American Institute of Mathematical Sciences

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

Global analysis of a stochastic TB model with vaccination and treatment

Tao Feng and  Zhipeng Qiu ,

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.

Keywords: Stochastic tuberculosis system, persistence in mean, stationary distribution, ergodicity.
Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 60H10.
Citation: Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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X. Han and P. E. Kloeden, Immune System Virus Model, Springer Singapore, Singapore, 2017. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

M. A. Nowak and R. M. May, Mathematical principles of immunology and virology, Nature Medicine, 410 (2001), 412-413.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. Han and P. E. Kloeden, Immune System Virus Model, Springer Singapore, Singapore, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

M. A. Nowak and R. M. May, Mathematical principles of immunology and virology, Nature Medicine, 410 (2001), 412-413.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Transfer diagram of the ODE TB model
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Figure 2.  Trajectory of the solution of system (2) and its corresponding deterministic model (1)
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Figure 3.  Trajectory of the solution of system (2) and its corresponding deterministic model (1)
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Figure 4.  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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