Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

Abstract

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. Te stability theory of delay differential equations is used to analyze the model. Te results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratio �0 is less than unity and unstable otherwise. Te stability of equilibrium with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. Te existence of Hopf-bifurcation is investigated and trans-versatility conditions are proved. Te model results suggest that, as the respective delays exceed some critical 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.