zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bifurcation and chaos in an epidemic model with nonlinear incidence rates. (English) Zbl 1187.92073
Summary: This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulations. It is verified that there are phenomena of transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.

34C23Bifurcation (ODE)
37N25Dynamical systems in biology
65C20Models (numerical methods)
34C60Qualitative investigation and simulation of models (ODE)
Full Text: DOI
[1] Berryman, A. A.; Millstein, J. A.: Are ecological systems chaotic-and if not, why not?, Trends ecol. Evolut. 4, 26-28 (1989)
[2] Blower, S. M.; Mclean, A. R.: Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco, Science 265, 1451-1454 (1994)
[3] Brauer, F.; Den Driessche, P. Van: Models for transmission of disease with immigration of infectives, Math. biosci. 171, 143-154 (2001) · Zbl 0995.92041 · doi:10.1016/S0025-5564(01)00057-8
[4] Donnelly, C. A.; Ghani, A. C.; Leung, G. M.; Hedley, A. J.; Fraser, C.; Riley, S.; Abu-Raddad, L. J.; Ho, L. M.; Thach, T. Q.; Chau, P.; Chan, K. P.; Lam, T. H.; Tse, L. Y.; Tsang, T.; Liu, S. H.; Kong, J. H. B.; Lau, E. M. C.; Fer-Guson, N. M.; Anderson, R. M.: Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong, Lancet 361, 1761-1766 (2003)
[5] Elbasha, E. H.; Gumel, A. B.: Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits, Bull. math. Biol. 68, 577-614 (2006)
[6] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector field, (1983) · Zbl 0515.34001
[7] Gumel, A. B.; Mccluskey, C. C.; Den Driessche, P. Van: Mathematical study of a staged-progression HIV model with imperfect vaccine, Bull. math. Biol. 68, 2105-2128 (2006) · Zbl 1296.92124
[8] Halloran, M. E.; Longini, I. M.; Nizam, A.; Yang, Y.: Containing bioterrorist smallpox, Science 298, 1428-1432 (2002)
[9] Hassell, M. P.; Comins, H. N.; May, R. M.: Spatial structure and chaos in insect population dynamics, Nature 353, 255-258 (1991)
[10] H.W. Hethcote, Three basic epidemiological models, in: S.A. Levin, T.G. Hallam, L. Gross (Eds.), Applied Mathematical Ecology, Biomathematics, vol. 18, Springer, Berlin, 1989, pp. 119 -- 143.
[11] House, T.; Keeling, M. J.: Deterministic epidemic models with explicit household structure, Math. biosci. 213, 29-39 (2008) · Zbl 1135.92029 · doi:10.1016/j.mbs.2008.01.011
[12] Hyman, J. M.; Li, J.; Stanley, E. A.: The differential infectivity and staged progression models for the transmission of HIV, Math. biosci. 208, 77-109 (1999) · Zbl 0942.92030 · doi:10.1016/S0025-5564(98)10057-3
[13] Kermack, W. O.; Mckendrick, A. G.: A contribution to the mathematical theory of epidemics, Proc. roy. Soc. A 115, 700-721 (1927) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[14] Kribs-Zaleta, C.; Valesco-Hernandez, J.: A simple vaccination model with multiple endemic states, Math. biosci. 164, 183-201 (2000) · Zbl 0954.92023 · doi:10.1016/S0025-5564(00)00003-1
[15] Lane, H. C.; Montagne, J. L.; Fauci, A. S.: Bioterrorism: a clear and present danger, Nature med. 7, 1271-1273 (2001)
[16] Liu, W. M.; Hethcote, H. W.; Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. Biol. 25, 359-380 (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162
[17] Liu, W. M.; Levin, S. A.; Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. Biol. 23, 187-204 (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956
[18] Mclean, A. R.; Blower, S. M.: Imperfect vaccines and herd immunity to HIV, Proc. roy. Soc. lond. B 253, 9-13 (1993)
[19] Mukhopadhyay, B.; Bhattacharyya, R.: Analysis of a spatially extended nonlinear SEIS epidemic model with distinct incidence for exposed and infectives, Nonlinear anal.: real world appl. 9, 585-598 (2008) · Zbl 1145.35390 · doi:10.1016/j.nonrwa.2006.12.003
[20] Ramani, A.; Carstea, A. S.; Willox, R.; Grammaticosb, B.: Oscillatingepidemics: a discrete-time model, Physica A 333, 278-292 (2004)
[21] Riley, S.; Ferguson, N. M.: Smallpox transmission and control: spatial dynamics in great britain, Proc. natl. Acad. sci. USA 103, 12637-12642 (2006)
[22] Willox, R.; Grammaticos, B.; Carstea, A. S.; Ramani, A.: Epidemic dynamics: discrete-time and cellular automaton models, Physica A 328, 13-22 (2003) · Zbl 1026.92042 · doi:10.1016/S0378-4371(03)00552-1
[23] World Health Organisation, Epidemic and Pandemic Alert and Response (EPR)avian Influenza, <http://www.who.int/csr/disease/avianinfluenza/>.