322
Views
17
CrossRef citations to date
0
Altmetric
Original Articles

Mathematical analysis of an epidemic model with isolation and optimal controls

, &
Pages 1318-1336 | Received 04 Nov 2014, Accepted 27 Feb 2016, Published online: 22 Jun 2016

References

  • R.H. Anderson and R.M. May, Infectious Disease of Humans, Oxford University Press, Oxford, 1991.
  • J. Arino, K.L. Cooke, P.V.D. Driessche, and J. Velasco-Hernandez, An epidemiology model that includes a leaky vaccine with a general waning function, Dyn. Sys. Ser. B. 4 (2) (2004), pp. 479–495.
  • M. Bartl, P. Li, and S. Schuster, Modelling the optimal timing in metabolic pathway activation – use of Pontryagin’s maximum principle and role of the golden section, Biosystems. 101 (1) (2010), pp. 67–77. doi: 10.1016/j.biosystems.2010.04.007
  • R. Bhattacharyya and B. Mukhopadhyay, On an epidemiological model with nonlinear infection incidence: Local and global perspective, Appl. Math. Model. 35 (2011), pp. 3166–3174. doi: 10.1016/j.apm.2010.12.014
  • G. Birkoff and G.C. Rota, Ordinary Differential Equations, Ginn, Boston, MA, 1982.
  • B. Buonomo, A. d’Onofrio, and D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci. 216 (2008), pp. 9–16. doi: 10.1016/j.mbs.2008.07.011
  • L. Cai, X. Li, M. Ghosh, and B. Guo, Stability analysis of an HIV/AIDS epidemic model with treatment, J. Comput. Appl. Math. 229 (2009), pp. 313–323. doi: 10.1016/j.cam.2008.10.067
  • K. Chakraborty, K. Das, S. Haldar, and T.K. Kar, A mathematical study of an eco-epidemiological system on disease persistence and extinction perspective, Appl. Math. Comput. 254 (2015), pp. 99–112.
  • P.V.D. Driessche and J. Waltmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48. doi: 10.1016/S0025-5564(02)00108-6
  • J.C. Eckalbar and W.L. Eckalbar, Dynamics of an epidemic model with quadratic treatment, Nonlinear Anal. Real. 12 (1) (2011), pp. 320–332. doi: 10.1016/j.nonrwa.2010.06.018
  • G. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag, New York, NY, 1983.
  • H.W. Hethcote, W. Wang, L. Han, and Z. Ma, A predator-prey model with infected prey, Theor. Popul. Biol. 66 (2004), pp. 259–268. doi: 10.1016/j.tpb.2004.06.010
  • S. Jana, P. Haldar, and T.K. Kar, Complex dynamics of an epidemic model with vaccination and treatment control, Int. J. Dynam. Cont. (2015). doi:10.1007/s40435-015-0189-7.
  • H.R. Joshi, Optimal control of an HIV immunology model, Optim. Con. Appl. Methods. 23 (2002), pp. 199–213. doi: 10.1002/oca.710
  • E. Jung, S. Lenhart, and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B. 2 (4) (2002), pp. 473–482. doi: 10.3934/dcdsb.2002.2.473
  • T.K. Kar, A. Ghorai, and S. Jana, Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide, J. Theor. Biol. 310 (2012), pp. 187–198. doi: 10.1016/j.jtbi.2012.06.032
  • T.K. Kar, and B. Ghosh, Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator, Biosystems. 109 (2012), pp. 220–232. doi: 10.1016/j.biosystems.2012.02.003
  • T.K. Kar and S. Jana, A theoretical study on mathematical modeling of an infectious disease with application of optimal control, BioSystems. 111 (1) (2013a), pp. 37–50. doi: 10.1016/j.biosystems.2012.10.003
  • T.K. Kar and S. Jana, Application of three controls optimally in a vector-borne disease a mathematical study, Commun. Nonlinear Sci. Num. Simu. 18 (2013b), pp. 2868–2884. doi: 10.1016/j.cnsns.2013.01.022
  • T.K. Kar, S. Jana, and A. Ghorai, Effect of isolation in an infectious disease, Int. J. Ecol. Econ. Stat. 29 (2) (2013), pp. 87–106. (CESER).
  • T.K. Kar and P.K. Mondal, Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear Anal. Real. 12 (2011), pp. 2058–2068. doi: 10.1016/j.nonrwa.2010.12.021
  • M.J. Keeling and P. Rohani, Modelling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2008.
  • W.O. Kermack and A.G. McKendric, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A. 115 (1927), pp. 700–721. doi: 10.1098/rspa.1927.0118
  • A.A. Lashari and G. Zaman, Optimal control of a vector-borne disease with horizontal transmission, Nonlinear Anal. Real. 13 (2012), pp. 203–212. doi: 10.1016/j.nonrwa.2011.07.026
  • S. Lenhart and J.T. Workman, Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series, Chapman & Hall/CRC, London, 2007.
  • D.L. Lukes, Differential equations: Classical to controlled, in Mathematics in Science and Engineering, Vol. 162, Academic Press, New York, NY, 1982.
  • P.K. Mondal, S. Jana, and T.K. Kar, A theoretical approach on controlling agricultural pest by biological controls, Acta Biotheor. 62 (2014), pp. 47–67. doi: 10.1007/s10441-013-9206-4
  • K.O. Okosun, R. Ouifki, and N. Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, Biosystems. 106 (2011), pp. 136–145. doi: 10.1016/j.biosystems.2011.07.006
  • L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, NY, 1962.
  • J.M. Tchuenche, S.A. Khamis, F.B. Agusto, and S.C. Mpeshe, Optimal control and sensitivity analysis of an influenza model with treatment and vaccination, Acta Biotheor. 59 (2011), pp. 1–28.
  • D.H. Thomasey and M. Martcheva, Serotype replacement of vertically transmitted diseases through perfect vaccination, J. Biol. Sys. 16 (2) (2008), pp. 255–277. doi: 10.1142/S0218339008002484
  • R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. Real. 10 (5) (2009), pp. 3175–3189. doi: 10.1016/j.nonrwa.2008.10.013
  • G. Zaman, Y.H. Kang, and I.H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, BioSystems. 93 (2008), pp. 240–249. doi: 10.1016/j.biosystems.2008.05.004
  • J.Z. Zhang, Z. Jin, Q.X. Liu, and Z.Y. Zhang, Analysis of a delayed SIR model with non-linear incidence rate, Discrete Dyn. Nat. Soc. (2008), pp. 1–16. Article ID 636153. doi:10.1155/2008/636153.
  • Y. Zhou, K. Yang, K. Zhou, and Y. Liang, Optimal vaccination policies for an SIR model with limited resources, Acta Biotheor. 62 (2014), pp. 171–181. doi: 10.1007/s10441-014-9216-x

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.