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)
Global properties of $SIR$ and $SEIR$ epidemic models with multiple parallel infectious stages. (English) Zbl 1169.92041
Summary: We consider global properties of compartment $SIR$ and $SEIR$ models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number $R _{0}$, this state can be either endemic ($R _{0}>1$), or infection-free ($R _{0}\leq 1$).

34D23Global stability of ODE
34D20Stability of ODE
37N25Dynamical systems in biology
Full Text: DOI
[1] Barbashin, E.A., 1970. Introduction to the Theory of Stability. Wolters-Noordhoff, Groningen. · Zbl 0198.19703
[2] Georgescu, P., Hsieh, Y.-H., 2006. Global stability for a virus dynamics model with nonlinaer incidence of infection and removal. SIAM J. Appl. Math. 67(2), 337--353. · Zbl 1109.92025 · doi:10.1137/060654876
[3] Guo, H., Li, M.Y., 2006. Global dynamics of a staged progression model for infectious diseases. Math. Biosci. Eng. 3(3), 513--525. · Zbl 1092.92040
[4] Korobeinikov, A., 2004a. Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. J. IMA 21(2), 75--83. · Zbl 1055.92051 · doi:10.1093/imammb/21.2.75
[5] Korobeinikov, A., 2004b. Global properties of basic virus dynamics models. Bull. Math. Biol. 66(4), 879--883. · doi:10.1016/j.bulm.2004.02.001
[6] Korobeinikov, A., 2006. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68(3), 615--626. · doi:10.1007/s11538-005-9037-9
[7] Korobeinikov, A., 2007. Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871--1886. · Zbl 1298.92101 · doi:10.1007/s11538-007-9196-y
[8] Korobeinikov, A., 2008. Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate. Math. Med. Biol. J. IMA, to appear. · Zbl 1171.92034
[9] Korobeinikov, A., Maini, P.K., 2004. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. Eng. 1(1), 57--60. · Zbl 1062.92061
[10] Korobeinikov, A., Maini, P.K., 2005. Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. J. IMA 22, 113--128. · Zbl 1076.92048 · doi:10.1093/imammb/dqi001
[11] Korobeinikov, A., Petrovskii, S.V., 2008. Toward a general theory of ecosystem stability: plankton-nutrient interaction as a paradigm. In: Hosking, R.J., Venturino, E. (Eds.), Aspects of Mathematical Modelling, pp. 27--40. Birkhäuser, Basel. · Zbl 1196.92042
[12] Korobeinikov, A., Wake, G.C., 2002. Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models. Appl. Math. Lett. 15(8), 955--961. · Zbl 1022.34044 · doi:10.1016/S0893-9659(02)00069-1
[13] La Salle, J., Lefschetz, S., 1961. Stability by Liapunov’s Direct Method. Academic, New York. · Zbl 0098.06102
[14] Okuonghae, D., Korobeinikov, A., 2006. Dynamics of tuberculosis: the effect of direct observation therapy strategy (DOTS) in Nageria. Math. Model. Nat. Phenom. Epidemiol. 2(1), 99--111.
[15] van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29--48. · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6