# American Institute of Mathematical Sciences

June  2019, 24(6): 2923-2939. doi: 10.3934/dcdsb.2018292

## Global analysis of a stochastic TB model with vaccination and treatment

 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Zhipeng Qiu

Received  September 2017 Revised  February 2018 Published  June 2019 Early access  October 2018

Fund Project: Z. Qiu is supported by the National Natural Science Foundation of China (NSFC) grant No. 11671206, T. Feng is supported by the Scholarship Foundation of China Scholarship Council grant No. 201806840120, the Postgraduate Research & Practice Innovation Program of Jiangsu Province grant No. KYCX18 0370 and the Fundamental Research Funds for the Central Universities grant No. 30918011339.

In this paper, a stochastic model is formulated to describe the transmission dynamics of tuberculosis. The model incorporates vaccination and treatment in the intervention strategies. Firstly, sufficient conditions for persistence in mean and extinction of tuberculosis are provided. In addition, sufficient conditions are obtained for the existence of stationary distribution and ergodicity. Moreover, numerical simulations are given to illustrate these analytical results. The theoretical and numerical results show that large environmental disturbances can suppress the spread of tuberculosis.

Citation: Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292
##### References:
 [1] S. Bowong and J. J. Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 4010-4021.  doi: 10.1016/j.cnsns.2009.02.017. [2] Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic sirs epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502.  doi: 10.1016/j.jde.2015.08.024. [3] Z. Chang, X. Meng and X. Lu, Analysis of a novel stochastic sirs epidemic model with two different saturated incidence rates, Physica A: Statistical Mechanics and its Applications, 472 (2017), 103-116.  doi: 10.1016/j.physa.2017.01.015. [4] S. Choi, E. Jung and S.-M. Lee, Optimal intervention strategy for prevention tuberculosis using a smoking-tuberculosis model, Journal of Theoretical Biology, 380 (2015), 256-270.  doi: 10.1016/j.jtbi.2015.05.022. [5] T. Feng, X. Meng, L. Liu and S. Gao, Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model, Journal of Inequalities and Applications, 2016 (2016), Paper No. 327, 29 pp. doi: 10.1186/s13660-016-1265-z. [6] D. Gao and N. Huang, A note on global stability for a tuberculosis model, Applied Mathematics Letters, 73 (2017), 163-168.  doi: 10.1016/j.aml.2017.05.004. [7] S. Gao, L. Chen and Z. Teng, Pulse vaccination of an seir epidemic model with time delay, Nonlinear Analysis: Real World Applications, 9 (2008), 599-607.  doi: 10.1016/j.nonrwa.2006.12.004. [8] X. Han and P. E. Kloeden, Immune System Virus Model, Springer Singapore, Singapore, 2017. [9] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with beddington-deangelis functional response, Applied Mathematics Letters, 22 (2009), 1690-1693.  doi: 10.1016/j.aml.2009.06.004. [10] H. Huo and M. Zou, Modelling effects of treatment at home on tuberculosis transmission dynamics, Applied Mathematical Modelling, 40 (2016), 9474-9484.  doi: 10.1016/j.apm.2016.06.029. [11] D. Jiang, C. Ji, N. Shi and J. Yu, The long time behavior of di sir epidemic model with stochastic perturbation, Journal of Mathematical Analysis and Applications, 372 (2010), 162-180.  doi: 10.1016/j.jmaa.2010.06.003. [12] D. Jiang, Q. Liu, N. Shi, T. Hayat, A. Alsaedi and P. Xia, Dynamics of a stochastic hiv-1 infection model with logistic growth, Physica A: Statistical Mechanics and its Applications, 469 (2017), 706-717.  doi: 10.1016/j.physa.2016.11.078. [13] D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, Journal of Mathematical Analysis and Applications, 303 (2005), 164-172.  doi: 10.1016/j.jmaa.2004.08.027. [14] R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [15] P. E. Kloeden and C. Pötzsche, Nonautonomous bifurcation scenarios in sir models, Mathematical Methods in the Applied Sciences, 38 (2015), 3495-3518.  doi: 10.1002/mma.3433. [16] A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for sir, sirs, and sis epidemiological models, Applied Mathematics Letters, 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1. [17] X. Leng, T. Feng and X. Meng, Stochastic inequalities and applications to dynamics analysis of a novel sivs epidemic model with jumps, Journal of Inequalities and Applications, 2017 (2017), Paper No. 138, 25 pp. doi: 10.1186/s13660-017-1418-8. [18] G. Li and Z. Jin, Global stability of a seir epidemic model with infectious force in latent, infected and immune period, Chaos, Solitons & Fractals, 25 (2005), 1177-1184.  doi: 10.1016/j.chaos.2004.11.062. [19] M. Liu, C. Bai and K. Wang, Asymptotic stability of a two-group stochastic seir model with infinite delays, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3444-3453.  doi: 10.1016/j.cnsns.2014.02.025. [20] Q. Liu, The threshold of a stochastic susceptible-infective epidemic model under regime switching, Nonlinear Analysis: Hybrid Systems, 21 (2016), 49-58.  doi: 10.1016/j.nahs.2016.01.002. [21] Q. Liu and Q. Chen, Analysis of the deterministic and stochastic sirs epidemic models with nonlinear incidence, Physica A: Statistical Mechanics and its Applications, 428 (2015), 140-153.  doi: 10.1016/j.physa.2015.01.075. [22] Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Dynamics of a stochastic tuberculosis model with constant recruitment and varying total population size, Physica A: Statistical Mechanics and its Applications, 469 (2017), 518-530.  doi: 10.1016/j.physa.2016.11.053. [23] X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stochastic Processes and Their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [24] X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis, Journal of Mathematical Analysis and Applications, 433 (2016), 227-242.  doi: 10.1016/j.jmaa.2015.07.056. [25] A. Miao, J. Zhang, T. Zhang and B. Pradeep, Threshold dynamics of a stochastic model with vertical transmission and vaccination, Computational and Mathematical Methods in Medicine, 2017 (2017), Art. ID 4820183, 10 pp. doi: 10.1155/2017/4820183. [26] D. Moualeu, A. N. Yakam, S. Bowong and A. Temgoua, Analysis of a tuberculosis model with undetected and lost-sight cases, Communications in Nonlinear Science and Numerical Simulation, 41 (2016), 48-63.  doi: 10.1016/j.cnsns.2016.04.012. [27] E. G. Nicholson, A. M. Geltemeyer and K. C. Smith, Practice guideline for treatment of latent tuberculosis infection in children, Journal of Pediatric Health Care, 29 (2015), 302-307. [28] M. A. Nowak and R. M. May, Mathematical principles of immunology and virology, Nature Medicine, 410 (2001), 412-413. [29] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [30] Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072. [31] Q. Yang and X. Mao, Extinction and recurrence of multi-group seir epidemic models with stochastic perturbations, Nonlinear Analysis: Real World Applications, 14 (2013), 1434-1456.  doi: 10.1016/j.nonrwa.2012.10.007. [32] D. Zhao, T. Zhang and S. Yuan, The threshold of a stochastic sivs epidemic model with nonlinear saturated incidence, Physica A: Statistical Mechanics and its Applications, 443 (2016), 372-379.  doi: 10.1016/j.physa.2015.09.092. [33] Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a sir epidemic model with stochastic perturbations, Applied Mathematics and Computation, 244 (2014), 118-131.  doi: 10.1016/j.amc.2014.06.100. [34] X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 1557-1568.  doi: 10.1016/j.cnsns.2013.09.010.

show all references

##### References:
 [1] S. Bowong and J. J. Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 4010-4021.  doi: 10.1016/j.cnsns.2009.02.017. [2] Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic sirs epidemic model with infectious force under intervention strategies, Journal of Differential Equations, 259 (2015), 7463-7502.  doi: 10.1016/j.jde.2015.08.024. [3] Z. Chang, X. Meng and X. Lu, Analysis of a novel stochastic sirs epidemic model with two different saturated incidence rates, Physica A: Statistical Mechanics and its Applications, 472 (2017), 103-116.  doi: 10.1016/j.physa.2017.01.015. [4] S. Choi, E. Jung and S.-M. Lee, Optimal intervention strategy for prevention tuberculosis using a smoking-tuberculosis model, Journal of Theoretical Biology, 380 (2015), 256-270.  doi: 10.1016/j.jtbi.2015.05.022. [5] T. Feng, X. Meng, L. Liu and S. Gao, Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model, Journal of Inequalities and Applications, 2016 (2016), Paper No. 327, 29 pp. doi: 10.1186/s13660-016-1265-z. [6] D. Gao and N. Huang, A note on global stability for a tuberculosis model, Applied Mathematics Letters, 73 (2017), 163-168.  doi: 10.1016/j.aml.2017.05.004. [7] S. Gao, L. Chen and Z. Teng, Pulse vaccination of an seir epidemic model with time delay, Nonlinear Analysis: Real World Applications, 9 (2008), 599-607.  doi: 10.1016/j.nonrwa.2006.12.004. [8] X. Han and P. E. Kloeden, Immune System Virus Model, Springer Singapore, Singapore, 2017. [9] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with beddington-deangelis functional response, Applied Mathematics Letters, 22 (2009), 1690-1693.  doi: 10.1016/j.aml.2009.06.004. [10] H. Huo and M. Zou, Modelling effects of treatment at home on tuberculosis transmission dynamics, Applied Mathematical Modelling, 40 (2016), 9474-9484.  doi: 10.1016/j.apm.2016.06.029. [11] D. Jiang, C. Ji, N. Shi and J. Yu, The long time behavior of di sir epidemic model with stochastic perturbation, Journal of Mathematical Analysis and Applications, 372 (2010), 162-180.  doi: 10.1016/j.jmaa.2010.06.003. [12] D. Jiang, Q. Liu, N. Shi, T. Hayat, A. Alsaedi and P. Xia, Dynamics of a stochastic hiv-1 infection model with logistic growth, Physica A: Statistical Mechanics and its Applications, 469 (2017), 706-717.  doi: 10.1016/j.physa.2016.11.078. [13] D. Jiang and N. Shi, A note on nonautonomous logistic equation with random perturbation, Journal of Mathematical Analysis and Applications, 303 (2005), 164-172.  doi: 10.1016/j.jmaa.2004.08.027. [14] R. Khasminskii, Stochastic Stability of Differential Equations, Stochastic Modelling and Applied Probability, 66. Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [15] P. E. Kloeden and C. Pötzsche, Nonautonomous bifurcation scenarios in sir models, Mathematical Methods in the Applied Sciences, 38 (2015), 3495-3518.  doi: 10.1002/mma.3433. [16] A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for sir, sirs, and sis epidemiological models, Applied Mathematics Letters, 15 (2002), 955-960.  doi: 10.1016/S0893-9659(02)00069-1. [17] X. Leng, T. Feng and X. Meng, Stochastic inequalities and applications to dynamics analysis of a novel sivs epidemic model with jumps, Journal of Inequalities and Applications, 2017 (2017), Paper No. 138, 25 pp. doi: 10.1186/s13660-017-1418-8. [18] G. Li and Z. Jin, Global stability of a seir epidemic model with infectious force in latent, infected and immune period, Chaos, Solitons & Fractals, 25 (2005), 1177-1184.  doi: 10.1016/j.chaos.2004.11.062. [19] M. Liu, C. Bai and K. Wang, Asymptotic stability of a two-group stochastic seir model with infinite delays, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3444-3453.  doi: 10.1016/j.cnsns.2014.02.025. [20] Q. Liu, The threshold of a stochastic susceptible-infective epidemic model under regime switching, Nonlinear Analysis: Hybrid Systems, 21 (2016), 49-58.  doi: 10.1016/j.nahs.2016.01.002. [21] Q. Liu and Q. Chen, Analysis of the deterministic and stochastic sirs epidemic models with nonlinear incidence, Physica A: Statistical Mechanics and its Applications, 428 (2015), 140-153.  doi: 10.1016/j.physa.2015.01.075. [22] Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Dynamics of a stochastic tuberculosis model with constant recruitment and varying total population size, Physica A: Statistical Mechanics and its Applications, 469 (2017), 518-530.  doi: 10.1016/j.physa.2016.11.053. [23] X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stochastic Processes and Their Applications, 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0. [24] X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis, Journal of Mathematical Analysis and Applications, 433 (2016), 227-242.  doi: 10.1016/j.jmaa.2015.07.056. [25] A. Miao, J. Zhang, T. Zhang and B. Pradeep, Threshold dynamics of a stochastic model with vertical transmission and vaccination, Computational and Mathematical Methods in Medicine, 2017 (2017), Art. ID 4820183, 10 pp. doi: 10.1155/2017/4820183. [26] D. Moualeu, A. N. Yakam, S. Bowong and A. Temgoua, Analysis of a tuberculosis model with undetected and lost-sight cases, Communications in Nonlinear Science and Numerical Simulation, 41 (2016), 48-63.  doi: 10.1016/j.cnsns.2016.04.012. [27] E. G. Nicholson, A. M. Geltemeyer and K. C. Smith, Practice guideline for treatment of latent tuberculosis infection in children, Journal of Pediatric Health Care, 29 (2015), 302-307. [28] M. A. Nowak and R. M. May, Mathematical principles of immunology and virology, Nature Medicine, 410 (2001), 412-413. [29] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [30] Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence, Journal of Mathematical Analysis and Applications, 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072. [31] Q. Yang and X. Mao, Extinction and recurrence of multi-group seir epidemic models with stochastic perturbations, Nonlinear Analysis: Real World Applications, 14 (2013), 1434-1456.  doi: 10.1016/j.nonrwa.2012.10.007. [32] D. Zhao, T. Zhang and S. Yuan, The threshold of a stochastic sivs epidemic model with nonlinear saturated incidence, Physica A: Statistical Mechanics and its Applications, 443 (2016), 372-379.  doi: 10.1016/j.physa.2015.09.092. [33] Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a sir epidemic model with stochastic perturbations, Applied Mathematics and Computation, 244 (2014), 118-131.  doi: 10.1016/j.amc.2014.06.100. [34] X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 1557-1568.  doi: 10.1016/j.cnsns.2013.09.010.
Transfer diagram of the ODE TB model
Trajectory of the solution of system (2) and its corresponding deterministic model (1)
Trajectory of the solution of system (2) and its corresponding deterministic model (1)
The pictures on the left are trajectories of the solution of system (2). The pictures on the right are the distribution density functions of system (2)
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