Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays
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Date
2019-02-03
Authors
Kamugisha Mbabazi, Fulgensia
Mugisha, Joseph Y. T.
Kimathi, Mark
Journal Title
Journal ISSN
Volume Title
Publisher
Hindawi
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 control reproduction ratio Ro is less than unity and unstable otherwise. The stability of equilibria with delays shows 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 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.
Description
Keywords
Model of pneumococcal pneumonia, Stability theory of delay differential equations, Hopf-bifurcation