Optimal treatment of an SIR epidemic model with time delay

BioSystems

Volumn 98, Issue 1, 2009, Pages 43-50

  Zaman, Gul ^a   Kang, Yong Han ^b   Jung, Il Hyo ^b  

^a NATIONAL UNIVERSITY OF SCIENCES AND TECHNOLOGY   (Pakistan)

^b Pusan National University   (South Korea)

Author keywords

Ebola outbreak; Epidemic model; Existence; Numerical simulation; Optimality; Time delay

Indexed keywords

DISEASE SPREAD; DISEASE TREATMENT; INFECTIOUS DISEASE; NUMERICAL MODEL;

ACQUIRED IMMUNE DEFICIENCY SYNDROME; ARTICLE; CONGO; EBOLA VIRUS; EPIDEMIC; ERADICATION THERAPY; HOST; HUMAN; HUMAN IMMUNODEFICIENCY VIRUS INFECTION; INFECTION CONTROL; INFLUENZA; MATHEMATICAL ANALYSIS; MEASLES; MUMPS; POLIOMYELITIS; PROBABILITY; SEVERE ACUTE RESPIRATORY SYNDROME; SIMULATION; SMALLPOX; STATISTICAL MODEL; VIRUS INFECTION;

AFRICA; CENTRAL AFRICA; CONGO; SUB-SAHARAN AFRICA;

EBOLA VIRUS;

EID: 68949184850     PISSN: 03032647     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.biosystems.2009.05.006     Document Type: Article

References (32)

1. Anderson R.M., and May R.M. Population biology of infectious diseases: Part I. Nature 280 5721 (1979) 361-367
2. Bachar M., and Dorfmayr A. HIV treatment models with time delay. C. R. Biol. 327 (2004) 983-994
3. Baker C.T.H., Paul C.A.H., and Will D.R. Issues in the numerical solution of evolutionary delay differential equations. Adv. Comput. Math. 3 (1995) 171-196
4. Birkhoff G., and Rota G.C. Ordinary Differential Equations, fourth ed (1989), John Wiley & Sons, New York
5. Brauer F., and Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology (2001), Springer, New York, USA
6. Chwell G., Hengartner N.W., Castillo-Chavez C., Fenimore P.W., and Hyman J.M. The basic reproductive number of Ebola and the effects of public health measures: the case of Congo and Uganda. J. Theor. Biol. 229 (2004) 119-126
7. Cooke K., van den Driessche P., and Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39 (1999) 332-352
8. Halanay A. Optimal control for systems with time lag. SIAM J. Control 6 (1968) 215-234
9. Hethcote H.W., and van den Driessche P. Two SIS epidemiologic models with delays. J. Math. Biol. 40 (2000) 3-26
10. Kamien M.I., and Schwartz N.L. Dynamics Optimization: The calculus of Variations and Optimal Control in Economics and Management (2000), Elsevier Science, The Netherland
11. + T-cells. C. R. Biol. 327 (2004) 1009-1016
12. Kermack W.O., and Mckendrick A.G. Contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115 (1927) 700-721
13. Khan A.S., Tshioko F.K., Heymann D.L., Le Guenno B., Nabeth P., Kerstiens D.L., Fleerackers Y., Kilmarx P.H., Rodier G.R., Nkulu O., Rollin P.E., Sanchez A., Zaki S.R., Swanepoel R., Tomori O., Nichol S.T., Peters C.J., Muyembe-Tamfum J.J., and Ksiazek T.G. The reemergence of Ebola hemorrhagic fever, Democratic Republic of the Congo. J. Inform. Dis. 179 (1995) 76-86
14. Lekone P.E., and Finkenstädt B.F. Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study. Biometrics 62 (2006) 1170-1177
15. Lenhart S., and John T.W. Optimal Control Applied to Biological Models (2007), Chapman & Hall/CRC, Mathematical and Computational Biology Series, London, UK
16. Linda J.S.A., and Amy M.B. Comparison of deterministic and stochastic SIS and SIR models in discrete time. Math. Biosci. 163 (2000) 1-33
17. Linda J.S.A. An Introduction to Mathematical Biology (2007), Pearson Education Ltd., USA pp. 123-127
18. Lukes D.L. Differential Equations: Classical to Controlled. Mathematics in Science and Engineering, vol. 162 (1982), Academic Press, New York p. 182
19. Ma Z., Liu J., and Li J. Stability analysis for differential infectivity epidemic models. Nonlinear Anal. Real World Appl. 4 (2003) 841-856
20. Ma W., Takeuchi Y., Hara T., and Beretta E. Permanence of an SIR epidemic model with distributed time delays. Tohoku Math. J. 54 (2002) 581-591
21. May R.M. Parasitic infections as regulators of animal populations. Am. Sci. 71 (1983) 36-45
22. Mills C.E., Robins J.M., and Lipsitch M. Transmissibility of 1918 pandemic influenza. Nature 432 7019 (2004) 904-906
23. Milner F.A., and Pugliese A. Periodic solutions: a robust numerical method for an S-I-R model of epidemics. J. Math. Biol. 39 (1999) 471-492
24. Smith H.L. A structured population model and related function-differential equation: global attractors and uniform persistence. J. Dyn. Diff. Eqns. 6 (1994) 71-99
25. Smith H.L., and Thieme H. Monotone semiflows in scalar non-quasi-monotone functional deferential equations. J. Math. Anal. Appl. 21 (1990) 673-692
26. So J.W.-H., Wu J., and Zou X. Structured population on two patches: modelling dispersion and delay. J. Math. Biol. 43 (2001) 37-51
27. Takeuchi Y., Ma W., and Beretta E. Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal. 42 (2000) 931-947
28. Tuckwell H.C., and Toubiana L. Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density dependence. Biosystems 90 (2007) 546-559
29. Wallinga J., and Teunis P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am. J. Epidemiol. 160 6 (2004) 509-516
30. Yoshida N., and Hara T. Global stability of a delayed SIR epidemic model with density dependent birth and death rates. J. Comput. Appl. Math. 201 (2007) 339-347
31. Zaman G., Kang Y.H., and Jung I.H. Stability analysis and optimal vaccination of an SIR epidemic model. Biosystems 93 (2008) 240-249
32. Zaman G., Kang Y.H., and Jung I.H. Optimal vaccination and treatment in the SIR epidemic model. Proc. KSIAM 3 (2007) 31-33