## Abstract and Applied Analysis

### 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 control reproduction ratio ${R}_{\mathrm{0}}$ 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
Accepted: 18 December 2018
First available in Project Euclid: 15 March 2019

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

#### References

• J. F. Brundage and G. D. Shanks, “Deaths from bacterial pneumonia during 1918-19 influenza pandemic,” Emerging Infectious Diseases, vol. 14, no. 8, pp. 1193–1199, 2008.
• A. Domínguez, P. Ciruela, J. J. García-García et al., “Effectiveness of 7-valent pneumococcal conjugate vaccine in the prevention of invasive pneumococcal disease in children aged 7–59 months. A matched case-control study,” PloS One, vol. 12, no. 8, pp. 1–15, 2017.
• M. Lipsitch, “Bacterial vaccines and serotype replacement: lessons from haemophilus influenzae and prospects for streptococcus pneumoniae,” Emerging Infectious Diseases, vol. 5, no. 3, pp. 336–345, 1999.
• F. K. Mbabazi, J. Mugisha, and M. Kimathi, “Modeling the within-host co-infection of influenza A virus and pneumococcus,” Applied Mathematics and Computation, vol. 339, pp. 488–506, 2018.
• S. P. Sethi, “Optimal Quarantine Programmes for Controlling an Epidemic Spread,” Journal of the Operational Research Society, vol. 29, no. 3, pp. 265–268, 1978.
• G.-Q. Sun, S.-L. Wang, Q. Ren, Z. Jin, and Y.-P. Wu, “Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak,” Scientific Reports, vol. 5, Article ID 11246, pp. 1–10, 2015.
• L. Li, Z. Jin, and J. Li, “Periodic solutions in a herbivore-plant system with time delay and spatial diffusion,” Applied Mathematical Modelling, vol. 40, no. 7-8, pp. 4765–4777, 2016.
• K. L. Cooke and P. van den Driessche, “Analysis of an SEIRS epidemic model with two delays,” Journal of Mathematical Biology, vol. 35, no. 2, pp. 240–260, 1996.
• R. Xu and Z. Ma, “Global stability of a delayed SEIRS epidemic model with saturation incidence rate,” Nonlinear Dynamics, vol. 61, no. 1-2, pp. 229–239, 2010.
• S. Saha and G. Samanta, “Modelling and optimal control of HIV/AIDS prevention through PrEP and limited treatment,” Physica A: Statistical Mechanics and its Applications, vol. 516, pp. 280–307, 2019.
• G. L. Rodgers and K. P. Klugman, “Surveillance of the impact of pneumococcal conjugate vaccines in developing countries,” Human Vaccines & Immunotherapeutics, vol. 12, no. 2, pp. 417–420, 2016.
• K. L. O'Brien, L. J. Wolfson, J. P. Watt et al., “Burden of disease caused by Streptococcus pneumoniae in children younger than 5 years: global estimates,” The Lancet, vol. 374, no. 9693, pp. 893–902, 2009.
• P.-Y. Iroh Tam, A. E. Sadoh, and S. K. Obaro, “A meta-analysis of antimicrobial susceptibility profiles for pneumococcal pneumonia in sub-Saharan Africa,” Paediatrics and International Child Health, vol. 38, no. 1, pp. 7–15, 2018.
• M. T. Waheed, M. Sameeullah, F. A. Khan et al., “Need of cost-effective vaccines in developing countries: What plant biotechnology can offer?” SpringerPlus, vol. 5, no. 1, pp. 1–9, 2016.
• V. Rémy, N. Largeron, S. Quilici, and S. Carroll, “The economic value of vaccination: why prevention is wealth,” Journal of Market Access & Health Policy, vol. 3, no. 1, pp. 1–4, 2015.
• G. Samanta and S. P. Bera, “Analysis of a Chlamydia epidemic model with pulse vaccination strategy in a random environment,” Nonlinear Analysis-Modelling and Control, vol. 23, no. 4, pp. 457–474, 2018.
• G. P. Samanta, P. Sen, and A. Maiti, “A delayed epidemic model of diseases through droplet infection and direct contact with saturation incidence and pulse vaccination,” Systems Science & Control Engineering, vol. 4, no. 1, pp. 320–333, 2016.
• J. Gjorgjieva, K. Smith, G. Chowell, F. Sánchez, J. Snyder, and C. Castillo-Chavez, “The Role of Vaccination in the Control of SARS,” Mathematical Biosciences and Engineering, vol. 2, no. 4, pp. 753–769, 2005.
• P. R. S. Rao and M. N. Kumar, “A dynamic model for infectious diseases: the role of vaccination and treatment,” Chaos, Solitons & Fractals, vol. 75, pp. 34–49, 2015.
• Q. Liu, B. Li, and M. Sun, “Global Dynamics of an SIRS Epidemic Model with Distributed Delay on Heterogeneous Network,” Mathematical Problems in Engineering, vol. 2017, Article ID 6376502, 9 pages, 2017.
• M. Li and X. Liu, “An SIR epidemic model with time delay and general nonlinear incidence rate,” Abstract and Applied Analysis, vol. 2014, Article ID 131257, 8 pages, 2014.
• H. Zhao and M. Zhao, “Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporatingmedia coverage with time delay,” Journal of Biological Dynamics, vol. 11, no. 1, pp. 8–24, 2017.
• A. K. Misra, S. N. Mishra, A. L. Pathak, P. Misra, and R. Naresh, “Modeling the effect of time delay in controlling the carrier dependent infectious disease - Cholera,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11547–11557, 2012.
• L. Zuo and M. Liu, “Effect of awareness programs on the epidemic outbreaks with time delay,” Abstract and Applied Analysis, vol. 2014, Article ID 940841, 8 pages, 2014.
• T. Zhang, X. Meng, and T. Zhang, “SVEIRS: A new epidemic disease model with time delays and impulsive effects,” Abstract and Applied Analysis, vol. 2014, Article ID 542154, 15 pages, 2014.
• G. O. Agaba, Y. N. Kyrychko, and K. B. Blyuss, “Time-delayed SIS epidemic model with population awareness,” Ecological Complexity, vol. 31, pp. 50–56, 2017.
• S. Sharma, A. Mondal, A. K. Pal, and G. P. Samanta, “Stability analysis and optimal control of avian influenza virus A with time delays,” International Journal of Dynamics and Control, vol. 6, no. 3, pp. 1351–1366, 2018.
• K. L. Sutton, H. T. Banks, and C. Castillo-Chavez, “Estimation of invasive pneumococcal disease dynamics parameters and the impact of conjugate vaccination in,” Tech. Rep. CRSC-TR07-15, North Carolina State University, Center for Research in Scientific Computation, Raleigh, NC, USA, 2007.
• Uganda Bureau of Statistics, “The national population and housing census 2014,” National Analytical Report, Uganda Bureau of Statistics, Kampala, Uganda, 2017.
• C. Ngari, G. Pokhariyal, and J. Koske, “Analytical Model for Childhood Pneumonia, a Case Study of Kenya,” British Journal of Mathematics & Computer Science, vol. 12, pp. 1–28, 2016.
• C. G. Ngari, D. M. Malonza, and G. G. Muthuri, “A model for childhood pneumonia dynamics,” Journal of Life Sciences Research, vol. 1, no. 2, pp. 31–40, 2014.
• A. Melegaro, Y. H. Choi, R. George, W. J. Edmunds, E. Miller, and N. J. Gay, “Dynamic models of pneumococcal carriage and the impact of the Heptavalent Pneumococcal Conjugate Vaccine on invasive pneumococcal disease,” BMC Infectious Diseases, vol. 10, no. 90, pp. 1–15, 2010.
• A. Lindstrand, Impact of pneumococcal conjugate vaccine onpneumococcal disease, carriage and serotype distribution: com-parative studies in Sweden and Uganda, Inst för folkhäl-sovetenskap/Dept of Public Health Sciences, Solna, Sweden, 2016.
• N. J. A. White, C. V. Spain, C. C. Johnson, L. M. Kinlin, and D. N. Fisman, “Let the sun shine in: effects of ultraviolet radiation on invasive pneumococcal disease risk in Philadelphia,” BMC Infectious Diseases, vol. 9, no. 1, pp. 1–11, 2009.
• K. Källander, H. Hildenwall, P. Waiswa, E. Galiwango, S. Petersona, and G. Pariyob, “Delayed care seeking for fatal pneumonia in children aged under five years in Uganda: A case-series study,” Bulletin of the World Health Organization, vol. 86, no. 5, pp. 332–338, 2008.
• G.-H. Li and Y.-X. Zhang, “Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates,” PLoS ONE, vol. 12, no. 4, pp. 1–28, 2017.
• O. Sharomi, C. N. Podder, A. B. Gumel, E. H. Elbasha, and J. Watmough, “Role of incidence function in vaccine-induced backward bifurcation in some HIV models,” Mathematical Biosciences, vol. 210, no. 2, pp. 436–463, 2007.
• C. Bottomley, A. Roca, P. C. Hill, B. Greenwood, and V. Isham, “A mathematical model of serotype replacement in pneumococcal carriage following vaccination,” Journal of the Royal Society Interface, vol. 10, no. 89, pp. 1–8, 2013.
• G. T. Tilahun, O. D. Makinde, and D. Malonza, “Modelling and optimal control of pneumonia disease with cost-effective strategies,” Journal of Biological Dynamics, vol. 11, no. suppl. 2, pp. 400–426, 2017.
• M. Bodnar, “The nonnegativity of solutions of delay differential equations,” Applied Mathematics Letters, vol. 13, no. 6, pp. 91–95, 2000.
• X. Yang, L. Chen, and J. Chen, “Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models,” Computers & Mathematics with Applications, vol. 32, no. 4, pp. 109–116, 1996.
• L. D. Nagy and D. D. Lisa, Epidemic Models with Pulse Vaccination And Time Delay [M.S. thesis], University of Waterloo, Waterloo, Canada, 2011.
• P. Van Den Driessche and J. Watmough, “Further notes on the basic reproduction number,” in Lecture Notes in Mathematics, F. Brauer, P. Van den Driessche, and J. Wu, Eds., vol. 1945, pp. 159–178, Springer, Berlin, Germany, 2008.
• Y. Song and J. Wei, “Bifurcation analysis for Chen's system with delayed feedback and its application to control of chaos,” Chaos, Solitons & Fractals, vol. 22, no. 1, pp. 75–91, 2004.
• K. Wesley, R. K. Titus, B. Jacob, and L. C. Robert, “Modeling the effects of time delay on HIV-1 in vivo dynamics in the presence of ARVs,” Science Journal of Applied Mathematics and Statistics, vol. 3, no. 4, pp. 204–213, 2015.
• K. L. Sutton, H. T. Banks, and C. Castillo-Chávez, “Estimation of invasive pneumococcal disease dynamics parameters and the impact of conjugate vaccination in Australia,” Mathematical Biosciences and Engineering, vol. 5, no. 1, pp. 175–204, 2008. \endinput