Stability and bifurcations in an epidemic model with varying immunity period

Bulletin of Mathematical Biology

Volumn 72, Issue 2, 2010, Pages 490-505

  Blyuss, Konstantin B ^a   Kyrychko, Yuliya N ^a  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Delay differential equations; Epidemic model; Varying temporary immunity

Indexed keywords

ADAPTIVE IMMUNITY; ALGORITHM; ARTICLE; BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; COMMUNICABLE DISEASE; COMPUTER SIMULATION; ENDEMIC DISEASE; HUMAN; IMMUNOLOGY; TIME;

ADAPTIVE IMMUNITY; ALGORITHMS; BASIC REPRODUCTION NUMBER; COMMUNICABLE DISEASES; COMPUTER SIMULATION; ENDEMIC DISEASES; HUMANS; MODELS, BIOLOGICAL; TIME FACTORS;

EID: 77951210926     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-009-9458-y     Document Type: Article

References (32)

1. Anderson, R. M., May, R. M., 1991. Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford.
2. Arino, J., van den Driessche, P., 2006. Time delays in epidemic models: modeling and numerical considerations. In: Arino, O., Hbid, M. L., Ait Dads, E. (Eds.), Delay Differential Equations and Applications, pp. 539-578. Springer, Berlin.
3. Arino, J., Cooke, K. L., van den Driessche, P., Velasco-Hernández, J., 2004. An epidemiology model that includes a leaky vaccine with a general waining function. Discrete Contin. Dyn. Syst. B 2, 479-495.
4. Arvin, A., 2005. Aging, immunity, and the varicella-zoster virus. N. Engl. J. Med. 352, 2266-2267.
5. Bayer, O., Heininger, U., Heiligensetzer, C., von Kries, R., 2007. Metaanalysis of vaccine effectiveness in varicella outbreaks. Vaccine 25, 6655-6660.
6. Beretta, E., Hara, T., Ma, W., Takeuchi, Y., 2001. Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4107-4115.
7. Brauer, F., van den Driessche, P., Wang, L., 2008. Oscillations in a patchy environment disease model. Math. Biosci. 215, 1-10.
8. Breda, D., Maset, S., Vermiglioa, R., 2006. Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions. Appl. Numer. Math. 56, 318-331.
9. Bungay, S. D., Campbell, S. A., 2007. Patterns of oscillation in a ring of identical cells with delayed coupling. Int. J. Bifurc. Chaos 17, 3109-3125.
10. Cooke, K. L., van den Driessche, P., 1996. Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 35, 240-260.
11. Dayan, G. H., Quinlisk, M. P., Parker, A. A., Barskey, A. E., et al., 2008. Recent resurgence of mumps in the United States. N. Engl. J. Med. 358, 1580-1589.
12. De Wals, P., Trottier, P., Pépin, J., 2006. Relative efficacy of different immunization schedules for the prevention of serogroup C meningococcal disease: a model-based evaluation. Vaccine 24, 3500-3504.
13. Derrick, W. R., van den Driessche, P., 1993. A disease transmission model in a nonconstant population. J. Math. Biol. 31, 495-512.
14. Engelborghs, K., Luzyanina, T., Samaey, G., 2001. DDE-BIFTOOL v. 2. 00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report No. TW-330, Department of Computer Science K. U. Leuven, Belgium.
15. Galanis, E., King, A. S., Varughese, P., Halperin, S. A., 2006. Changing epidemiology and emerging risk groups for pertussis. Can. J. Med. Assoc. 174, 451-452.
16. Gao, S., Teng, Z., Nieto, J. J., Torres, A., 2007. Analysis of an SIR epidemic model with pulse vaccination and distributed time delay. J. Biomed. Biotechnol. 64870.
17. Gomes, M. G. M., White, L. J., Medley, G. F., 2004. Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives. J. Theor. Biol. 228, 539-549.
18. Hethcote, H. W., van den Driessche, P., 1991. Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271-287.
19. Hethcote, H. W., van den Driessche, P., 1995. An SIS epidemic model with variable population size and a delay. J. Math. Biol. 34, 177-194.
20. Jianga, Z., Wei, J., 2008. Stability and bifurcation analysis in a delayed SIR model. Chaos Solitons Fractals 35, 609-619.
21. Jokinen, S., Österlund, P., Julkunen, I., Davidkin, I., 2007. Cellular immunity to mumps virus in young adults 21 years after measles-mumps-rubella vaccination. J. Infect. Dis. 196, 861-867.
22. Korobeinikov, A., Maini, P. K., 2005. Non-linear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113-128.
23. Krauskopf, B., 2005. Bifurcation analysis of lasers with delay. In: Kane, D. M., Shore, K. A. (Eds.), Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, pp. 147-183. Wiley, New York.
24. Kuang, Y., 1993. Delay Differential Equations with Applications in Population Biology. Academic Press, New York.
25. Kyrychko, Y. N., Blyuss, K. B., 2005. Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate. Nonlinear Anal. RWA 6, 495-507.
26. Leuridan, E., van Damme, P., 2007. Passive transmission and persistence of naturally acquired or vaccine-induced maternal antibodies against measles in newborns. Vaccine 25, 6296-6304.
27. Lu, C. Y., Chiang, B. L., Chi, W. K., Chang, M. H., Ni, Y. H., Hsu, H. M., Twu, S. J., et al., 2004. Waning immunity to plasma-derived Hepatitis B vaccine and the need for boosters 15 years after neonatal vaccination. Hepatology 40, 1414-1420.
28. Mathews, J. D., McCaw, C. T., McVernon, J., McBryde, E. S., McCaw, J. M., 2007. A biological model for influenza transmission: pandemic planning implications of asymptomatic infection and immunity. PLoS ONE 2, e1220.
29. Mossonga, J., Muller, C. P., 2003. Modelling measles re-emergence as a result of waning of immunity in vaccinated populations. Vaccine 21, 4597-4603.
30. Orosz, G., Krauskopf, B., Wilson, R. E., 2005. Bifurcations and multiple traffic jams in a car-following model with reaction-time delay. Physica D 211, 277-293.
31. Ruan, S., 2006. Delay differential equations in single species dynamics. In: Arino, O., Hbid, M., Ait Dads, E. (Eds.), Delay Differential Equations with Applications. NATO Science Series II, vol. 205, pp. 477-517. Springer, Berlin.
32. Wena, L., Yang, X., 2008. Global stability of a delayed SIRS model with temporary immunity. Chaos Solitons Fractals 38, 221-226.