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)
The analysis and application of an HBV model. (English) Zbl 1243.34054
Summary: A mathematical model is formulated to describe the spread of hepatitis B. The stability of equilibria and persistence of disease are analyzed. The results shows that the dynamics of the model is completely determined by the basic reproductive number $\rho _{0}$. If $\rho _{0}< 1$, the disease-free equilibrium is globally stable. When $\rho _{0}> 1$, the disease-free equilibrium is unstable and the disease is uniformly persistent. Furthermore, under certain conditions, it is proved that the endemic equilibrium is globally attractive. Numerical simulations are conducted to demonstrate our theoretical results. The model is applied to HBV transmission in China. The parameter values of the model are estimated based on available HBV epidemic data in China. The simulation results matches the HBV epidemic data in China approximately.

34C20Transformation and reduction of ODE and systems, normal forms
34C60Qualitative investigation and simulation of models (ODE)
Full Text: DOI
[1] World Health Organization. 2008, hepatitis B. World Health Organization Fact Sheet N^\circ&nbsp; 204. &lt;http://www.who.int/mediacentre/factsheets/fs204/en/index.html&gt;.
[2] Seeger, C.; Mason, W.: Hepatitis B virus biology, Microbiol. mol. Biol. rev. 64, 51-68 (2000)
[3] Candotti, D.; Opare-Sem, O.; Rezvan, H.; Sarkodie, F.; Allain, J. P.: Molecular and serological characterization of hepatitis B virus in deferred ghanaian blood donors with and without elevated alanine aminotransferase, J. viral. Hepat. 13, 715-724 (2006)
[4] Kane, M.: Global programme for control of hepatitis B infection, Vaccine 13, S47-S49 (1995)
[5] Chinese Center for Disease Control and Prevention. &lt;http://www.chinacdc.cn/n272442/n272530/n3479265/n3479303/37095.html&gt;.
[6] Medley, G. F.; Lindop, N. A.; Edmunds, W. J.; Nokes, D. James: Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control, Nat. med. 7, 619-624 (2001)
[7] Ministry of Health of the People’s Republic of China. National report of notifiable diseases, 2004-2009. &lt;http://www.moh.gov.cn/publicfiles//business/htmlfiles/wsb/pyqxx/list.htm&gt;.
[8] Hou, J. L.; Liu, Z. H.; Gu, F.: Epidemiology and prevention of hepatitis B virus infection, Int. J. Med. sci. 2, 50-57 (2005)
[9] Wang, Z.; Zhang, J.; Yang, H.: Quantitative analysis of HBV DNA level and hbeag titer in hepatitis B surface antigen positive mothers and their babies: hbeag passage through the placenta and the rate of decay in babies, J. med. Virol. 71, 360-366 (2003)
[10] Xu, D.; Yan, Y.; Choi, B. C.: Risk factors and mechanism of transplacental transmission of hepatitis B virus: a case control study, J. med. Virol. 67, 20-26 (2002)
[11] John, T. J.; Cooksley, G.: Hepatitis B vaccine boosters: is there a clinical need in high endemicity populations, J. gastroenterol. Hepatol. 20, 5-10 (2005)
[12] O’leary, C.; Hong, Z.; Zhang, F.; Dawood, M.; Smart, G.; Kaita, K.; Wu, J.: A mathematical model to study the effect of hepatitis B virus vaccine and antivirus treatment among the canadian inuit population, Eur. J. Clin. microbiol. Infect. dis. 29, 63-72 (2010)
[13] Zou, L.; Zhang, W. N.; Ruan, S. G.: Modeling the transmission dynamics and control of hepatitis B virus in China, J. theor. Biol. 262, 330-338 (2010)
[14] Thornley, S.; Bullen, C.; Roberts, M.: Hepatitis B in a high prevalence New Zealand population: a mathematical model applied to infection control policy, J. theor. Biol. 254, 599-603 (2008)
[15] Hahne’, S.; Ramsay, M.; Balogum, K.: Incidence and routes of transmission of hepatitis B virus in england and wales, 1995 -- 2000: implications for immunisation policy, J. clin. Virol. 29, 211-220 (2004)
[16] Mclean, A. R.; Blumberg, B. S.: Modelling the impact of mass vaccination against hepatitis B.I. Model formulation and parameter estimation, Proc. R. Soc. lond. B. 256, 7-15 (1994)
[17] Zhao, S. J.; Xu, Z. Y.; Lu, Y.: A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int. J. Epidemiol. 29, 744-752 (2000)
[18] Hoofnagle, J. H.; Doo, E.; Liang, T. J.; Fleischer, R.; Lok, A. S.: Management of hepatitis B: summary of a clinical research workshop, Hepatology 45, 1056-1075 (2007)
[19] Hyams, K. C.: Risks of chronicity following acute hepatitis B virus infection: a review, Clin. infect. Dis. 20, 992-1000 (1995)
[20] 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 · doi:10.1016/S0025-5564(02)00108-6
[21] Lasalle, J. P.: The stability of dynamical systems, regional conference series in applied mathematics, (1976) · Zbl 0364.93002
[22] Smith, H. L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, mathematical surveys and monographs, Amer. math. Soc. prov. 41 (1995) · Zbl 0821.34003
[23] Zhao, X. Q.: Dynamical systems in population biology, (2003) · Zbl 1023.37047
[24] Jia, J. D.: Hepatitis B in China: from guideline to practice, Virol. sin. 23, 152-155 (2008)
[25] National Bureau of Statistics of China. China Statistical Yearbook 2009, Birth rate, Death rate and Natural growth rate of population.&lt;ttp://www.stats.gov.cn/tjsj/ndsj/2009/indexch.htm&gt;.
[26] Edmunds, W. J.; Medley, G. F.; Nokes, D. J.: The transmission dynamics and control of hepatitis B virus in the gambia, Stat. med. 15, 2215-2233 (1996)