Sequences of epidemic waves have been observed in past influenza pandemics, such as the Spanish influenza. Possible explanations may be sought either in mechanisms altering the structure of the network of contacts, such as those induced by changes in the rates of movement of people or by public health measures, or in the genetic drift of the influenza virus, since the appearance of new strains can reduce or eliminate herd immunity. The pandemic outbreaks may also be influenced by coinfection with other acute respiratory infections (ARI) that increase transmissibility of influenza virus (by coughing, sneezing, running nose). In fact, some viruses (e.g., Rhinovirus and Adenovirus) have been found to induce "clouds" of bacteria and increase the transmissibility of Staphylococcus aureus. Moreover, Rhinovirus and Adenovirus were detected in patients during past pandemics, and their presence is linked to superspreading events. In this paper, by assuming increased transmissibility in coinfected individuals, we propose and study a model where multiple pandemic waves are triggered by coinfection with ARI. The model agrees well with mortality excess data during the 1918 pandemic influenza, thereby providing indications for potential pandemic mitigation.