It is common for many persons to have a cough for weeks, or even several months after a respiratory infection. However, your son might well have asthma with worsening the past 3 weeks precipitated by exposure to irritants or allergens or by an acute respiratory infection. Distinguishing between these possibilities will require examination by a doctor or his assistant. He should also have pulmonary function tests to establish (or not) the diagnosis of asthma. There is a form of asthma called cough-variant asthma in which the main symptom is cough rather than wheeze.
Good luck
It could be bronchitis. Which can come on with asthma. It can only be treated with antibiotics. I get it twice a year at least. But I also have a nebulizer breathing treatment so that helps calm it. He will continue to cough until his airway is clear.