A theoretical study on mathematical modelling of an infectious disease with application of optimal control

BioSystems

Volumn 111, Issue 1, 2013, Pages 37-50

  Kar, T K ^a   Jana, Soovoojeet ^a  

^a INDIAN INSTITUTE OF ENGINEERING SCIENCE AND TECHNOLOGY   (India)

Author keywords

Epidemic model; Global stability; Optimal control; Transcritical bifurcation; Treatment; Vaccination

Indexed keywords

COST-BENEFIT ANALYSIS; DISEASE CONTROL; DISEASE TREATMENT; EPIDEMIC; INFECTIOUS DISEASE; MATHEMATICAL ANALYSIS; NUMERICAL MODEL; THEORETICAL STUDY; VACCINATION;

ARTICLE; HEALTH CARE COST; INFECTION; INFECTION CONTROL; MATHEMATICAL MODEL; THEORETICAL STUDY; VACCINATION;

EID: 84872389705     PISSN: 03032647     EISSN: 18728324     Source Type: Journal    
DOI: 10.1016/j.biosystems.2012.10.003     Document Type: Article

References (35)

1. Amaku M., Coutinho F.A.B., Massad E. Why dengue and yellow fever coexist in some areas of the world and not in others?. Biosystems 2011, 106:111-120.
2. Anderson R.H., May R.M. Infectious Disease of Humans 1991, Oxford University Press, Oxford, U.K.
3. Arino J., Cooke K.L., Driessche P.V.D., Velasco-Hernández J. An epidemiology model that includes a leaky vaccine with a general waning function. Dyn. Sys. Ser. B 2004, 4(2):479-495.
4. Bailey N.T.J. The Mathematical Theory of Infectious Disease and its Applications 1975, Charles Griffin and Company, London.
5. Bartl M., Li P., Schuster S. Modelling the optimal timing in metabolic pathway activation- Use of Pontryagin's Maximum Principle and role of the Golden section. Biosystems 2010, 101(1):67-77.
6. Bernoulli D. Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir. Histoire l'Acad. Roy. Sci. (Paris) avec Mém. Math. Phys. Mém. 1760, 1-45.
7. Birkoff G., Rota G.C. Ordinary Differential Equations 1982, Ginn, Boston.
8. Buonomo B., d'Onofrio A., Lacitignola D. Global stability of an SIR epidemic model with information dependent vaccination. Math. Biosci. 2008, 216:9-16.
9. Buonomo B., Lacitignola D. On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal. Model. Control 2011, 16(1):30-46.
10. Cai L., Guo S., Li X., Ghosh M. Global dynamics of a dengue epidemic mathematical model. Chaos Soliton. Fract. 2009, 42:2297-2304.
11. Cai L., Li X., Ghosh M., Guo B. Stability analysis of an HIV/AIDS epidemic model with treatment. J. Comput. Appl. Math. 2009, 229:313-323.
12. Daley D.J., Gani J. Epidemic Modelling 1999, Cambridge University Press, Cambridge, U.K.
13. Eckalbar J.C., Eckalbar W.L. Dynamics of an epidemic model with quadratic treatment. Nonlinear Analysis: Real World Applications 2011, 12(1):320-332.
14. Guckenheimer G., Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1983, Springer Verlag, New York.
15. Hethcote H.W. The mathematics of infectious diseases. SIAM Review 2000, 42:599-653.
16. Joshi H.R. Optimal control of an HIV immunology model. Optim. Con. Appl. Methods 2002, 23:199-213.
17. Jung E., Lenhart S., Feng Z. Optimal control of treatments in a two-strain tuberculosis model. Discrete Contin. Dyn. Syst. Ser. B 2002, 2(4):473-482.
18. Kar T.K., Mondal P.K. Global dynamics and bifurcation in delayed SIR epidemic model. Nonlinear Anal.-Real. 2011, 12:2058-2068.
19. Kar T.K., Batabyal A. Stability analysis and optimal control of an SIR epidemic model with vaccination. Biosystems 2011, 104:127-135.
20. Kar T.K., Ghorai A., Jana S. Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide. J. Theor. Biol. 2012, 310:187-198.
21. Kar T.K., Ghosh B. Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator. Biosystems 2012, 109:220-232.
22. Keeling M.J., Rohani P. Modelling Infectious Diseases in Humans and Animals 2008, Princeton University Press, Princeton, New Jersey.
23. Kermack W.O., McKendric A.G. Contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 1927, 115:700-721.
24. Lenhart S., Workman J.T. Optimal Control Applied to Biological Model 2007, Mathematical and Computational Biology Series Chapman & Hall/CRC.
25. Li M.Y., Muldowney J.S. A geometric approach to global stability problems. SIAM J. Math. Anal. 1996, 27(4):1070-1083.
26. Lukes D.L. Differential equations: classical to controlled. Mathematics in SCIENCE and Engineering, vol. 162 1982, Academic Press, New York.
27. Martin R.H. Logarithmic norms and projections applied to linear differential systems. J. Math. Anal. Appl. 1974, 45:432-454.
28. Michael E., Malecela-Lazaro M.N., Simonsen P.E., Pedersen E.M., Barker G., Kumar A., Kazura J.W. Mathematical modelling and the control of lymphatic filariasis. Lancet Infect. Dis. 2004, 4:223-234.
29. Okosun K.O., Ouifki R., Marcus N. Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity. Biosystems 2011, 106:136-145.
30. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The Mathematical Theory of Optimal Processes 1962, Wiley, New York.
31. Thomasey D.H., Martcheva M. Serotype replacement of vertically transmitted diseases through perfect vaccination. J. Biol. sys. 2008, 16(2):255-277.
32. Wang W. Backward bifurcation of an epidemic model with treatment. Math. Biosci. 2006, 201:58-71.
33. Zaman G., Kang Y.H., Jung I.H. Stability analysis and optimal vaccination of an SIR epidemic model. Biosystems 2008, 93:240-249.
34. Zhang F.-F., Jin Z., Sun G.-Q. Bifurcation analysis of a delayed epidemic model. Appl. Math. Comput. 2010, 216:753-767.
35. Zhang J.Z., Jin Z., Liu Q.X., Zhang Z.Y. Analysis of a delayed SIR model with non-linear incidence rate. Discrete Dyn. Nat. Soc 2008, 1-16. Article ID 636153. 10.1155/2008/636153.