We explore a standard epidemiological model, known as the SIRD model, to study the COVID-19 infection in India, and a few other countries around the world. We use (a) the stable cumulative infection of various countries and (b) the number of infection versus the tests carried out to evaluate the model. The time-dependent infection rate is set in the model to obtain the best fit with the available data. The model is simulated aiming to project the probable features of the infection in India, various Indian states, and other countries. India imposed an early lockdown to contain the infection that can be treated by its healthcare system. We find that with the current infection rate and containment measures, the total active infection in India would be maximum at the end of June or beginning of July 2020. With proper containment measures in the infected zones and social distancing, the infection is expected to fall considerably from August. If the containment measures are relaxed before the arrival of the peak infection, more people from the susceptible population will fall sick as the infection is expected to see a threefold rise at the peak. If the relaxation is given a month after the peak infection, a second peak with a moderate infection will follow. However, a gradual relaxation of the lockdown started well ahead of the peak infection, leads to a nearly twofold increase of the peak infection with no second peak. The model is further extended to incorporate the infection arising from the population showing no symptoms. The preliminary finding suggests that random testing needs to be carried out within the asymptomatic population to contain the spread of the disease. Our model provides a semi-quantitative overview of the progression of COVID-19 in India, with model projections reasonably replicating the current progress. The projection of the model is highly sensitive to the choice of the parameters and the available data.