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 an epidemic model with latent stage and vaccination. (English) Zbl 1220.34069
Summary: For an epidemic model with latent stage and vaccination for the newborns and susceptibles, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 1, then the disease-free equilibrium is globally asymptotically stable, that is, the disease dies out eventually; if R 0 >1, then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region, that is, the disease persists in the population. We propose an approach for determining the Lyapunov function and proving the negative definiteness or semidefiniteness of its derivative. Our proof shows that, for a given Lyapunov function, its derivative should be arranged in different forms for the different values of parameters to prove the negative definiteness or semidefiniteness of its derivative.
MSC:
34C60Qualitative investigation and simulation of models (ODE)
92D30Epidemiology
34D20Stability of ODE
34D23Global stability of ODE
References:
[1]Freedman, H. I.; So, J. W. -H.: Global stability and persistence of simple food chains, Math. biosci. 76, 69-86 (1985) · Zbl 0572.92025 · doi:10.1016/0025-5564(85)90047-1
[2]Goh, B. -S.: Management and analysis of biological populations, (1980)
[3]Ma, Z.: Mathematical modeling and study of population biology, (1989)
[4]Guo, H.; Li, M. Y.: Global dynamics of a staged progression model for infectious diseases, Math. biosci. Eng. 3, 513-525 (2006) · Zbl 1092.92040 · doi:10.3934/mbe.2006.3.513
[5]Mccluskey, C. C.: Global stability for a class of mass action systems allowing for latency in tuberculosis, J. math. Anal. appl. 338, 518-535 (2008) · Zbl 1131.92042 · doi:10.1016/j.jmaa.2007.05.012
[6]Mccluskey, C. C.: Lyapunov functions for tuberculosis models with fast and slow progression, Math. biosci. Eng. 4, 603-614 (2006) · Zbl 1113.92057 · doi:10.3934/mbe.2006.3.603
[7]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
[8]Bame, N.; Bowong, S.; Mbang, J.: Global stability analysis for SEIS models with n latent classes, Math. biosci. Eng. 5, 20-33 (2008) · Zbl 1155.34325 · doi:10.3934/mbe.2008.5.20
[9]Liu, L.; Zhou, Y.; Wu, J.: Global dynamics in a TB model incorporating case detection and two treatment stages, Rocky mt. J. math. 38, 1541-1559 (2008) · Zbl 1194.92040 · doi:10.1216/RMJ-2008-38-5-1541
[10]Melesse, D. Y.; Gumel, A. B.: Global asymptotic properties of an SEIRS model with multiple infectious stages, J. math. Anal. appl. 366, 202-217 (2010) · Zbl 1184.92043 · doi:10.1016/j.jmaa.2009.12.041
[11]Guo, H.; Li, M. Y.; Shuai, Z.: A graph-theoretic approach to the method of global Lyapunov functions, Proc. amer. Math. soc. 136, 2793-2802 (2008) · Zbl 1155.34028 · doi:10.1090/S0002-9939-08-09341-6
[12]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
[13]Korobeinikov, A.; Maini, P. K.: Non-linear incidence and stability of infectious disease models, Math. med. Biol. 22, 113-128 (2005) · Zbl 1076.92048 · doi:10.1093/imammb/dqi001
[14]Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. math. Biol. 68, 615-626 (2006)
[15]Korobeinikov, A.: Global properties of infectious disease models with nonlinear incidence, Bull. math. Biol. 69, 1871-1886 (2007)
[16]Yuan, Z.; Wang, L.: Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear anal. Real world appl. 11, 995-1004 (2010)
[17]Lasalle, J. P.: The stability of dynamical systems. Regional conference series in applied mathematics, (1976)
[18]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