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)
The domain of attraction for the endemic equilibrium of an SIRS epidemic model. (English) Zbl 1217.92076
Summary: A new method is adopted to construct a Lyapunov function for the endemic equilibrium of the J. Mena-Lorca and H. W. Hethcote [J. Math. Biol. 30, No. 7, 693–716 (1992; Zbl 0748.92012)] SIRS epidemic model with bilinear incidence and constant recruitment. On the basis of the Lyapunov function, the domain of the attraction of the endemic equilibrium is estimated by solving a linear matrix inequality (LMI) optimization problem with multivariate polynomial objective function and constraints.
MSC:
92D30Epidemiology
37N25Dynamical systems in biology
34D20Stability of ODE
90C90Applications of mathematical programming
92-08Computational methods (appl. to natural sciences)
References:
[1]Aiello, O. E.; Da Silva, M. A. A.: New approach to dynamical Monte Carlo method: application to epidemic model, Physica A 327, 525-534 (2003) · Zbl 1026.92035 · doi:10.1016/S0378-4371(03)00504-1
[2]Brauer, F.; Castillo-Chavez, C.: Mathematical models in population biology and epidemiology, (2001)
[3]Ball, F.; Neal, P.: A general model for stochastic SIR epidemics with two levels of mixing, Math. biosci. 180, 73-102 (2002) · Zbl 1015.92034 · doi:10.1016/S0025-5564(02)00125-6
[4]Balint, S.; Balint, A.; Negru, V.: The optimal Lyapunov function in diagonlizable case, An. univ. Timisoara, seria. St. mat. 24, 1-7 (1986) · Zbl 0615.34050
[5]Balint, S.: Considerations concerning the maneuvering of some physical systems, An. univ. Timisoara, seria. St. mat. 23, 8-16 (1985) · Zbl 0664.93039
[6]D’onofio, A.: On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. math. Lett. 18, 729-732 (2005) · Zbl 1064.92041 · doi:10.1016/j.aml.2004.05.012
[7]El-Doma, M.: Stability analysis for a general age-dependent vaccination model, Math. comput. Modell. 24, 109-117 (1996) · Zbl 0894.92027 · doi:10.1016/0895-7177(96)00131-8
[8]Fan, J. S.: A new extracting formula and a new distinguishing means on the variable cubic equiation, Nat. sci. J. Hainan teachers coll. 2, 91-98 (1989)
[9]Greenhalgh, D.; Khan, Q. J. A.; Lewis, F. I.: Hopf bifurcation in two SIRS density dependent epidemic models, Math. comput. Modell. 39, 1261-1283 (2004) · Zbl 1065.92042 · doi:10.1016/j.mcm.2004.06.007
[10]Hethcote, H.; Ma, Z. E.; Liao, S. B.: Effects of quarantine in six endemic models for infectious diseases, Math. biosci. 180, 141-160 (2002) · Zbl 1019.92030 · doi:10.1016/S0025-5564(02)00111-6
[11]D. Henrion, J.B. Lasserre, Gloptipoly: global optimization over polynomials with Matlab and SeDuMi, 2002. http://homepages.laas.fr/henrion/.
[12]Hormander, L.: An introduction to complex analysis in several variables, (1989)
[13]Kaslik, E.; Balint, A. M.; Balint, St.: Methods for determination and approximation of the domain of attraction, Nonlinear anal. 60, 703-717 (2005) · Zbl 1066.34053 · doi:10.1016/j.na.2004.09.046
[14]Kermack, W. O.; Mckendrick, A. G.: Contributions to the mathematical theory of epidemic. Part-I, Proc. R soc. 115A, 700-721 (1927) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[15]La Salle, J.; Lefschetz, S.: Stability by liapunovs direct method, (1961) · Zbl 0098.06102
[16]Lasserre, J. B.: Global optimization with polynomials and the problem of moments, SIAM J. Optim. 11, 796-817 (2001) · Zbl 1010.90061 · doi:10.1137/S1052623400366802
[17]Lasserre, J. B.: An explicit equivalent positive semidefinite program for 0 – 1 nonlinear programs, SIAM J. Optim. 12, 756-769 (2002) · Zbl 1007.90046 · doi:10.1137/S1052623400380079
[18]Mena-Lorca, J.; Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population size, J. math. Biol. 30, 693-716 (1992) · Zbl 0748.92012
[19]Song, B. J.; Castillo-Chavez, C.; Aparicio, J. P.: Tuberculosis models with fast and slow dynamics: the role of close and casual contacts, Math. biosci. 180, 187-205 (2002) · Zbl 1015.92025 · doi:10.1016/S0025-5564(02)00112-8
[20]J.F. Sturm, Using SeDuMi1.02, a Matlab toolbox for optimization over symmetric cones, 1999. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.6954.
[21]Zhang, Z. H.; Suo, Y. H.; Peng, J. G.; Lin, W. H.: Singular perturbation approach to stability of an SIRS epidemic system, Nonlinear anal. Rear world appl. 5, 2688-2699 (2009) · Zbl 1162.92039 · doi:10.1016/j.nonrwa.2008.07.009