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)
Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates. (English) Zbl 1223.92057
Summary: We introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . This shows that the basic reproduction number R 0 is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.
MSC:
92D30Epidemiology
34D23Global stability of ODE
65C20Models (numerical methods)
References:
[1]Lajmanovich, A.; York, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci. 28, 221-236 (1976) · Zbl 0344.92016 · doi:10.1016/0025-5564(76)90125-5
[2]Anderson, R. M.; May, R. M.: Population biology of infectious diseases I, Nature 280, 361-367 (1979)
[3]Huang, W.; Cooke, K. L.; Castillo-Chavez, C.: Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. math. 52, 835-854 (1992) · Zbl 0769.92023 · doi:10.1137/0152047
[4]Li, M. Y.; Graef, J. R.; Wang, L.; Karsai, J.: Global dynamics of a SEIR model with varying total population size, Math. biosci. 160, 191-213 (1999) · Zbl 0974.92029 · doi:10.1016/S0025-5564(99)00030-9
[5]Thieme, H. R.: Mathematics in population biology, (2003)
[6]Xiao, D.; Ruan, S.: Global analysis of an epidemic model with nonmonotone incidence rate, Math. biosci. 208, 419-429 (2007) · Zbl 1119.92042 · doi:10.1016/j.mbs.2006.09.025
[7]Guo, H.; Li, M. Y.; Shuai, Z.: Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. appl. Math. Q 14, 259-284 (2006) · Zbl 1148.34039
[8]Guo, H.; Li, M. Y.; Shuai, Z.: A graphtheoretic approach to the method of global Lyapunov functions, Proc. am. Math. soc. 136, 2793-2802 (2008) · Zbl 1155.34028 · doi:10.1090/S0002-9939-08-09341-6
[9]Li, M. Y.; Shuai, Z. S.; Wang, C. C.: Global stability of multi-group epidemic models with distributed delays, J. math. Anal. appl. 361, 38-47 (2010) · Zbl 1175.92046 · doi:10.1016/j.jmaa.2009.09.017
[10]Li, M. Y.; Shuai, Z. S.: Global-stability problem for coupled systems of differential equations on networks, J. differ. Eqn. 248, 1-20 (2010) · Zbl 1190.34063 · doi:10.1016/j.jde.2009.09.003
[11]Mccluskey, C. C.: Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear anal. Real world appl. 11, 55-59 (2010) · Zbl 1185.37209 · doi:10.1016/j.nonrwa.2008.10.014
[12]Yuan, Z. H.; Wang, L.: Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear anal. Real world appl. 11, 995-1004 (2010)
[13]Yuan, Z. H.; Zou, X. F.: Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population, Nonlinear anal. Real world appl. 11, 3479-3490 (2010) · Zbl 1208.34134 · doi:10.1016/j.nonrwa.2009.12.008
[14]Korobeinikov, A.: Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. math. Biol. 71, 75-83 (2009) · Zbl 1169.92041 · doi:10.1007/s11538-008-9352-z
[15]Georgescu, P.; Hsieh, Y. H.; Zhang, H.: A Lyapunov functional for a stage structured predator – prey model with nonlinear predation rate, Nonlinear anal. Real world appl. 11, 3653-3665 (2010) · Zbl 1206.34065 · doi:10.1016/j.nonrwa.2010.01.012
[16]Gao, S. J.; Teng, Z. D.; Xie, D. H.: The effects of pulse vaccination on SEIR model with two time delays, Appl. math. Comput. 201, 282-292 (2008) · Zbl 1143.92024 · doi:10.1016/j.amc.2007.12.019
[17]Kyrychko, Y. N.; Blyuss, K. B.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal. Real world appl. 6, 495-507 (2005) · Zbl 1144.34374 · doi:10.1016/j.nonrwa.2004.10.001
[18]Wang, X.; Tao, Y. D.; Song, X. Y.: Pulse vaccination on SEIR epidemic model with nonlinear incidence rate, Appl. math. Comput. 210, 398-404 (2009) · Zbl 1162.92323 · doi:10.1016/j.amc.2009.01.004
[19]Cai, L. M.; Lin, X. Z.: Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. math. Model. 33, 2919-2926 (2009) · Zbl 1205.34049 · doi:10.1016/j.apm.2008.01.005
[20]Freedman, H. I.; Tang, M. X.; Ruan, S. G.: Uniform persistence and flows near a closed positively invariant set, J. dyn. Differ. eqn. 6, 583-600 (1994) · Zbl 0811.34033 · doi:10.1007/BF02218848