem by using the analytical results obtained in the study. The first integrals and the associated exact analytical solutions are investigated of the model with respect to algebraic relations among the model parameters. Then, for both cases, the dynamical behaviors of the model based on the analytical solutions are analyzed, and the graphical representations of the closed-form solutions are demonstrated and compared. In addition, it is shown that the SIRD-CAAP model can be decoupled based on its first integrals for all cases from the mathematical perspective point of view. Furthermore, the periodicity properties and the classification of the regimes of the solutions with respect to the model parameter constraints are discussed. Finally, the COVID-19 applications are given using the data related to the different countries.