Officially, Alex is not in the labor force.
Given that he is not actually employed (doesn't have an actual job), he cannot be considered part of the labor force. If he had some kind of job other than taking care of his children (even if it's a freelance job he can do from his home), then he would be in the labor force. However, this way, he is not and fall into the same category as unemployed students, for example.
