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 stability of two-group SIR model with random perturbation. (English) Zbl 1184.34064
The authors study a two-group SIR model introduced by {\it H. Guo, M. Y. Li} and {\it Z. Shuai} [Can. Appl. Math. Q. 14, No. 3, 259--284 (2006; Zbl 1148.34039)] with respect to white noise stochastic perturbations around its positive endemic equilibrium. They prove a stability result given in Theorem 4.3 which show that the endemic equilibrium is stochastically asymptotically stable in the large. The proof of this theorem is based on a Lyapunov function. Some suggestive numerical simulations are included.

34F05ODE with randomness
34D23Global stability of ODE
Full Text: DOI
[1] Guo, H. B.; 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
[2] Kermack, W. O.; Mckendrick, A. G.: Contributions to the mathematical theory of epidemics (Part I), Proc. R. Sm. A 115, 700-721 (1927) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[3] Meng, X. Z.; Chen, L. S.: The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Appl. math. Comput. 197, 528-597 (2008) · Zbl 1131.92056 · doi:10.1016/j.amc.2007.07.083
[4] Zhang, F. P.; Li, Z. Z.; Zhang, F.: Global stability of an SIR epidemic model with constant infectious period, Appl. math. Comput. 199, 285-291 (2008) · Zbl 1136.92336 · doi:10.1016/j.amc.2007.09.053
[5] Hsu, Sze-Bi; Roeger, Lih-Ing W.: The final size of a SARS epidemic model without quarantine, J. math. Anal. appl. 333, 557-566 (2007) · Zbl 1113.92055 · doi:10.1016/j.jmaa.2006.11.026
[6] Zhang, Z. H.; Peng, J. G.: A SIRS epidemic model with infection-age dependence, J. math. Anal. appl. 331, 1396-1414 (2007) · Zbl 1116.35024 · doi:10.1016/j.jmaa.2006.09.061
[7] Wei, H. M.; Li, X. Z.; Martcheva, M.: An epidemic model of a vector-borne disease with direct transmission and time delay, J. math. Anal. appl. 342, 895-908 (2008) · Zbl 1146.34059 · doi:10.1016/j.jmaa.2007.12.058
[8] Roy, M.; Holt, R. D.: Effects of predation on host -- pathogen dynamics in SIR models, Theor. popul. Biol. 73, 319-331 (2008) · Zbl 1209.92054 · doi:10.1016/j.tpb.2007.12.008
[9] Tchuenche, J. M.; Nwagwo, A.; Levins, R.: Global behaviour of an SIR epidemic model with time delay, Math. methods appl. Sci. 30, 733-749 (2007) · Zbl 1112.92055 · doi:10.1002/mma.810
[10] Zhang, T. L.; Teng, Z. D.: Permanence and extinction for a nonautonomous SIRS epidemic model with time delay, Appl. math. Model. 33, 1058-1071 (2009) · Zbl 1168.34358 · doi:10.1016/j.apm.2007.12.020
[11] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal. 47, 4107-4115 (2001) · Zbl 1042.34585 · doi:10.1016/S0362-546X(01)00528-4
[12] 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
[13] Beretta, E.; Capasso, V.: Global stability results for a multigroup SIR epidemic model, Mathematical ecology, 317-342 (1986) · Zbl 0684.92015
[14] Hethcote, H. W.: An immunization model for a heterogeneous population, Theor. popul. Biol. 14, 338-349 (1978) · Zbl 0392.92009
[15] Rass, L.; Radcliffe, J.: Global asymptotic convergence results for multitype models, Int. J. Appl. math. Comput. sci. 10, 63-79 (2000) · Zbl 0978.92026
[16] Koide, C.; Seno, H.: Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease, Math. comput. Model. 23, 67-91 (1996) · Zbl 0846.92025 · doi:10.1016/0895-7177(96)00004-0
[17] Mao, X.; Marion, G.; Renshaw, E.: Asymptotic behaviour of the stochastic Lotka -- Volterra model, J. math. Anal. appl. 287, 141-156 (2003) · Zbl 1048.92027 · doi:10.1016/S0022-247X(03)00539-0
[18] Mao, X.; Marion, G.; Renshaw, E.: Environmental noise suppresses explosion in population dynamics, Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[19] Mao, X.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057
[20] Arnolld, L.; Horsthemke, W.; Stuxki, J. W.: The influence of external real and white noise on the Lotka -- Volterra model, Biomed. J. 21, 451-471 (1976) · Zbl 0433.92019 · doi:10.1002/bimj.4710210507
[21] Khasminskii, R. Z.; Klerbaner, F. C.: Long term behavior of solution of the Lotka -- Volterra system under small random perturbations, Ann. appl. Probab. 11, 952-963 (2001) · Zbl 1061.34513 · doi:10.1214/aoap/1015345354
[22] Bahar, A.; Mao, X.: Stochastic delay Lotka -- Volterra model, J. math. Anal. appl. 292, 364-380 (2004) · Zbl 1043.92034 · doi:10.1016/j.jmaa.2003.12.004
[23] Jiang, D. Q.; Shi, N. Z.: A note on nonautonomous logistic equation with random perturbation, J. math. Anal. appl. 303, 164-172 (2005) · Zbl 1076.34062 · doi:10.1016/j.jmaa.2004.08.027
[24] Jiang, D. Q.; Shi, N. Z.; Li, X. Y.: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. Anal. appl. 303, 164-172 (2008) · Zbl 1140.60032
[25] Gard, T. C.: Persistence in stochastic food web models, Bull. math. Biol. 46, 357-370 (1984) · Zbl 0533.92028
[26] Imhof, L.; Walcher, S.: Exclusion and persistence in deterministic and stochastic chemostat models, J. differential equations 217, 26-53 (2005) · Zbl 1089.34041 · doi:10.1016/j.jde.2005.06.017
[27] Dalal, N.; Greenhalgh, D.; Mao, X. R.: A stochastic model for internal HIV dynamics, J. math. Anal. appl. 341, 1084-1101 (2008) · Zbl 1132.92015 · doi:10.1016/j.jmaa.2007.11.005
[28] Dalal, N.; Greenhalgh, D.; Mao, X. R.: A stochastic model of AIDS and condom use, J. math. Anal. appl. 325, 36-53 (2007) · Zbl 1101.92037 · doi:10.1016/j.jmaa.2006.01.055
[29] Tornatore, E.; Buccellato, S. M.; Vetro, P.: Stability of a stochastic SIR system, Phys. A 354, 111-126 (2005)
[30] Carletti, M.: On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. biosci. 175, 117-131 (2002) · Zbl 0987.92027 · doi:10.1016/S0025-5564(01)00089-X
[31] Beretta, E.; Kolmanovskii, V.; Shaikhet, L.: Stability of epidemic model with time delays influenced by stochastic perturbations, Math. comput. Simulation 45, 269-277 (1998) · Zbl 1017.92504 · doi:10.1016/S0378-4754(97)00106-7