Can we use differential equations for population that is discrete?
You can make the continuous approximation when the population size is large. As mentioned by arboviral, there are algorithms that allow you to perform stochastic simulations with discrete variables. However, these are computationally much more intensive than integration of ODEs. Moreover, analytical solutions for the master-equations (time evolution of probabilities) are very difficult to calculate. Therefore, whenever possible, people go for ODE-based continuous models.

These models would give incorrect representation of the dynamics for small populations and fail to explain phenomena such as extinction. However, they can explain the dynamics of large populations fairly well. So the choice of the modelling approach depends on the questions you want to ask and the complexity/computational cost of the model.