- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Computational and Mathematical Methods in Medicine
Volume 2012 (2012), Article ID 826052, 8 pages
Global Stability Analysis of SEIR Model with Holling Type II Incidence Function
1Department of Mathematics, The Hashemite University, Zarqa 13115, Jordan
2Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
Received 27 June 2012; Revised 9 August 2012; Accepted 12 September 2012
Academic Editor: Jacek Waniewski
Copyright © 2012 Mohammad A. Safi and Salisu M. Garba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. M. Anderson and R. M. May, Eds., Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York, NY, USA, 1991.
- E. Beretta and D. Breda, “An SEIR epidemic model with constant latency time and infectious period,” Mathematical Biosciences and Engineering, vol. 8, no. 4, pp. 931–952, 2011.
- K. Dietz, “Transmission and control of arbovirus disease,” in Epidemiology, K. L. Cooke, Ed., p. 104, SIAM, Philadelphia, Pa, USA, 1975.
- H. W. Hethcote, “Mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000.
- C. Rebelo, A. Margheri, and N. Bacaër, “Persistence in seasonally forced epidemiological models,” Journal of Mathematical Biology, vol. 64, no. 6, pp. 933–949, 2012.
- Z. Feng and J. X. Velasco-Hernández, “Competitive exclusion in a vector-host model for the dengue fever,” Journal of Mathematical Biology, vol. 35, no. 5, pp. 523–544, 1997.
- S. M. Garba and A. B. Gumel, “Mathematical recipe for HIV elimination in Nigeria,” Journal of the Nigerian Mathematical Society, vol. 29, pp. 1–66, 2010.
- S. M. Garba, A. B. Gumel, and J. M. S. Lubuma, “Dynamically-consistent non-standard finite difference method for an epidemic model,” Mathematical and Computer Modelling, vol. 53, no. 1-2, pp. 131–150, 2011.
- A. B. Gumel, “Global dynamics of a two-strain avian influenza model,” International Journal of Computer Mathematics, vol. 86, no. 1, pp. 85–108, 2009.
- H. W. Hethcote and H. R. Thieme, “Stability of the endemic equilibrium in epidemic models with subpopulations,” Mathematical Biosciences, vol. 75, no. 2, pp. 205–227, 1985.
- M. A. Safi and A. B. Gumel, “Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine,” Computers and Mathematics with Applications, vol. 61, no. 10, pp. 3044–3070, 2011.
- M. A. Safi and A. B. Gumel, “Qualitative study of a quarantine/isolation model with multiple disease stages,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1941–1961, 2011.
- M. A. Safi, M. Imran, and A. B. Gumel, “Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation,” Theory in Biosciences, vol. 131, no. 1, pp. 19–30, 2012.
- 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.
- T. Zhang and Z. Teng, “On a nonautonomous SEIRS model in epidemiology,” Bulletin of Mathematical Biology, vol. 69, no. 8, pp. 2537–2559, 2007.
- L. D. Wang and J. Q. Li, “Global stability of an epidemic model with nonlinear incidence rate and differential infectivity,” Applied Mathematics and Computation, vol. 161, no. 3, pp. 769–778, 2005.
- C. Vargas-De-León, “On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,” Chaos, Solitons & Fractals, vol. 44, no. 12, pp. 1106–1110, 2011.
- J. K. Hale, Ordinary di Erential Equations, John Wiley & Sons, New York, NY, USA, 1969.
- J. C. Kamgang and G. Sallet, “Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE),” Mathematical Biosciences, vol. 213, no. 1, pp. 1–12, 2008.
- M. Qiao, A. Liu, and U. Foryś, “Qualitative analysis of the SICR epidemic model with impulsive vaccinations,” Mathematical Methods in the Applied Sciences. In press.
- R. M. Anderson and R. M. Verlag, Population Biology of Infectious Diseases, Springer, New York, NY, USA, 1982.
- V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1-2, pp. 43–61, 1978.
- J. Hou and Z. Teng, “Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates,” Mathematics and Computers in Simulation, vol. 79, no. 10, pp. 3038–3054, 2009.
- J. Zu, W. Wang, and B. Zu, “Evolutionary dynamics of prey-predator systems with holling type II,” Mathematical Biosciences and Engineering, vol. 4, no. 2, pp. 221–237, 2007.
- W. M. Liu, S. A. Levin, and Y. Iwasa, “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,” Journal of Mathematical Biology, vol. 23, no. 2, pp. 187–204, 1985.
- S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135–163, 2003.
- M. A. Safi and A. B. Gumel, “The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 215–235, 2011.
- J. Zu, W. Wang, and B. Zu, “Evolutionary dynamics of prey-predator systems with Holling type II functional response,” Mathematical Biosciences and Engineering, vol. 4, no. 4, article 755, 2007.
- G. Huang and Y. Takeuchi, “Global analysis on delay epidemiological dynamic models with nonlinear incidence,” Journal of Mathematical Biology, vol. 63, no. 1, pp. 125–139, 2011.
- H. R. Thieme, “Global asymptotic stability in epidemic models,” in Equadiff, H. W. Knobloch and K. Schmidt, Eds., vol. 1017 of Lecture notes in Mathematics, pp. 608–615, Springer, Heidelberg, Germany, 1983.
- H. R. Thieme, “Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations,” Mathematical Biosciences, vol. 111, no. 1, pp. 99–130, 1992.
- M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, “Global dynamics of a SEIR model with varying total population size,” Mathematical Biosciences, vol. 160, no. 2, pp. 191–213, 1999.
- M. Y. Li, H. L. Smith, and L. Wang, “Global dynamics of an seir epidemic model with vertical transmission,” SIAM Journal on Applied Mathematics, vol. 62, no. 1, pp. 58–69, 2001.
- G. Li and Z. Jin, “Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period,” Chaos, Solitons and Fractals, vol. 25, no. 5, pp. 1177–1184, 2005.
- J. Zhang and Z. Ma, “Global dynamics of an SEIR epidemic model with saturating contact rate,” Mathematical Biosciences, vol. 185, no. 1, pp. 15–32, 2003.
- A. Korobeinikov, “Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,” Bulletin of Mathematical Biology, vol. 71, no. 1, pp. 75–83, 2009.
- K. Fylkesnes, R. M. Musonda, M. Sichone, Z. Ndhlovu, F. Tembo, and M. Monze, “Declining HIV prevalence and risk behaviours in Zambia: evidence from surveillance and population-based surveys,” AIDS, vol. 15, no. 7, pp. 907–916, 2001.
- D. Low-Beer and R. L. Stoneburner, “Behaviour and communication change in reducing HIV: is Uganda unique?” African Journal of AIDS Research, vol. 2, no. 1, pp. 9–21, 2003.
- N. Hussaini, M. Winter, and A. B. Gumel, “Qualitative assessment of the role of public health education program on HIV transmission dynamics,” Mathematical Medicine and Biology, vol. 28, no. 3, pp. 245–270, 2011.
- E. C. Green, D. T. Halperin, V. Nantulya, and J. A. Hogle, “Uganda's HIV prevention success: the role of sexual behavior change and the national response,” AIDS and Behavior, vol. 10, no. 4, pp. 335–346, 2006.
- H. R. Thieme, Mathematics in Population, Princeton University Press, 2003.
- H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995.
- O. Diekmann, J. A. Heesterbeek, and J. A. Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990.
- P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.
- J. P. LaSalle, The Stability of Dynamical Systems, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1976.
- H. I. Freedman, S. Ruan, and M. Tang, “Uniform persistence and flows near a closed positively invariant set,” Journal of Dynamics and Differential Equations, vol. 6, no. 4, pp. 583–600, 1994.
- N. P. Bhatia and G. P. Szeg, Dynamical Systems: Stability Theory and Applications, vol. 35 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1967.