- 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
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 595487, 21 pages
Stability Analysis and Optimal Control of a Vector-Borne Disease with Nonlinear Incidence
1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12 Campus, Islamabad 44000, Pakistan
2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
3Department of Mathematics, Vaal University of Technology, Andries Potgieter Boulevard, Private Bag X021, Vanderbijlpark 1900, South Africa
Received 29 July 2012; Revised 17 September 2012; Accepted 17 September 2012
Academic Editor: M. De la Sen
Copyright © 2012 Muhammad Ozair et al. 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.
- L. Cai and X. Li, “Analysis of a simple vector-host epidemic model with direct transmission,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 679613, 12 pages, 2010.
- L. Cai, S. Guo, X. Li, and M. Ghosh, “Global dynamics of a dengue epidemic mathematical model,” Chaos, Solitons and Fractals, vol. 42, no. 4, pp. 2297–2304, 2009.
- S. Liu, Y. Pei, C. Li, and L. Chen, “Three kinds of TVS in a SIR epidemic model with saturated infectious force and vertical transmission,” Applied Mathematical Modelling, vol. 33, no. 4, pp. 1923–1932, 2009.
- 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.
- X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433–443, 2008.
- D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419–429, 2007.
- W. Wang, “Epidemic models with nonlinear infection forces,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 267–279, 2006.
- 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.
- W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–380, 1987.
- L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3531–3541, 2010.
- S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series, Chapman and Hall/CRC, London, UK, 2007.
- I. H. Jung, Y. H. Kang, and G. Zaman, “Optimal treatment of an SIR epidemic model with timedelay,” Biosystems, vol. 98, no. 1, pp. 43–50, 2009.
- A. A. Lashari, S. Aly, K. Hattaf, G. Zaman, I. H. Jung, and X. Z. Li, “Presentation of malaria epidemics using multiple optimal controls,” Journal of Applied Mathematics, vol. 2012, Article ID 946504, 17 pages, 2012.
- K. Blayneh, Y. Cao, and H.-D. Kwon, “Optimal control of vector-borne diseases: treatment and prevention,” Discrete and Continuous Dynamical Systems A, vol. 11, no. 3, pp. 587–611, 2009.
- K. O. Okosun and O. D. Makinde, “Impact of chemo-therapy on optimal control of malaria disease with infected immigrants,” BioSystems, vol. 104, no. 1, pp. 32–41, 2011.
- A. A. Lashari and G. Zaman, “Global dynamics of vector-borne diseases with horizontal transmission in host population,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 745–754, 2011.
- A. A. Lashari and G. Zaman, “Optimal control of a vector borne disease with horizontal transmission,” Nonlinear Analysis. Real World Applications, vol. 13, no. 1, pp. 203–212, 2012.
- T. K. Kar and A. Batabyal, “Stability analysis and optimal control of an SIR epidemic model with vaccination,” BioSystems, vol. 104, no. 2-3, pp. 127–135, 2011.
- J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1976.
- M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070–1083, 1996.
- A. Fonda, “Uniformly persistent semidynamical systems,” Proceedings of the American Mathematical Society, vol. 104, no. 1, pp. 111–116, 1988.
- G. Butler, H. I. Freedman, and P. Waltman, “Uniformly persistent systems,” Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 425–430, 1986.
- N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” Bulletin of Mathematical Biology, vol. 70, no. 5, pp. 1272–1296, 2008.
- L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, vol. 4, Gordon & Breach Science Publishers, New York, NY, USA, 1986.
- W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, NY, USA, 1975.