Modeling Influenza–Streptococcus pneumoniae Co-Infection: Multistage Progression and Competitive Exclusion

Co-infection between influenza and Streptococcus pneumoniae is an important public health concern, since influenza can increase susceptibility to secondary bacterial invasion and enhance bacterial transmission. We develop and analyze a compartmental model of influenza–pneumococcal co-infection with eleven epidemiological compartments, incorporating imperfect vaccination, quarantine, and a three-stage pneumococcal progression from colonization to invasive disease. We establish the positivity and boundedness of solutions, determine the equilibria, and derive, via the next-generation matrix method, the sub-model reproduction numbers R0I and R0P together with the composite threshold R0=max{R0I,R0P}. Each single infection undergoes a forward bifurcation: the disease-free equilibrium is locally asymptotically stable when the corresponding reproduction number is below unity, and a unique endemic equilibrium—globally asymptotically stable in the pneumococcal sub-model, established by a Goh–Volterra Lyapunov function—emerges above it. When both thresholds exceed unity, the two infections compete for the shared susceptible pool and undergo competitive exclusion: one infection persists while the other, together with the co-infected class, is eliminated. We show that the outcome is governed not by the disease-free reproduction numbers but by the invasion reproduction numbers evaluated at the single-infection boundary equilibria, so that even equal disease-free thresholds do not yield coexistence. Numerical simulations confirm the analytical results and quantify the effect of quarantine and vaccination, providing a framework for assessing control of interacting respiratory infections.