Abstract and Applied Analysis

Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

Fulgensia Kamugisha Mbabazi, Joseph Y. T. Mugisha, and Mark Kimathi

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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 R0 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.

Article information

Source
Abstr. Appl. Anal., Volume 2019 (2019), Article ID 3757036, 21 pages.

Dates
Received: 12 September 2018
Accepted: 18 December 2018
First available in Project Euclid: 15 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1552615229

Digital Object Identifier
doi:10.1155/2019/3757036

Citation

Mbabazi, Fulgensia Kamugisha; Mugisha, Joseph Y. T.; Kimathi, Mark. Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays. Abstr. Appl. Anal. 2019 (2019), Article ID 3757036, 21 pages. doi:10.1155/2019/3757036. https://projecteuclid.org/euclid.aaa/1552615229


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