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)
Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models. (English) Zbl 1161.92046
Summary: The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcations. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a W. E. Ricker [Stock recruitment. J. Fish Res. Board Canada II 5, 559–623 (1954)] recruitment function in an SIS model and obtain a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on the initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.
37G15Bifurcations of limit cycles and periodic orbits
37G35Attractors and their bifurcations
39A11Stability of difference equations (MSC2000)
37N25Dynamical systems in biology
[1]Allen L.J.S., Burgin A.M.: Comparison of deterministic and stochastic SIS and SIR models in discrete-time. Math. Biosci. 163, 1–33 (2000) · Zbl 0978.92024 · doi:10.1016/S0025-5564(99)00047-4
[2]Allen L.J.S.: Some discrete-time SI, SIR and SIS epidemic models. Math. Biosci. 124, 83–105 (1994) · Zbl 0807.92022 · doi:10.1016/0025-5564(94)90025-6
[3]Alligood K., Sauer T., Yorke J.A.: Chaos: An Introduction to Dynamical Systems. Springer, New York (1996)
[4]Anderson R.M., May R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1992)
[5]Bailey N.T.J.: The Mathematical Theory of Infectious Diseases and its Applications. Griffin, London (1975)
[6]Berezovsky F., Karev C., Song B., Castillo-Chavez C.: A simple model with surprising dynamics. Math. Biosci. Eng. 2, 133–152 (2005)
[7]Berezovsky F., Novozhilov S., Karev G.: Population models with singular equilbrium. Math. Biosci. 208, 270–299 (2007) · Zbl 1116.92049 · doi:10.1016/j.mbs.2006.10.006
[8]Beverton R.J.H., Holt S.J.: On the Dynamics of Exploited Fish Populations. Fish. Invest. Ser. II, H. M. Stationery Office, London (1957)
[9]Castillo-Chavez C., Yakubu A.: Dispersal, disease and life-history evolution. Math. Biosci. 173, 35–53 (2001) · Zbl 1005.92029 · doi:10.1016/S0025-5564(01)00065-7
[10]Castillo-Chavez C., Yakubu A.: Discrete-time S-I-S models with complex dynamics. Nonlinear Anal. 47, 4753–4762 (2001) · Zbl 1042.37544 · doi:10.1016/S0362-546X(01)00587-9
[11]Castillo-Chavez C., Yakubu A.A.: Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments. In: Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., Yakubu, A.-A. (eds) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods and Theory, pp. 165–181. Springer, New York (2002)
[12]Cull P.: Local and global stability for population models. Biol. Cybern. 54, 141–149 (1986) · Zbl 0607.92018 · doi:10.1007/BF00356852
[13]Elaydi S.N., Yakubu A.-A.: Global stability of cycles: Lotka-Volterra competition model with stocking. J. Difference Equ. Appl. 8, 537–549 (2002) · Zbl 1048.39002 · doi:10.1080/10236190290027666
[14]Feng Z., Castillo-Chavez C., Capurro A.F.: A model for tuberculosis with exogenous reinfection. Theor. Pop. Biol. 57, 235–247 (2000) · Zbl 0972.92016 · doi:10.1006/tpbi.2000.1451
[15]Franke J.E., Yakubu A.-A.: Population models with periodic recruitment functions and survival rates. J. Difference Equ. Appl. 11, 1169–1184 (2005) · Zbl 1079.92063 · doi:10.1080/10236190500386275
[16]Franke J.E., Yakubu A.-A.: Discrete-Time SIS Epidemic Model In a Seasonal Environment. SIAM J. Appl. Math. 66(5), 1563–1587 (2006) · Zbl 1108.37303 · doi:10.1137/050638345
[17]Hadeler K.P., Castillo-Chavez C.: A core group model for disease transmission. Math. Bisoci. 128, 41–55 (1995) · Zbl 0832.92021 · doi:10.1016/0025-5564(94)00066-9
[18]Hadeler K.P., van den Driessche P.: Backward bifurcation in epidemic control. Math. Biosci. 146, 15–35 (1997) · Zbl 0904.92031 · doi:10.1016/S0025-5564(97)00027-8
[19]Hassell M.P., Lawton J.H., May R.M.: Patterns of dynamical behavior in single species populations. J. Animal Ecol. 45, 471–486 (1976) · doi:10.2307/3886
[20]Hsu S.-B., Hwang T.-W., Kuang Y.: Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system. J. Math. Biol. 432, 489–506 (2001) · Zbl 0984.92035 · doi:10.1007/s002850100079
[21]Hwang T.-W., Kuang Y.: Deterministic extinction effect in parasites on host populations. J. Math. Biol. 46, 17–30 (2003) · Zbl 1015.92042 · doi:10.1007/s00285-002-0165-7
[22]Kermack W.O., McKendrick A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 138, 55–83 (1932) · Zbl 0005.30501 · doi:10.1098/rspa.1932.0171
[23]Kuang Y., Beretta E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998) · Zbl 0895.92032 · doi:10.1007/s002850050105
[24]May R.M., Oster G.F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–579 (1976) · doi:10.1086/283092
[25]May R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–469 (1977) · doi:10.1038/261459a0
[26]May R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974)
[27]Nicholson A.J.: Compensatory reactions of populations to stresses, and their evolutionary significance. Aust. J. Zool. 2, 1–65 (1954) · doi:10.1071/ZO9540001
[28]Ricker W.E.: Stock recruitment. J. Fish. Res. Board Canada II 5, 559–623 (1954)
[29]Rios-Soto, K., Castillo-Chavez, C., Neubert, M., Titi, E., Yakubu, A.: Epidemic spread in populations at demographic equilibrium. Contemporary Mathematics, AMS volume 410, Mathematical Studies on Human Disease Dynamic: Emerging Paradigms and Challenges, pp. 297–309 (2006)
[30]Ross R.: The Prevention of Malaria. Murray, London (1911)
[31]Sacker R.S.: A new approach to the perturbation theory of invariant surfaces. Comm. Pure Appl. Math. 18, 717–732 (1965) · Zbl 0133.35501 · doi:10.1002/cpa.3160180409
[32]van den Driessche P., Watmough J.: A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40, 525–540 (2000) · Zbl 0961.92029 · doi:10.1007/s002850000032
[33]Yakubu A.-A.: Allee effects in a discrete-time SIS epidemic model with infected newborns. J. Difference Equ. Appl. 13, 341–356 (2007) · Zbl 1118.92056 · doi:10.1080/10236190601079076
[34]Yakubu A.-A., Fogarty M.: Spatially discrete metapopulation models with directional dispersal. Math. Biosci. 204, 68–101 (2006) · Zbl 1104.92071 · doi:10.1016/j.mbs.2006.05.007