American Institute of Mathematical Sciences

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September  2015, 20(7): 2217-2232. doi: 10.3934/dcdsb.2015.20.2217

Global stability of the dengue disease transmission models

Jing-Jing Xiang 1, , Juan Wang 2, and  Li-Ming Cai 2,

1. 

School of Science, Xi'an University of Architecture & Technology, Xi'an, 710055, China
2. 

Department of Mathematics, Xinyang Normal University, Xinyang 464000, China

Received  January 2015 Revised  May 2015 Published  July 2015

In this paper, we further investigate the global stability of the dengue transmission models. By using persistence theory, it is showed that the disease of system uniformly persists when the basic reproduction number is larger than unity. By constructing suitable Lyapunov function methods and LaSalle Invariance Principle, we show that the unique endemic equilibrium of the model is always globally asymptotically stable as long as it exists.
Keywords: basic reproduction number, Lyapunov function, global stability., Mass action incidence.
Mathematics Subject Classification: Primary: 34D23; Secondary: 93D05, 92D2.
Citation: Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217
References:
[1]

S. Bhatt, P. W. Gething, O. J. Brady, et al., The global distribution and burden of dengue, Nature, 496 (2013), 504-507. doi: 10.1038/nature12060.

[2]

L. Cai, S. Guo, X. Li and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos, Solitons and Fractals, 42 (2009), 2297-2304. doi: 10.1016/j.chaos.2009.03.130.

[3]

L. Cai, M. Martcheva and X. Li, epidemic models with age of infection, indirect transmission and incomplete treatment, Discrete and Continuous Dynamical Systems Series B, 18 (2013), 2239-2265. doi: 10.3934/dcdsb.2013.18.2239.

[4]

L. Cai, M. Martcheva and X. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, Journal of Biological Dynamics, 7 (2013), 47-67. doi: 10.1080/17513758.2013.772253.

[5]

L. Cai, X. Li and M. Ghosh, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T-cells, Appl. Math. Model., 35 (2011), 3587-3595. doi: 10.1016/j.apm.2011.01.033.

[6]

CDC, Centers for disease control and prevention, Dengue Homepage, http://www.cdc.gov/Dengue/epidemiology/index.html

[7]

L. Esteva and C. Vargas, Coexistence of different serotypes of dengue virus, J. Math. Biol., 46 (2003), 31-47. doi: 10.1007/s00285-002-0168-4.

[8]

Z. Feng and X. Jorge Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544. doi: 10.1007/s002850050064.

[9]

S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002.

[10]

S. B. Halstead, Pathogenesis of dengue: Challenges to molecular biology, Science, 239 (1988), 476-481. doi: 10.1126/science.3277268.

[11]

J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proceedings of the American Mathematical Society, 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.

[12]

D. Kalajdzievska and M. Y. Li, Modeling the effects of carriers on transmission dynamics of infectious diseases, Math. Biosci. Eng., 8 (2011), 711-722. doi: 10.3934/mbe.2011.8.711.

[13]

M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.

[14]

J. Li, Y. Xiao, F. Zhang and Y. Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Anal. RWA., 13 (2012), 2006-2016. doi: 10.1016/j.nonrwa.2011.12.022.

[15]

Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Sciences Press, Beijing, 2004.

[16]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal: Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[17]

P. Pongsumpun and I. M. Tang, Transmission of dengue hemorrhagic fever in an age structured population, Math. Comput. Model., 37 (2003), 949-961. doi: 10.1016/S0895-7177(03)00111-0.

[18]

A. J. Tatem, S. I. Hay and D. J. Rogers, Global traffic and disease vector dispersal, Proc. Natl. Acad. Sci. USA., 103 (2006), 6242-6247. doi: 10.1073/pnas.0508391103.

[19]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026.

[20]

H. M. Yang and C. P. Ferreira, Assessing the effects of vector control on dengue transmission, Appl. Math. Computat., 198 (2008), 401-413. doi: 10.1016/j.amc.2007.08.046.

show all references

References:
[1]

S. Bhatt, P. W. Gething, O. J. Brady, et al., The global distribution and burden of dengue, Nature, 496 (2013), 504-507. doi: 10.1038/nature12060.

[2]

L. Cai, S. Guo, X. Li and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos, Solitons and Fractals, 42 (2009), 2297-2304. doi: 10.1016/j.chaos.2009.03.130.

[3]

L. Cai, M. Martcheva and X. Li, epidemic models with age of infection, indirect transmission and incomplete treatment, Discrete and Continuous Dynamical Systems Series B, 18 (2013), 2239-2265. doi: 10.3934/dcdsb.2013.18.2239.

[4]

L. Cai, M. Martcheva and X. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, Journal of Biological Dynamics, 7 (2013), 47-67. doi: 10.1080/17513758.2013.772253.

[5]

L. Cai, X. Li and M. Ghosh, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T-cells, Appl. Math. Model., 35 (2011), 3587-3595. doi: 10.1016/j.apm.2011.01.033.

[6]

CDC, Centers for disease control and prevention, Dengue Homepage, http://www.cdc.gov/Dengue/epidemiology/index.html

[7]

L. Esteva and C. Vargas, Coexistence of different serotypes of dengue virus, J. Math. Biol., 46 (2003), 31-47. doi: 10.1007/s00285-002-0168-4.

[8]

Z. Feng and X. Jorge Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544. doi: 10.1007/s002850050064.

[9]

S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002.

[10]

S. B. Halstead, Pathogenesis of dengue: Challenges to molecular biology, Science, 239 (1988), 476-481. doi: 10.1126/science.3277268.

[11]

J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proceedings of the American Mathematical Society, 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.

[12]

D. Kalajdzievska and M. Y. Li, Modeling the effects of carriers on transmission dynamics of infectious diseases, Math. Biosci. Eng., 8 (2011), 711-722. doi: 10.3934/mbe.2011.8.711.

[13]

M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.

[14]

J. Li, Y. Xiao, F. Zhang and Y. Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Anal. RWA., 13 (2012), 2006-2016. doi: 10.1016/j.nonrwa.2011.12.022.

[15]

Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Sciences Press, Beijing, 2004.

[16]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal: Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[17]

P. Pongsumpun and I. M. Tang, Transmission of dengue hemorrhagic fever in an age structured population, Math. Comput. Model., 37 (2003), 949-961. doi: 10.1016/S0895-7177(03)00111-0.

[18]

A. J. Tatem, S. I. Hay and D. J. Rogers, Global traffic and disease vector dispersal, Proc. Natl. Acad. Sci. USA., 103 (2006), 6242-6247. doi: 10.1073/pnas.0508391103.

[19]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026.

[20]

H. M. Yang and C. P. Ferreira, Assessing the effects of vector control on dengue transmission, Appl. Math. Computat., 198 (2008), 401-413. doi: 10.1016/j.amc.2007.08.046.

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