The flu is an upper respiratory disease and bronchitis is a lower respiratory disease.
Infection of the respiratory tract by various pathogens such as viruses and bacteria, or physical and chemical irritation resulting in damage to the airway mucosa, can lead to increased sensitivity of the airway mucosa, followed by cough symptoms when the respiratory tract is irritated by cold air, fumes, etc.
Usually, with treatment or rest, the infection can be removed, the airways can repair themselves and the cough symptoms will improve.
However, if the cough persists after a respiratory infection, especially if it has been present for more than three weeks, it is advisable to seek medical attention to determine whether the cough is simply a result of the infection or a combination of other respiratory conditions.