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)
An SIS patch model with variable transmission coefficients. (English) Zbl 1218.92064
Summary: An SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number $\cal R_0$ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if $\cal R_0 \leqslant 1$, and the disease is uniformly persistent and there exists at least one endemic equilibrium if $\cal R_0 >1$. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as $\cal R_0 >1$. Numerical calculations are performed to illustrate some results for the case with two patches.

37N25Dynamical systems in biology
65C40Computational Markov chains (numerical analysis)
Full Text: DOI
[1] Cosner, C.; Beier, J. C.; Cantrell, R. S.; Impoinvil, D.; Kapitanski, L.; Potts, M. D.; Troyo, A.; Ruan, S.: The effects of human movement on the persistence of vector-borne diseases, J. theor. Biol. 258, 550 (2009)
[2] Cui, J.; Takeuchi, Y.; Saito, Y.: Spreading disease with transport-related infection, J. theor. Biol. 239, 376 (2006)
[3] Cui, J.; Tao, X.; Zhu, H.: A SIS infection model incorporating media coverage, Rocky mount. J. math. 38, 1323 (2008) · Zbl 1170.92024 · doi:10.1216/RMJ-2008-38-5-1323
[4] D. Gao, S. Ruan, A multi-patch malaria model with logistic growth populations, submitted for publication.
[5] Greenhalgh, D.: Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. comput. Model. 25, No. 2, 85 (1997) · Zbl 0877.92023 · doi:10.1016/S0895-7177(97)00009-5
[6] Hadeler, K. P.; Thieme, H. R.: Monotone dependence of the spectral bound on the transition rates in linear compartmental models, J. math. Biol. 57, 697 (2008) · Zbl 1161.92043 · doi:10.1007/s00285-008-0185-z
[7] Jin, Y.; Wang, W.: The effect of population dispersal on the spread of a disease, J. math. Anal. appl. 308, 343 (2005) · Zbl 1065.92044 · doi:10.1016/j.jmaa.2005.01.034
[8] Kamgang, J. C.; Sallet, G.: Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE), Math. biosci. 213, 1 (2008) · Zbl 1135.92030 · doi:10.1016/j.mbs.2008.02.005
[9] Liu, R.; Wu, J.; Zhu, H.: Media/psychological impact on multiple outbreaks of emerging infectious diseases, Comput. math. Methods med. 8, 153 (2007) · Zbl 1121.92060 · doi:10.1080/17486700701425870
[10] A. Mummert, H. Weiss, Get the news out loudly and quickly: modeling the influence of the media on limiting infectious disease outbreaks, 2010. Available from: <arXiv:1006.5028v2>.
[11] Salmani, M.; Den Driessche, P. Van: A model for disease transmission in a patchy environment, Discrete contin. Dyn. syst. Ser. B 6, 185 (2006) · Zbl 1088.92050 · doi:10.3934/dcdsb.2006.6.185
[12] Seibert, P.; Suarez, R.: Global stabilization of nonlinear cascade systems, Syst. control lett. 14, 347 (1990) · Zbl 0699.93073 · doi:10.1016/0167-6911(90)90056-Z
[13] Smith, H. L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical surveys and monographs 41 (1995) · Zbl 0821.34003
[14] Smith, H. L.; Waltman, P.: Perturbation of a globally stable steady state, Proc. amer. Math. soc. 127, 447 (1999) · Zbl 0924.58087 · doi:10.1090/S0002-9939-99-04768-1
[15] Smith, H. L.; Zhao, X. -Q.: Dynamics of a periodically pulsed bio-reactor model, J. differ. Equat. 155, 368 (1999) · Zbl 0930.35085 · doi:10.1006/jdeq.1998.3587
[16] Sun, C.; Wei, Y.; Arino, J.; Khan, K.: Effect of media-induced social distancing on disease transmission in a two patch setting, Math. biosci. 230, 87 (2011) · Zbl 1211.92051 · doi:10.1016/j.mbs.2011.01.005
[17] Thieme, H. R.: Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. anal. 24, 407 (1993) · Zbl 0774.34030 · doi:10.1137/0524026
[18] Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[19] Vidyasagar, M.: Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE trans. Automat. control 25, 773 (1980) · Zbl 0478.93044 · doi:10.1109/TAC.1980.1102422
[20] Wang, W.; Mulone, G.: Threshold of disease transmission in a patch environment, J. math. Anal. appl. 285, 321 (2003) · Zbl 1021.92039 · doi:10.1016/S0022-247X(03)00428-1
[21] Wang, W.; Zhao, X. -Q.: An epidemic model in a patchy environment, Math. biosci. 190, 97 (2004) · Zbl 1048.92030 · doi:10.1016/j.mbs.2002.11.001
[22] Zhao, X. -Q.: Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Can. appl. Math. quart. 3, 473 (1995) · Zbl 0849.34034
[23] Zhao, X. -Q.; Jing, Z. -J.: Global asymptotic behavior in some cooperative systems of functional differential equations, Can. appl. Math. quart. 4, 421 (1996) · Zbl 0888.34038
[24] Zhao, X. -Q.: Dynamical systems in population biology, (2003) · Zbl 1023.37047
[25] Zhang, Z.; Ding, T.; Huang, W.; Dong, Z.: Qualitative theory of differential equations, Qualitative theory of differential equations 101 (1992) · Zbl 0779.34001