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)
Stochastic models of some endemic infections. (English) Zbl 0991.92026
Summary: Stochastic models are established and studied for several endemic infections with demography. Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models. The approximations are valid for sufficiently large population sizes. Conditions for validity of the approximations are given for each of the models. These are also conditions for validity of the corresponding deterministic models. It is noted that some deterministic models are unacceptable approximations of the stochastic models for a large range of realistic parameter values.

MSC:
92D30Epidemiology
60J70Applications of Brownian motions and diffusion theory
WorldCat.org
Full Text: DOI
References:
[1] Nåsell, I.: The threshold concept in stochastic epidemic and endemic models. Epidemic models: their structure and relation to data (1995) · Zbl 0850.92047
[2] Nåsell, I.: On the time to extinction in recurrent epidemics. J. roy. Statist. soc. B, part 2 61, 309 (1999) · Zbl 0917.92023
[3] Bailey, N. T. J.: The mathematical theory of infectious diseases and its applications. (1975) · Zbl 0334.92024
[4] Hethcote, H. W.: The mathematics of infectious diseases. SIAM rev. 42, No. 4, 599 (2001)
[5] Mena-Lorca, J.; Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population sizes. J. math. Biol. 30, 693 (1992) · Zbl 0748.92012
[6] Hethcote, H. W.: Qualitative analysis of communicable disease models. Math. biosci. 28, 335 (1976) · Zbl 0326.92017
[7] Jacquez, J. A.; Simon, C. A.: The stochastic SI model with recruitment and deaths. I. comparison with the closed SIS model. Math. biosci. 117, 77 (1993) · Zbl 0785.92025
[8] Nåsell, I.: The quasi-stationary distribution of the closed endemic SIS model. Adv. appl. Prob. 28, 895 (1996) · Zbl 0854.92020
[9] Nåsell, I.: On the quasi-stationary distribution of the stochastic logistic epidemic. Math. biosci. 156, 21 (1999) · Zbl 0954.92024
[10] I. Nåsell, Measles outbreaks are not chaotic, IMA Volumes in Mathematics and its Applications, Mathematical Approaches for Emerging and Reemerging Infectious Diseases, Part II: Models, vol. 126, Methods and Theory, 2002, Springer, in press · Zbl 1022.92036
[11] Andersson, H.; Britton, T.: Stochastic epidemics in dynamic populations: quasi-stationarity and extinction. J. math. Biol. 41, No. 6, 559 (2000) · Zbl 1002.92018
[12] Allen, L. J. S.; Burgin, A. M.: Comparison of deterministic and stochastic SIS and SIR models in discrete time. Math. biosci. 163, 11 (2000) · Zbl 0978.92024