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In this paper, a mathematical model of pneumococcal pneumonia with timedelays is proposed. The stability theory of delay differential equations isused to analyze the model. The results show that the disease-free equilibriumis asymptotically stable if the control reproduction ratio R0 is less thanunity and unstable otherwise. The stability of equilibria with delays showsthat the endemic equilibrium is locally stable without delays and stable if thedelays are under conditions. The existence of Hopf-bifurcation is investigatedand transversality conditions proved. The model results suggest that as therespective delays exceed some critical value past the endemic equilibrium, thesystem loses stability through the process of local birth or death ofoscillations. Further, a decrease or an increase in the delays leads toasymptotic stability or instability of the endemic equilibrium respectively.The analytical results, are supported by numerical simulations.  Keywords: Time delay, Pneumococcal pneumonia, Vaccination, Stability,Hopf-bifurcation

Hopf-bifurcation analysis of pneumococcal pneumonia with time delays