Backward bifurcation and optimal control in transmission dynamics of West Nile virus

Bulletin of Mathematical Biology

Volumn 72, Issue 4, 2010, Pages 1006-1028

  Blayneh, Kbenesh W ^a   Gumel, Abba B ^b   Lenhart, Suzanne ^c   Clayton, Tim ^d  

^a FLORIDA A AND M UNIVERSITY   (United States)

^b UNIVERSITY OF MANITOBA   (Canada)

^c UNIVERSITY OF TENNESSEE   (United States)

^d Mathematics and Natural Science Department   (United States)

Author keywords

Bifurcation; Equilibria; Optimal control; Stability; West Nile virus

Indexed keywords

ANIMAL; ARTICLE; BIOLOGICAL MODEL; BIRD; CULEX; DISEASE CARRIER; DISEASE TRANSMISSION; GROWTH, DEVELOPMENT AND AGING; HUMAN; VIROLOGY; WEST NILE FEVER; WEST NILE FLAVIVIRUS;

ANIMALS; BIRDS; CULEX; DISEASE TRANSMISSION, INFECTIOUS; HUMANS; INSECT VECTORS; MODELS, BIOLOGICAL; WEST NILE FEVER; WEST NILE VIRUS;

AVES; WEST NILE VIRUS;

EID: 77952674174     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-009-9480-0     Document Type: Article

References (37)

1. Adams, B. M., Banks, H. T., Davidian, M., Kwon, H.-D., Tran, H. T., Wynne, S. N., Rosenberg, E. S., 2005. HIV dynamics: Modeling, data analysis and optimal treatment protocols. J. Comput. Appl. Math. 184, 10-49.
2. Bowman, C., Gumel, A. B., van den Driessche, P., Wu, J., Zhu, H., 2005. A mathematical model for assessing control strategies against West Nile virus. Bull. Math. Biol. 67, 1107-1133.
3. Blayneh, K., Cao, Y., Kwon, Hee-Dae, 2009. Optimal control of vector-borne diseases: treatment and prevention. Discrete Contin. Dyn. Syst. Ser. 11(3), 587-611.
4. Burt, F. J., Grobbelaar, A. A., Leman, P. A., Anthony, F. S., Gibson, G. V. F., Swanepoel, R., 2002. Phylogenetic Relationships of Southern African West Nile virus isolates. CDC: Emerging Infectious Diseases 8(8). http://www.medscape.com/viewarticle/440765.
5. Carr, J., 1981. Applications of Centre Manifold Theory. Springer, New York.
6. Castillo-Chavez, C., Song, B., 2004. Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361-404.
7. Castillo-Chavez, C., Feng, Z., Huang, W., 2002. In: Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., Yakubu, A.-A. (Eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases. Springer, New York.
8. Center for Disease Control and Prevention (2005). West Nive Virus Fact Sheet, September 27, 2005. www.cdc.gov/ncidod/westnile/wnv_factsheet.htm.
9. Cruz-Pacheco, G., Esteva, L., Montano-Hirose, J. A., Vargas, D., 2005. Modelling the dynamics of West Nile virus. Bull. Math. Biol. 67, 1157-1172.
10. Culshaw, R. V., 2004. Optimal HIV treatment by maximising immune response. J. Math. Biol. 48, 545-562.
11. Cushing, J. M., 1998. An Introduction to Structured Population Dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 71, SIAM, Philadelphia.
12. Darensburg, T., Kocic, V., 2004. On the discrete model of West Nile-like epidemics. Proc. Dyn. Appl. 4, 358-366.
13. Diekmann, O., Heesterbeek, J. A. P., 2000. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York.
14. Dushoff, J., Wenzhang, H., Castillo-Chavez, C., 1998. Backwards bifurcations and catastrophe in simple models of fatal diseases. J. Math. Biol. 36, 227-248.
15. Fleming, W. H., Rishel, R. W., 1975. Deterministic and Stochastic Optimal Control. Springer, New York.
16. Gourley, S. A., Liu, L. R., Wu, J., 2007. Some vector borne diseases with structured host populations: Extinction and spatial spread. SIAM J. Appl. Math. 67, 408-433.
17. Huhn, D. G., James, J. S., Montgomery, P. S., Dworkin, S. M., 2003. West Nile virus in the United States: an update on an emerging infectious disease. Am. Fam. Phys. 68(4), 653-675.
18. Jang, S. R.-J., 2007. On a discrete West Nile epidemic model. Comput. Appl. Math. 26, 397-414.
19. Joshi, H. R., 2003. Optimal control of HIV immunology model. Optim. Control Appl. Methods 23(4), 199-213.
20. Jianga, J., Qiub, Z., Wu, J., Zhu, H., 2009. Threshold conditions for West Nile virus outbreaks. Bull. Math. Biol. doi: 10. 1007/s11538-008-9374-6.
21. Jung, E., Lenhart, S., Feng, Z., 2002. Optimal control of treatments in a two-strain tuberculosis model. Discrete Contin. Dyn. Syst. Ser. 2(4), 473-482.
22. Kenkre, V. M., Parmenter, R. R., Peixoto, I. D., Sadasiv, L., 2006. A theoretic framework for the analysis of the West Nile virus epidemic. Comput. Math. 42, 313-324.
23. Kilpatric, A. M., Kramer, L. D., Jones, M. J., Marra, P. P., Daszak, P., Fonseca, D. M., 2007. Genetic influences on mosquito feeding behavior and the emergence of zoonotic pathogens. Am. J. Trop. Med. Hyg. 77(4), 667-671.
24. Kirschner, E. D., Lenhart, S., Serbin, S., 1997. Optimal control of the chemotherapy of HIV. J. Math. Biol. 35, 775-792.
25. Lenhart, S., Workman, J. T., 2007. Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series. Chapman & Hall/CRC Press, London/Boca Raton.
26. Lewis, M., Renclawowicz, J., van den Driessche, P., 2006a. Traveling waves and spread rates for a West Nile virus model. Bull. Math. Biol. 66, 3-23.
27. Lewis, M. A., Renclawowicz, J., van den Driesssche, P., Wonham, M., 2006b. A comparison of continuous and discrete-time West Nile virus models. Bull. Math. Biol. 68, 491-509.
28. Lukes, D. L., 1982. Differential Equations: Classical to Controlled. Mathematics in Science and Engineering. Academic Press, New York.
29. NGwa, G. A., Shu, W. S., 2000. A mathematical model for endemic malaria with variable human and mosquito populations. Math. Comput. Model. 32, 747-763.
30. Nosal, B., Pellizzari, R., 2003. West Nile virus. CMAJ 168(11), 1443-1444.
31. Ontero, J., Anderson, F., Andreadis, T. G., Main, A. J., Kline, D. L., 2004. Prevalence of West Nile virus in tree canopy-in habiting Culex pipiens and associated mosquitoes. Am. J. Trop. Med. Hyg. 71(1), 112-119.
32. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., Mishchenko, E. F., 1986. The Mathematical Theory of Optimal Process, vol. 4. Gordon & Breach, New York.
33. Peterson, L. R., Marfin, A. A., 2002. West Nile virus: a primer for the clinician. Ann. Intern. Med. 137(3), 173-179.
34. Sharomi, O., Podder, C. N., Gumel, A. B., Elbasha, E. H., Watmough, J., 2007. Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Math. Biosci. 210, 436-463.
35. Thomas, D. M., Urena, B., 2001. A model describing the evolution of West Nile-like encephalitis in New York city. Math. Comput. Model. 34, 771-781.
36. van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29-48.
37. Wonham, M. J., de-Camino-Beck, T., Lewis, M. A., 2004. An epidemiological model for West Nile virus: Invasion analysis and control applications. Proc. R. Soc. Lond. 271, 501-507.