American Institute of Mathematical Sciences

Advanced
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 & 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. doi: 10.1038/nature12060. Google Scholar

[2]

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

[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. doi: 10.3934/dcdsb.2013.18.2239. Google Scholar

[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. doi: 10.1080/17513758.2013.772253. Google Scholar

[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. doi: 10.1016/j.apm.2011.01.033. Google Scholar

[6]

CDC, Centers for disease control and prevention, Dengue Homepage,, , (). Google Scholar

[7]

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

[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. doi: 10.1007/s002850050064. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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. doi: 10.1016/j.nonrwa.2011.12.022. Google Scholar

[15]

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

[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. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar

[17]

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

[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. doi: 10.1073/pnas.0508391103. Google Scholar

[19]

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

[20]

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

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. doi: 10.1038/nature12060. Google Scholar

[2]

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

[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. doi: 10.3934/dcdsb.2013.18.2239. Google Scholar

[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. doi: 10.1080/17513758.2013.772253. Google Scholar

[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. doi: 10.1016/j.apm.2011.01.033. Google Scholar

[6]

CDC, Centers for disease control and prevention, Dengue Homepage,, , (). Google Scholar

[7]

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

[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. doi: 10.1007/s002850050064. Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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. doi: 10.1016/j.nonrwa.2011.12.022. Google Scholar

[15]

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

[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. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar

[17]

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

[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. doi: 10.1073/pnas.0508391103. Google Scholar

[19]

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

[20]

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

[1]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57
[2]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[3]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[4]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[5]

Pierre Gabriel. Global stability for the prion equation with general incidence. Mathematical Biosciences & Engineering, 2015, 12 (4) : 789-801. doi: 10.3934/mbe.2015.12.789
[6]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[7]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[8]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837
[9]

Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207
[10]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239
[11]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971
[12]

Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016
[13]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
[14]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
[15]

Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525
[16]

Attila Dénes, Gergely Röst. Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
[17]

Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 61-74. doi: 10.3934/dcdsb.2011.15.61
[18]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133
[19]

Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134
[20]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences