Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Lyapunov functions for tuberculosis models with fast and slow progression

1. Department of Mathematics, Wilfrid Laurier University, 75 University Ave West, Waterloo, ON, N2L 3C5

The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptotically stable.
  Article Metrics

Keywords fast and slow progression.; Lyapunov function; global stability; tuberculosis

Citation: C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences and Engineering, 2006, 3(4): 603-614. doi: 10.3934/mbe.2006.3.603


This article has been cited by

  • 1. C. Connell McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, Journal of Mathematical Analysis and Applications, 2008, 338, 1, 518, 10.1016/j.jmaa.2007.05.012
  • 2. Horst R. Thieme, Hal L. Smith, Chemostats and epidemics: Competition for nutrients/hosts, Mathematical Biosciences and Engineering, 2013, 10, 5/6, 1635, 10.3934/mbe.2013.10.1635
  • 3. C. P. Bhunu, S. Mushayabasa, J. M. Tchuenche, A Theoretical Assessment of the Effects of Smoking on the Transmission Dynamics of Tuberculosis, Bulletin of Mathematical Biology, 2011, 73, 6, 1333, 10.1007/s11538-010-9568-6
  • 4. Xueyong Zhou, Xiangyun Shi, Huidong Cheng, Modelling and stability analysis for a tuberculosis model with healthy education and treatment, Computational and Applied Mathematics, 2013, 32, 2, 245, 10.1007/s40314-013-0008-8
  • 5. Fuxiang Li, Wanbiao Ma, Dynamics analysis of an HTLV-1 infection model with mitotic division of actively infected cells and delayed CTL immune response, Mathematical Methods in the Applied Sciences, 2018, 41, 8, 3000, 10.1002/mma.4797
  • 6. Elamin H. Elbasha, Global Stability of Equilibria in a Two-Sex HPV Vaccination Model, Bulletin of Mathematical Biology, 2008, 70, 3, 894, 10.1007/s11538-007-9283-0
  • 7. Asaf Khan, Gul Zaman, Optimal control strategy of SEIR endemic model with continuous age-structure in the exposed and infectious classes, Optimal Control Applications and Methods, 2018, 10.1002/oca.2437
  • 8. Bing Li, Shengqiang Liu, Jing’an Cui, Jia Li, A Simple Predator-Prey Population Model with Rich Dynamics, Applied Sciences, 2016, 6, 5, 151, 10.3390/app6050151
  • 9. Xinli Hu, Threshold dynamics for a Tuberculosis model with seasonality, Mathematical Biosciences and Engineering, 2011, 9, 1, 111, 10.3934/mbe.2012.9.111
  • 10. Chayu Yang, Paride O. Lolika, Steady Mushayabasa, Jin Wang, Modeling the spatiotemporal variations in brucellosis transmission, Nonlinear Analysis: Real World Applications, 2017, 38, 49, 10.1016/j.nonrwa.2017.04.006
  • 11. C. P. BHUNU, S. MUSHAYABASA, ASSESSING THE EFFECTS OF INTRAVENOUS DRUG USE ON HEPATITIS C TRANSMISSION DYNAMICS, Journal of Biological Systems, 2011, 19, 03, 447, 10.1142/S0218339011004056
  • 12. Bruno Buonomo, Deborah Lacitignola, Analysis of a tuberculosis model with a case study in Uganda, Journal of Biological Dynamics, 2010, 4, 6, 571, 10.1080/17513750903518441
  • 13. Lili Liu, Jinliang Wang, Xianning Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Analysis: Real World Applications, 2015, 24, 18, 10.1016/j.nonrwa.2015.01.001
  • 14. Julie Nadeau, C. Connell McCluskey, Global stability for an epidemic model with applications to feline infectious peritonitis and tuberculosis, Applied Mathematics and Computation, 2014, 230, 473, 10.1016/j.amc.2013.12.124
  • 15. Saul C. Mpeshe, Heikki Haario, Jean M. Tchuenche, A Mathematical Model of Rift Valley Fever with Human Host, Acta Biotheoretica, 2011, 59, 3-4, 231, 10.1007/s10441-011-9132-2
  • 16. Andrey Melnik, Andrei Korobeinikov, Global asymptotic properties of staged models with multiple progression pathways for infectious diseases, Mathematical Biosciences and Engineering, 2011, 8, 4, 1019, 10.3934/mbe.2011.8.1019
  • 17. D. Okuonghae, Lyapunov functions and global properties of some tuberculosis models, Journal of Applied Mathematics and Computing, 2015, 48, 1-2, 421, 10.1007/s12190-014-0811-4
  • 18. Xue-yong Zhou, Jing-an Cui, Zhong-hua Zhang, Global results for a cholera model with imperfect vaccination, Journal of the Franklin Institute, 2012, 349, 3, 770, 10.1016/j.jfranklin.2011.09.013
  • 19. Sylvie Diane Djiomba Njankou, Farai Nyabadza, Modelling the potential impact of limited hospital beds on Ebola virus disease dynamics, Mathematical Methods in the Applied Sciences, 2018, 10.1002/mma.4789
  • 20. S.A. Pedro, J.M. Tchuenche, HIV/AIDS dynamics: Impact of economic classes with transmission from poor clinical settings, Journal of Theoretical Biology, 2010, 267, 4, 471, 10.1016/j.jtbi.2010.09.019
  • 21. C. P. BHUNU, MODELING THE SPREAD OF STREET KIDS IN ZIMBABWE, Journal of Biological Systems, 2014, 22, 03, 429, 10.1142/S0218339014500168
  • 22. S. MUSHAYABASA, C. P. BHUNU, MODELING THE IMPACT OF VOLUNTARY TESTING AND TREATMENT ON TUBERCULOSIS TRANSMISSION DYNAMICS, International Journal of Biomathematics, 2012, 05, 04, 1250029, 10.1142/S1793524511001726
  • 23. Junli Liu, Tailei Zhang, Global stability for a tuberculosis model, Mathematical and Computer Modelling, 2011, 54, 1-2, 836, 10.1016/j.mcm.2011.03.033
  • 24. Rigobert C. Ngeleja, Livingstone S. Luboobi, Yaw Nkansah-Gyekye, Plague disease model with weather seasonality, Mathematical Biosciences, 2018, 10.1016/j.mbs.2018.05.013
  • 25. Xichao Duan, Sanling Yuan, Xuezhi Li, Global stability of an SVIR model with age of vaccination, Applied Mathematics and Computation, 2014, 226, 528, 10.1016/j.amc.2013.10.073
  • 26. Shu Liao, Jin Wang, Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices, Chaos, Solitons & Fractals, 2012, 45, 7, 966, 10.1016/j.chaos.2012.03.009
  • 27. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Mathematical Biosciences and Engineering, 2012, 9, 4, 819, 10.3934/mbe.2012.9.819
  • 28. Fuxiang Li, Wanbiao Ma, Zhichao Jiang, Dan Li, Stability and Hopf Bifurcation in a Delayed HIV Infection Model with General Incidence Rate and Immune Impairment, Computational and Mathematical Methods in Medicine, 2015, 2015, 1, 10.1155/2015/206205
  • 29. Rigobert Charles Ngeleja, Livingstone Luboobi, Yaw Nkansah-Gyekye, Stability Analysis of Bubonic Plague Model with the Causing Pathogen <i>Yersinia pestis</i> in the Environment, Advances in Infectious Diseases, 2016, 06, 03, 120, 10.4236/aid.2016.63016
  • 30. A. Mhlanga, C. P. Bhunu, S. Mushayabasa, HSV-2 and Substance Abuse amongst Adolescents: Insights through Mathematical Modelling, Journal of Applied Mathematics, 2014, 2014, 1, 10.1155/2014/104819
  • 31. Jean Jules Tewa, Samuel Bowong, Boulchard Mewoli, Mathematical analysis of two-patch model for the dynamical transmission of tuberculosis, Applied Mathematical Modelling, 2012, 36, 6, 2466, 10.1016/j.apm.2011.09.004
  • 32. Cruz Vargas-De-León, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos, Solitons & Fractals, 2011, 44, 12, 1106, 10.1016/j.chaos.2011.09.002
  • 33. Samuel Bowong, Jean Jules Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Communications in Nonlinear Science and Numerical Simulation, 2009, 14, 11, 4010, 10.1016/j.cnsns.2009.02.017
  • 34. S. Mushayabasa, C.P. Bhunu, C. Webb, M. Dhlamini, A mathematical model for assessing the impact of poverty on yaws eradication, Applied Mathematical Modelling, 2012, 36, 4, 1653, 10.1016/j.apm.2011.09.022
  • 35. Jianquan Li, Yanni Xiao, Fengqin Zhang, Yali Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Analysis: Real World Applications, 2012, 13, 5, 2006, 10.1016/j.nonrwa.2011.12.022
  • 36. Samuel Bowong, Jean Jules Tewa, Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate, Communications in Nonlinear Science and Numerical Simulation, 2010, 15, 11, 3621, 10.1016/j.cnsns.2010.01.007
  • 37. Hai-Feng Huo, Shuai-Jun Dang, Yu-Ning Li, Stability of a Two-Strain Tuberculosis Model with General Contact Rate, Abstract and Applied Analysis, 2010, 2010, 1, 10.1155/2010/293747
  • 38. Ram P. Sigdel, C. Connell McCluskey, Global stability for an SEI model of infectious disease with immigration, Applied Mathematics and Computation, 2014, 243, 684, 10.1016/j.amc.2014.06.020
  • 39. Samuel Bowong, Jurgen Kurths, Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality, Nonlinear Dynamics, 2012, 67, 3, 2027, 10.1007/s11071-011-0127-y
  • 40. C. Connell McCluskey, Using Lyapunov Functions to Construct Lyapunov Functionals for Delay Differential Equations, SIAM Journal on Applied Dynamical Systems, 2015, 14, 1, 1, 10.1137/140971683
  • 41. Jianquan Li, Yali Yang, Yicang Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Analysis: Real World Applications, 2011, 12, 4, 2163, 10.1016/j.nonrwa.2010.12.030
  • 42. Isaac Mwangi Wangari, Lewi Stone, Analysis of a Heroin Epidemic Model with Saturated Treatment Function, Journal of Applied Mathematics, 2017, 2017, 1, 10.1155/2017/1953036
  • 43. Victor Moreno, Baltazar Espinoza, Kamal Barley, Marlio Paredes, Derdei Bichara, Anuj Mubayi, Carlos Castillo-Chavez, The role of mobility and health disparities on the transmission dynamics of Tuberculosis, Theoretical Biology and Medical Modelling, 2017, 14, 1, 10.1186/s12976-017-0049-6
  • 44. Jean Claude Kamgang, Vivient Corneille Kamla, Stéphane Yanick Tchoumi, Modeling the Dynamics of Malaria Transmission with Bed Net Protection Perspective, Applied Mathematics, 2014, 05, 19, 3156, 10.4236/am.2014.519298
  • 45. Samuel Bowong, Optimal control of the transmission dynamics of tuberculosis, Nonlinear Dynamics, 2010, 61, 4, 729, 10.1007/s11071-010-9683-9
  • 46. Horst R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, Journal of Differential Equations, 2011, 250, 9, 3772, 10.1016/j.jde.2011.01.007
  • 47. Y. Ma, C. R. Horsburgh, L. F. White, H. E. Jenkins, Quantifying TB transmission: a systematic review of reproduction number and serial interval estimates for tuberculosis, Epidemiology and Infection, 2018, 1, 10.1017/S0950268818001760
  • 48. King-Yeung Lam, Xueying Wang, Tianran Zhang, Traveling Waves for a Class of Diffusive Disease-Transmission Models with Network Structures, SIAM Journal on Mathematical Analysis, 2018, 50, 6, 5719, 10.1137/17M1144258
  • 49. Rachel A. Nyang’inja, David N. Angwenyi, Cecilia M. Musyoka, Titus O. Orwa, Mathematical modeling of the effects of public health education on tungiasis—a neglected disease with many challenges in endemic communities, Advances in Difference Equations, 2018, 2018, 1, 10.1186/s13662-018-1875-5
  • 50. Luju Liu, Weiyun Cai, Yusen Wu, Global dynamics for an SIR patchy model with susceptibles dispersal, Advances in Difference Equations, 2012, 2012, 1, 10.1186/1687-1847-2012-131
  • 51. Hong Xiang, Ming-Xuan Zou, Hai-Feng Huo, Modeling the Effects of Health Education and Early Therapy on Tuberculosis Transmission Dynamics, International Journal of Nonlinear Sciences and Numerical Simulation, 2019, 0, 0, 10.1515/ijnsns-2016-0084
  • 52. Christian Kuehn, , Multiple Time Scale Dynamics, 2015, Chapter 20, 665, 10.1007/978-3-319-12316-5_20
  • 53. P. Magal, C.C. McCluskey, G.F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 2010, 89, 7, 1109, 10.1080/00036810903208122
  • 54. S. B. Chibaya, F. Nyabadza, Mathematical Modelling of the Potential Role of Supplementary Feeding for People Living with HIV/AIDS, International Journal of Applied and Computational Mathematics, 2019, 5, 3, 10.1007/s40819-019-0660-9
  • 55. Jean Claude Kamgang, Christopher Penniman Thron, Analysis of Malaria Control Measures’ Effectiveness Using Multistage Vector Model, Bulletin of Mathematical Biology, 2019, 10.1007/s11538-019-00637-6
  • 56. A. O. Egonmwan, D. Okuonghae, Mathematical analysis of a tuberculosis model with imperfect vaccine, International Journal of Biomathematics, 2019, 10.1142/S1793524519500736
  • 57. Isa Abdullahi Baba, Rabiu Aliyu Abdulkadir, Parvaneh Esmaili, Analysis of tuberculosis model with saturated incidence rate and optimal control, Physica A: Statistical Mechanics and its Applications, 2019, 123237, 10.1016/j.physa.2019.123237
  • 58. A. Mhlanga, Dynamical analysis and control strategies in modelling Ebola virus disease, Advances in Difference Equations, 2019, 2019, 1, 10.1186/s13662-019-2392-x
  • 59. Mohammad Hassan Nematollahi, Ramin Vatankhah, Mojtaba Sharifi, Nonlinear adaptive control of tuberculosis with consideration of the risk of endogenous reactivation and exogenous reinfection, Journal of Theoretical Biology, 2019, 110081, 10.1016/j.jtbi.2019.110081
  • 60. Pankaj Singh Rana, Nitin Sharma, Mathematical modeling and stability analysis of a SI type model for HIV/AIDS, Journal of Interdisciplinary Mathematics, 2020, 23, 1, 257, 10.1080/09720502.2020.1721921
  • 61. A. Mhlanga, Assessing the Impact of Optimal Health Education Programs on the Control of Zoonotic Diseases, Computational and Mathematical Methods in Medicine, 2020, 2020, 1, 10.1155/2020/6584323
  • 62. Ran Zhang, Lili Liu, Xiaomei Feng, Zhen Jin, Existence of traveling wave solutions for a diffusive tuberculosis model with fast and slow progression, Applied Mathematics Letters, 2021, 112, 106848, 10.1016/j.aml.2020.106848

Reader Comments

your name: *   your email: *  

Copyright Info: 2006, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved