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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 0 is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if 0 1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if 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 0 >1. Numerical calculations are performed to illustrate some results for the case with two patches.
MSC:
92D30Epidemiology
37N25Dynamical systems in biology
65C40Computational Markov chains (numerical analysis)
References:
[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: lt;arXiv:1006.5028v2gt;.
[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)
[25]Zhang, Z.; Ding, T.; Huang, W.; Dong, Z.: Qualitative theory of differential equations, Qualitative theory of differential equations 101 (1992) · Zbl 0779.34001