Cancer is a systematic disease, which can affect any part or tissue of the living body. In treating the cancer, different methods are employed including radiotherapy, which may be prescribed for every patient during the therapy. In radiotherapy, ionizing radiations are exploited to damage the cells and when the ionizing radiation interact with the cell, it can result in the initial and long-term biophysical effects. These initial effects that are caused due to the physical processes such as ionization and excitation as well as chemical radicals, can result in the DNA damage and structural changes in it. Understanding the damage process of the ionizing radiations requires a knowledge of the effects of the molecular, physical, and chemical interactions of the radiation in the cell and DNA. The damages of the ionizing radiation include single- and double-strand breaks as well as base damage. The damaged DNAs can be repaired through some processes within the cell. Those DNA damages, especially of double-strand-break type, that are mis-repaired or unrepaired, can result in the cell death. This process plays a key role in killing the cancer cells and curing the cancer. One can study the effect of ionizing radiations in causing the DNA damage through empirical methods and simulations. Monte Carlo simulations make it possible to calculate the DNA damage, damage types, and how numerous they are. The purpose of the present investigation is to performing simulations of the involved physical and chemical processes as well as calculating the DNA damage from different radiations more precisely and thoroughly, using Geant4-DNA code. Herewith, by simulating the physical and chemical processes of particles in water (equivalent to the cell environment), we calculate the initial DNA damage caused by the interaction of primary electrons of 100-4500 eV and primary protons of 0.5-20 MeV energy. Hence, by considering the shares of the direct damage of physical processes as well as the indirect damage of chemical processes, we calculate different types of simple and complex single- and double-strand breaks. In addition, we calculate the break efficiency specifically for double-strand breaks in DNA and cell, and hence compare the results with the experimental and other simulation results. The efficiency results of this investigation for protons and electrons above 500 eV show reasonable agreement with the experimental and majority of the simulation results. For initial electron radiations with energies below 500 eV, there are differences between our results and other experimental and simulation results. These differences are due to the differences of the ionization and excitation cross sections of electrons at low energies in various codes, as compared to the experiment, as well as the difference in models, processes, and chemical reaction-rates implemented in Geant4-DNA, as compared to other codes. Furthermore, other parameters such as the DNA geometry, threshold energy for direct-damage registration (investigated in this work too), and the simulation time can also be influential in creating these differences.