Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays
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...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2019/3757036 |
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| author | Fulgensia Kamugisha Mbabazi Joseph Y. T. Mugisha Mark Kimathi |
| author_facet | Fulgensia Kamugisha Mbabazi Joseph Y. T. Mugisha Mark Kimathi |
| author_sort | Fulgensia Kamugisha Mbabazi |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-96712b99202a468485ad8a7eedc25c19 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-96712b99202a468485ad8a7eedc25c192025-02-03T06:11:06ZengWileyAbstract and Applied Analysis1085-33751687-04092019-01-01201910.1155/2019/37570363757036Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time DelaysFulgensia Kamugisha Mbabazi0Joseph Y. T. Mugisha1Mark Kimathi2Department of Mathematics, Pan African University Institute of Basic Sciences, Technology and Innovation, P.O. Box 62000–00200, Nairobi, KenyaDepartment of Mathematics, Makerere University, P.O. Box 7062, Kampala, UgandaDepartment of Mathematics, Statistics and Actuarial Sciences, Machakos University, P.O. Box, 136–90100, Machakos, KenyaIn 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.http://dx.doi.org/10.1155/2019/3757036 |
| spellingShingle | Fulgensia Kamugisha Mbabazi Joseph Y. T. Mugisha Mark Kimathi Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays Abstract and Applied Analysis |
| title | Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays |
| title_full | Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays |
| title_fullStr | Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays |
| title_full_unstemmed | Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays |
| title_short | Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays |
| title_sort | hopf bifurcation analysis of pneumococcal pneumonia with time delays |
| url | http://dx.doi.org/10.1155/2019/3757036 |
| work_keys_str_mv | AT fulgensiakamugishambabazi hopfbifurcationanalysisofpneumococcalpneumoniawithtimedelays AT josephytmugisha hopfbifurcationanalysisofpneumococcalpneumoniawithtimedelays AT markkimathi hopfbifurcationanalysisofpneumococcalpneumoniawithtimedelays |