2017, 14(5-6): 1337-1360. doi: 10.3934/mbe.2017069

Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse

1. 

School of Mathematics and Statistics Central China Normal University Wuhan 430079, China

2. 

School of Mathematics and Computer Science Guizhou Education University Guiyang 550018, China

* Corresponding authorr: Shanjing Ren

Received  June 4, 2016 Revised  December 2016 Published  May 2017

Fund Project: This research was supported by the National Natural Science Foundation of China(N0.11371161), the Special Fund of Provincial Governor for Excellent Scientific Technology and Educational Talents(Grand No.QKJB[2012]19)

In this paper, an $SEIR$ epidemic model for an imperfect treatment disease with age-dependent latency and relapse is proposed. The model is well-suited to model tuberculosis. The basic reproduction number $R_{0}$ is calculated. We obtain the global behavior of the model in terms of $R_{0}$. If $R_{0} < 1$, the disease-free equilibrium is globally asymptotically stable, whereas if $R_{0}>1$, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present.

Citation: Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5-6) : 1337-1360. doi: 10.3934/mbe.2017069
References:
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R. A. Adams and J. J. Fournier, Sobolev Spaces Academic Press, New York, 2003.

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F. Brauer, Z. Shuai, P. van den Driessche, Dynamics of an age-of-infection cholera model, Mathematical Biosciences and Engineering, 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335.

[3]

Y. Chen, J. Yang, F. Zhang, The global stability of an SIRS model with infection age, Mathematical Biosciences and Engineering, 11 (2014), 449-469. doi: 10.3934/mbe.2014.11.449.

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P. van den Driessche, L. Wang, X. Zou, Modeling diseases with latency and relapse, Mathematical Biosciences and Engineering, 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.

[5]

P. van den Driessche, X. Zou, Modeling relapse in infectious diseases, Mathematical Biosciences, 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

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R. D. Demasse, A. Ducrot, An age-structured within-host model for multistrain malaria infections, Siam Journal on Applied Mathematics, 73 (2013), 572-593. doi: 10.1137/120890351.

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O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324.

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J. K. Hale, Asymptotic Behavior of Dissipative Systems American Mathematical Society, Providence, 1988.

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J. K. Hale, Functional Differential Equations Springer, New York, 1971.

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J. K. Hale, P. Waltman, Persistence in infinite dimensional systems, Siam Journal on Mathematical Analysis, 20 (1989), 388-395. doi: 10.1137/0520025.

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F. Hoppensteadt, Mathematical Theories of Populations: Deomgraphics, Genetics, and Epidemics SIAM, Philadelphia, 1975.

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M. Iannelli, Mathematical theory of age-structured population dynamics, Applied Mathematics Monagraphs CNR, 7 (1994).

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W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London Ser. A, 115 (1927), 700-721.

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W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics Ⅱ. The problem of endemicity, Proc.R. Soc.London Ser. A, 138 (1932), 55-83.

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W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics Ⅲ. Further studies of the problem of endemicity, Proc.R. Soc.London Ser. A, 141 (1933), 94-122.

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M. Y. Li, J. S. Muldowney, Global stability for the SEIR model in epidemilogy, Mathematical Biosciences, 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.

[17]

L. Liu, J. Wang, X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Analysis, 24 (2015), 18-35. doi: 10.1016/j.nonrwa.2015.01.001.

[18]

P. Magal, C. C. McCluskey, G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.

[19]

P. Magal, Compact attractors for time-periodic age-structured population models, Electronic Journal of Differential Equations, 65 (2001), 229-262.

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C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Mathematical Biosciences and Engineering, 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819.

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A. G. McKendrick, Application of mathematics to medical problems, Proceedings of the Edinburgh Mathematical Society, 44 (1925), 98-130. doi: 10.1017/S0013091500034428.

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H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 2011.

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C. Vargas-De-León, On the Global Stability of Infectious Diseases Models with Relapse, Abstr. Appl, 9 (2013), 50-61.

[24]

J. Wang, R. Zhang, T. Kuniya, A note on dynamics of an age-of-infection cholera model, Mathematical Biosciences and Engineering, 13 (2016), 227-247. doi: 10.3934/mbe.2016.13.227.

[25]

J. Wang, R. Zhang, T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, Mathematical Analysis and Applications, 432 (2015), 289-313. doi: 10.1016/j.jmaa.2015.06.040.

[26]

J. Wang, R. Zhang, T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, Applied Mathematics, 81 (2016), 321-343. doi: 10.1093/imamat/hxv039.

[27]

J. Wang, R. Zhang, T. Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes, Biological Dynamics, 9 (2015), 73-101. doi: 10.1080/17513758.2015.1006696.

[28]

L. Wang, Z. Liu, X. Zhang, Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination, Nonlinear Analysis: Real World Applications, 32 (2016), 136-158. doi: 10.1016/j.nonrwa.2016.04.009.

[29]

J. Wang, S. S. Gao, X. Z. Li, A TB model with infectivity in latent period and imperfect treatment, Discrete Dynamics in Nature and Society, 2012 (2012), 267-278. doi: 10.1155/2012/184918.

[30]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics Marcel Dekker, New York, 1985.

[31]

Y. Yang, S. Ruan, D. Xiao, Global stability of age-structured virus dynamics model with Beddinaton-Deangelis infection function, Mathematical Biosciences and Engineering, 12 (2015), 859-877. doi: 10.3934/mbe.2015.12.859.

[32]

J. H. Zhang, Qualitative Analysis and Data Simulation of Tuberculosis Transmission Models Ph. D thesis, Central China Normal University in Wuhan, 2014.

[33]

Centers for Disease Control and Prevention, Vaccines and Immunizations-Measles Epidemiology and Prevention of Vaccine-Preventable Diseases Available from: http://www.cdc.gov/vaccines/pubs/pinkbook/meas.html.

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces Academic Press, New York, 2003.

[2]

F. Brauer, Z. Shuai, P. van den Driessche, Dynamics of an age-of-infection cholera model, Mathematical Biosciences and Engineering, 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335.

[3]

Y. Chen, J. Yang, F. Zhang, The global stability of an SIRS model with infection age, Mathematical Biosciences and Engineering, 11 (2014), 449-469. doi: 10.3934/mbe.2014.11.449.

[4]

P. van den Driessche, L. Wang, X. Zou, Modeling diseases with latency and relapse, Mathematical Biosciences and Engineering, 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.

[5]

P. van den Driessche, X. Zou, Modeling relapse in infectious diseases, Mathematical Biosciences, 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[6]

R. D. Demasse, A. Ducrot, An age-structured within-host model for multistrain malaria infections, Siam Journal on Applied Mathematics, 73 (2013), 572-593. doi: 10.1137/120890351.

[7]

O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324.

[8]

J. K. Hale, Asymptotic Behavior of Dissipative Systems American Mathematical Society, Providence, 1988.

[9]

J. K. Hale, Functional Differential Equations Springer, New York, 1971.

[10]

J. K. Hale, P. Waltman, Persistence in infinite dimensional systems, Siam Journal on Mathematical Analysis, 20 (1989), 388-395. doi: 10.1137/0520025.

[11]

F. Hoppensteadt, Mathematical Theories of Populations: Deomgraphics, Genetics, and Epidemics SIAM, Philadelphia, 1975.

[12]

M. Iannelli, Mathematical theory of age-structured population dynamics, Applied Mathematics Monagraphs CNR, 7 (1994).

[13]

W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London Ser. A, 115 (1927), 700-721.

[14]

W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics Ⅱ. The problem of endemicity, Proc.R. Soc.London Ser. A, 138 (1932), 55-83.

[15]

W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics Ⅲ. Further studies of the problem of endemicity, Proc.R. Soc.London Ser. A, 141 (1933), 94-122.

[16]

M. Y. Li, J. S. Muldowney, Global stability for the SEIR model in epidemilogy, Mathematical Biosciences, 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.

[17]

L. Liu, J. Wang, X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Analysis, 24 (2015), 18-35. doi: 10.1016/j.nonrwa.2015.01.001.

[18]

P. Magal, C. C. McCluskey, G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.

[19]

P. Magal, Compact attractors for time-periodic age-structured population models, Electronic Journal of Differential Equations, 65 (2001), 229-262.

[20]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Mathematical Biosciences and Engineering, 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819.

[21]

A. G. McKendrick, Application of mathematics to medical problems, Proceedings of the Edinburgh Mathematical Society, 44 (1925), 98-130. doi: 10.1017/S0013091500034428.

[22]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 2011.

[23]

C. Vargas-De-León, On the Global Stability of Infectious Diseases Models with Relapse, Abstr. Appl, 9 (2013), 50-61.

[24]

J. Wang, R. Zhang, T. Kuniya, A note on dynamics of an age-of-infection cholera model, Mathematical Biosciences and Engineering, 13 (2016), 227-247. doi: 10.3934/mbe.2016.13.227.

[25]

J. Wang, R. Zhang, T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, Mathematical Analysis and Applications, 432 (2015), 289-313. doi: 10.1016/j.jmaa.2015.06.040.

[26]

J. Wang, R. Zhang, T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, Applied Mathematics, 81 (2016), 321-343. doi: 10.1093/imamat/hxv039.

[27]

J. Wang, R. Zhang, T. Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes, Biological Dynamics, 9 (2015), 73-101. doi: 10.1080/17513758.2015.1006696.

[28]

L. Wang, Z. Liu, X. Zhang, Global dynamics for an age-structured epidemic model with media impact and incomplete vaccination, Nonlinear Analysis: Real World Applications, 32 (2016), 136-158. doi: 10.1016/j.nonrwa.2016.04.009.

[29]

J. Wang, S. S. Gao, X. Z. Li, A TB model with infectivity in latent period and imperfect treatment, Discrete Dynamics in Nature and Society, 2012 (2012), 267-278. doi: 10.1155/2012/184918.

[30]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics Marcel Dekker, New York, 1985.

[31]

Y. Yang, S. Ruan, D. Xiao, Global stability of age-structured virus dynamics model with Beddinaton-Deangelis infection function, Mathematical Biosciences and Engineering, 12 (2015), 859-877. doi: 10.3934/mbe.2015.12.859.

[32]

J. H. Zhang, Qualitative Analysis and Data Simulation of Tuberculosis Transmission Models Ph. D thesis, Central China Normal University in Wuhan, 2014.

[33]

Centers for Disease Control and Prevention, Vaccines and Immunizations-Measles Epidemiology and Prevention of Vaccine-Preventable Diseases Available from: http://www.cdc.gov/vaccines/pubs/pinkbook/meas.html.

Figure 1.  Here is the Model of TB
Figure 2.  The time series of $S(t)$ and $I(t)$, and the age distributions of $e(t, a)$ and $r(t, c)$ when $\tau=12$
Figure 3.  he time series of $S(t)$ and $I(t)$, and the age distributions of $e(t, a)$ and $r(t, c)$ when $\tau=1$
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