# American Institute of Mathematical Sciences

October 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 04, 2016 Revised  December 30, 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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Diekmann, J. A. P. Heesterbeek and 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. doi: 10.1090/surv/025. [9] J. K. Hale, Functional Differential Equations, Springer, New York, 1971. doi: 10.1007/BFb0060406. [10] J. K. Hale and 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. doi: 10.1137/1.9781611970487. [12] M. Iannelli, Mathematical theory of age-structured population dynamics, Applied Mathematics Monagraphs CNR, 7 (1994). [13] W. O. Kermack and 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 and 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 and 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 and 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 and 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 and 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. doi: 10.1090/gsm/118. [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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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. doi: 10.1090/surv/025. [9] J. K. Hale, Functional Differential Equations, Springer, New York, 1971. doi: 10.1007/BFb0060406. [10] J. K. Hale and 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. doi: 10.1137/1.9781611970487. [12] M. Iannelli, Mathematical theory of age-structured population dynamics, Applied Mathematics Monagraphs CNR, 7 (1994). [13] W. O. Kermack and 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 and 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 and 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 and 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 and 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 and 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. doi: 10.1090/gsm/118. [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 and 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 and 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 and 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 and 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 and 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 and 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 and 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.
Here is the Model of TB
The time series of $S(t)$ and $I(t)$, and the age distributions of $e(t, a)$ and $r(t, c)$ when $\tau=12$
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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