Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

Abstract

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the controlreproductionratio𝑅0 is less thanunity and unstable otherwise.The stability of equilibria with delaysshows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions.The existence of Hopf-bifurcation is investigated and transversality conditions are proved.The model results suggest that,as the respective delays exceeds omecritical value past the endemic equilibrium,the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.