zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
A discrete epidemic model with stage structure. (English) Zbl 1066.92045
Summary: A discrete SIS epidemic model with stage structure is proposed where a disease spreads among mature individuals. A basic reproduction number $R_0$ of the model is formulated, which is more complicated to calculate than that of differential equation models because the attractor of the model in the disease free space may be composed of equilibria, period cycles, and even strange attractors. If the recruitment rate is of Beverton-Holt type, when $R_0 < 1$ and the recovery rate is equal to 0, the disease free equilibrium is globally stable, and $R_0$ is monotone for any parameter of the system. When the recruitment rate is of Ricker’s type, it is shown that the existence and extinction of the disease can emerge alternately with the change of the intrinsic growth rate. The method for finding basic reproduction numbers can be applied to other discrete epidemic models.

MSC:
92D30Epidemiology
39A11Stability of difference equations (MSC2000)
39A10Additive difference equations
WorldCat.org
Full Text: DOI
References:
[1] Brauer, F.; Den Driessche, P. Ven: Models for transmission of disease with immigration of infectives. Math biosci 171, 143-154 (2001) · Zbl 0995.92041
[2] Hethcote, H. W.: Qualitative analysis of communicable disease models. Math biosci 28, 335-356 (1976) · Zbl 0326.92017
[3] Hethcote, H. W.; Den Driessche, P. Van: Two SIS epidemiologic models with delays. J math biol 40, 3-26 (2000) · Zbl 0959.92025
[4] Cooke, K.; Den Driessche, P. Van: Analysis of an SEIRS epidemic model with two delays. J math biol 35, 240-260 (1996) · Zbl 0865.92019
[5] Britton, N. F.: Essential mathematical biology. (2003) · Zbl 1037.92001
[6] Diekman; Heesterbeek, J. A. P.; Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio R0 in models for infections diseases in heterogeneous populations. J math biol 28, 365-382 (1990) · Zbl 0726.92018
[7] Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math biosci 180, 29-48 (2002) · Zbl 1015.92036
[8] Wang, W.; Zhao, X.: An epidemic model in a patchy environment. Math biosci 190, 97-112 (2004) · Zbl 1048.92030
[9] Wang, W.; Ruan, S.: Simulating the SARS outbreak in Beijing with limited data. J theor biol 227, 369-379 (2004)
[10] Wang, W.: Population dispersal and disease spread. Discrete contin dyn syst ser B 4, 797-804 (2004) · Zbl 1115.92051
[11] Wang, W.; Ruan, S.: Bifurcation in an epidemic model with constant removal rate of the infective. J math anal appl 291, 775-793 (2004) · Zbl 1054.34071
[12] Wang, W.; Ma, Z.: Global dynamics of an epidemic model with time delay. Nonlinear anal real world appl 3, 365-373 (2002) · Zbl 0998.92038
[13] Wang, W.; Chen, L.: A predator-prey system with stage-structure for predator. Comput math appl 33, 83-91 (1997)
[14] Zhao, X.; Wang, W.: Fisher waves in an epidemic model. Discrete contin dyn syst ser B 4, 1117-1128 (2004) · Zbl 1097.34022
[15] Zhao, X.: Dynamical systems in population biology. (2000)
[16] Xiao, Y.; Chen, L.: On an SIS epidemic model with stage structure. J syst sci complex 2, 275-288 (2003) · Zbl 1138.92369
[17] Zhou, Y.; Paolo, F.: Dynamics of a discrete age-structure SIS models. Discrete contin dyn syst ser B 4, 841-850 (2004) · Zbl 1115.92053