A mathematical model of infection, inflammation and immune response in an idealized bronchial tree is developed. This work is based on a model from the literature that is extended to account for the propagation dynamics of an infection between the airways. The inflammation affects the size of the airways, the air flows distribution in the airway tree, and, consequently, the oxygen transfers to blood. We test different infections outcomes and propagation speed. In the hypotheses of our model, the inflammation can reduce notably and sometimes drastically the oxygen flow to blood. Our model predicts how the air flows and oxygen exchanges reorganize in the tree during an infection. Our results highlight the links between the localization of the infection and the amplitude of the loss of oxygen flow to blood. We show that a compensation phenomena due to the reorganization of the flow exists, but that it remains marginal unless the power produced the ventilation muscles is increased. Our model forms a first step towards a better understanding of the dynamics of bronchial infections.