Neural network regression problem, integer output neuron constraint
You are correct to approach this as a regression problem, mostly because you are interested in the order of your outputs. For example if there are 1000 people present and you predict 1005, it's a better prediction than 7005. If you were treating this as a classification problem, both of these would be interpreted as missclassifications. 

The most practical way to approach your constraint is to round the output of your neural network, i.e. if it outputs 3456.78, round it up to 3457. You need to have a linear activation function to be able to use the NN for regression.