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January  2022, 27(1): 199-228. doi: 10.3934/dcdsb.2021037

Analysis of an age-structured model for HIV-TB co-infection

Zhong-Kai Guo 1,, , Hai-Feng Huo 2,, and  Hong Xiang 2,

1. 

School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China
2. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

* Corresponding author: Zhong-Kai Guo and Hai-Feng Huo

Received  August 2020 Revised  November 2020 Published  January 2022 Early access  January 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), the HongLiu first-class disciplines Development Program of Lanzhou University of Technology, the Youth Science Fund of Lanzhou Jiaotong University(1200060930), and the Scientific Research Foundation of Lanzhou Jiaotong University(1520020410)

According to the report of the WHO, there is a strong relationship between AIDS and tuberculosis (TB). Therefore, it is very important to study how to control TB in the context of the global AIDS epidemic. In this paper, we establish an age structured mathematical model of HIV-TB co-infection to study the transmission dynamics of this co-infection, and consider awareness in the modeling. We give the basic reproduction numbers for each of the two diseases and find four equilibria, namely, disease-free equilibrium, TB-free equilibrium, HIV-free equilibrium and endemic disease equilibrium. Then we discuss the local stability of the equilibria according to the range of values of the two basic reproduction numbers, and find the endemic equilibrium is unstable. We also discuss the global stability of the disease-free equilibrium and the TB-free equilibrium. Based on the new HIV-positive cases and TB cases data in China, the best-fit parameter values and initial values of the model are identified by the MCMC algorithm. Then we perform uncertainty and sensitivity analysis to identify the parameters that have significant impact on the basic reproduction number $ \mathcal{R}_{T} $. Finally, combined with the established model, we give some measures that may help China achieve the goal of WHO of reducing the incidence of TB by 80% by 2030 compared to 2015.

Keywords: Age structure, co-infection, stability, MCMC algorithm, uncertainty and sensitivity analysis.
Mathematics Subject Classification: Primary: 37N25, 92B05; Secondary: 37B25.
Citation: Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Analysis of an age-structured model for HIV-TB co-infection. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 199-228. doi: 10.3934/dcdsb.2021037
References:
[1]

World Health Organization, Available from: https://www.who.int/health-topics/hiv-aids/.

[2]

Centers for Disease Control and Prevention, Available from: https://www.cdc.gov/tb/.

[3]

World Health Organization, Available from: https://www.who.int/tb/en/.

[4]

E. M. C. D'Agata, P. Magal, S. Ruan and G. Webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Differential Integral Equations, 19 (2006), 573-600. 

[5]

F. B. Agusto and A. Adekunle, Optimal control of a two-strain tuberculosis-HIV/AIDS co-infection model, Biosystems, 119 (2014), 20-44.  doi: 10.1016/j.biosystems.2014.03.006.

[6]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.

[7]

S. Gakkhar and N. Chavda, A dynamical model for HIV-TB co-infection, Applied Mathematics and Computation, 218 (2012), 9261-9270.  doi: 10.1016/j.amc.2012.03.004.

[8]

I. Ghosh, P. K. Tiwari, S. Samanta, I. M. Elmojtaba, N. Al-Salti and J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Mathematical Biosciences, 306 (2018), 160-169.  doi: 10.1016/j.mbs.2018.09.014.

[9]

Z.-K. Guo, H.-F. Huo and H. Xiang, Global dynamics of an age-structured malaria model with prevention, Mathematical Biosciences and Engineering, 16 (2019), 1625-1653.  doi: 10.3934/mbe.2019078.

[10]

H. Haario, M. Laine, A. Mira and E. Saksman, Dram: Efficient adaptive MCMC, Statistics and Computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.

[11]

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

[12]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giadini Editori e Stampatori, Pisa, 1994.

[13]

D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1, Theoretical Population Biology, 55 (1999), 94-109.  doi: 10.1006/tpbi.1998.1382.

[14]

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.

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[16]

A. Mallela, S. Lenhart and N. K. Vaidya, HIV-TB co-infection treatment: Modeling and optimal control theory perspectives, Journal of Computational and Applied Mathematics, 307 (2016), 143-161.  doi: 10.1016/j.cam.2016.02.051.

[17]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.

[18]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[19]

E. Massad, M. N. Burattini, F. A. B. Coutinho, H. M. Yang and S. M. Raimundo, Modeling the interaction between AIDS and tuberculosis, Mathematical and Computer Modelling, 17 (1993), 7-21.  doi: 10.1016/0895-7177(93)90013-O.

[20]

C. M. A. Pinto and A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Applied Mathematics and Computation, 242 (2014), 36-46.  doi: 10.1016/j.amc.2014.05.061.

[21]

M. Samsuzzoha, M. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915.  doi: 10.1016/j.apm.2012.03.029.

[22]

S. C. Shiboski and N. P. Jewell, Statistical analysis of the time dependence of HIV infectivity based on partner study data, Journal of the American Statistical Association, 87 (1992), 360-372.  doi: 10.1080/01621459.1992.10475215.

[23]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 118 2011. doi: 10.1090/gsm/118.

[24]

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

[25]

National Bureau of Statistics of China, Available from: http://www.stats.gov.cn/.

[26]

Chinese Center for Disease Control and Prevention, Available from: http://www.chinacdc.cn/.

show all references

References:
[1]

World Health Organization, Available from: https://www.who.int/health-topics/hiv-aids/.

[2]

Centers for Disease Control and Prevention, Available from: https://www.cdc.gov/tb/.

[3]

World Health Organization, Available from: https://www.who.int/tb/en/.

[4]

E. M. C. D'Agata, P. Magal, S. Ruan and G. Webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Differential Integral Equations, 19 (2006), 573-600. 

[5]

F. B. Agusto and A. Adekunle, Optimal control of a two-strain tuberculosis-HIV/AIDS co-infection model, Biosystems, 119 (2014), 20-44.  doi: 10.1016/j.biosystems.2014.03.006.

[6]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete and Continuous Dynamical Systems - Series B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.

[7]

S. Gakkhar and N. Chavda, A dynamical model for HIV-TB co-infection, Applied Mathematics and Computation, 218 (2012), 9261-9270.  doi: 10.1016/j.amc.2012.03.004.

[8]

I. Ghosh, P. K. Tiwari, S. Samanta, I. M. Elmojtaba, N. Al-Salti and J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Mathematical Biosciences, 306 (2018), 160-169.  doi: 10.1016/j.mbs.2018.09.014.

[9]

Z.-K. Guo, H.-F. Huo and H. Xiang, Global dynamics of an age-structured malaria model with prevention, Mathematical Biosciences and Engineering, 16 (2019), 1625-1653.  doi: 10.3934/mbe.2019078.

[10]

H. Haario, M. Laine, A. Mira and E. Saksman, Dram: Efficient adaptive MCMC, Statistics and Computing, 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.

[11]

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

[12]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giadini Editori e Stampatori, Pisa, 1994.

[13]

D. Kirschner, Dynamics of co-infection with M. tuberculosis and HIV-1, Theoretical Population Biology, 55 (1999), 94-109.  doi: 10.1006/tpbi.1998.1382.

[14]

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.

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[16]

A. Mallela, S. Lenhart and N. K. Vaidya, HIV-TB co-infection treatment: Modeling and optimal control theory perspectives, Journal of Computational and Applied Mathematics, 307 (2016), 143-161.  doi: 10.1016/j.cam.2016.02.051.

[17]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.

[18]

M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[19]

E. Massad, M. N. Burattini, F. A. B. Coutinho, H. M. Yang and S. M. Raimundo, Modeling the interaction between AIDS and tuberculosis, Mathematical and Computer Modelling, 17 (1993), 7-21.  doi: 10.1016/0895-7177(93)90013-O.

[20]

C. M. A. Pinto and A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Applied Mathematics and Computation, 242 (2014), 36-46.  doi: 10.1016/j.amc.2014.05.061.

[21]

M. Samsuzzoha, M. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915.  doi: 10.1016/j.apm.2012.03.029.

[22]

S. C. Shiboski and N. P. Jewell, Statistical analysis of the time dependence of HIV infectivity based on partner study data, Journal of the American Statistical Association, 87 (1992), 360-372.  doi: 10.1080/01621459.1992.10475215.

[23]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 118 2011. doi: 10.1090/gsm/118.

[24]

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

[25]

National Bureau of Statistics of China, Available from: http://www.stats.gov.cn/.

[26]

Chinese Center for Disease Control and Prevention, Available from: http://www.chinacdc.cn/.

Figure 1.  Estimated number of deaths from HIV/AIDS and TB in 2017. Deaths from TB among HIV-positive people are shown in grey
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Figure 2.  Flowchart of the TB-HIV co-infection
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Figure 3.  Data fitting: (a) the fitting results of the number of new TB cases reported from 2005 to 2017; (b)the fitting results of the number of new HIV-positive cases reported from 2005 to 2017. The solid black line represents the fitted data, and the red dots represent the actual data. The areas from the darkest to the lightest correspond to the 50%, 90%, 95% and 99% posterior limits of the model uncertainty
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Figure 4.  The distribution histogram of the basic reproduction number $ \mathcal{R}_{T} $
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Figure 5.  The PRCC values
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Figure 6.  The effect of changes in $ m_{5} $ on the number of new TB cases
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Figure 7.  The effect of changes in $ m_{4} $ on the number of new TB cases
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Figure 8.  The effect of changes in $ \beta_{T} $ on the number of new TB cases
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Figure 9.  The effect of changes in $ m_{5} $, $ \beta_{T} $ and $ m_{4} $ on the number of new TB cases
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Table 1.  Description of parameters of the model $ (1) $
Parameters Description
$ \Lambda $ the recruitment rate of the susceptible class
$ \mu $ the natural death rate of the population
$ \beta(a) $ the transmission coefficient of HIV class to susceptible class
$ \beta_{T} $ the transmission coefficient of active TB class to susceptible class
$ \delta_{i}(a) $ the death rate due to HIV
$ \sigma(\theta) $ the rate at which latent class progress into infectious class
$ \mu_{T} $ the death rate due to TB
$ \mu_{c} $ the death rate due to co-infection
$ \alpha $ the rate at which treatment infectious individuals progress into
susceptible class
$ \delta(\theta) $ the rate at which treatment latent individuals progress into
susceptible class
$ \delta_{T}(a) $ the transmission coefficient of TB infectious class to HIV class
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Parameters Description
$ \Lambda $ the recruitment rate of the susceptible class
$ \mu $ the natural death rate of the population
$ \beta(a) $ the transmission coefficient of HIV class to susceptible class
$ \beta_{T} $ the transmission coefficient of active TB class to susceptible class
$ \delta_{i}(a) $ the death rate due to HIV
$ \sigma(\theta) $ the rate at which latent class progress into infectious class
$ \mu_{T} $ the death rate due to TB
$ \mu_{c} $ the death rate due to co-infection
$ \alpha $ the rate at which treatment infectious individuals progress into
susceptible class
$ \delta(\theta) $ the rate at which treatment latent individuals progress into
susceptible class
$ \delta_{T}(a) $ the transmission coefficient of TB infectious class to HIV class
Table 2.  New TB cases and HIV-positive cases from 2005 to 2017 in China (persons)
Year 2005 2006 2007 2008 2009 2010 2011
TB cases 1,259,308 1,127,571 1,163,959 1,169,540 1,076,938 991,350 953,275
HIV-positive cases 30,887 38,262 42,633 51,525 57,473 61,622 73,196
Year 2012 2013 2014 2015 2016 2017
TB cases 951,508 904,434 889,381 864,015 836,236 835,193
HIV-positive cases 100,328 105,784 119,193 132,016 142,124 152,746
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Year 2005 2006 2007 2008 2009 2010 2011
TB cases 1,259,308 1,127,571 1,163,959 1,169,540 1,076,938 991,350 953,275
HIV-positive cases 30,887 38,262 42,633 51,525 57,473 61,622 73,196
Year 2012 2013 2014 2015 2016 2017
TB cases 951,508 904,434 889,381 864,015 836,236 835,193
HIV-positive cases 100,328 105,784 119,193 132,016 142,124 152,746
Table 3.  The parameters values and initial values of the model $ (3) $
Parameter Mean Std 95% CI Source
$ \Lambda $ 16439333 - - [25]
$ \mu $ 1/74.7 - - [25]
$ S(0) $ 1307560000 - - [25]
$ \alpha $ 0.002489009 0.000296466 [0.002483197, 0.00249482] MCMC
$ m_{5} $ 0.073713359 0.002394865 [0.073666415, 0.073760303] MCMC
$ \delta $ 0.054719731 0.002310545 [0.05467444, 0.054765022] MCMC
$ m_{3} $ 3.07 $ \times 10^{-9} $ 7.63$ \times 10^{-11} $ [3.070478$ \times 10^{-9} $, 3.073469$ \times 10^{-9} $ ] MCMC
$ \beta $ 1.591747005 0.02403578 [1.591275856, 1.592218155] MCMC
$ \beta_{T} $ 1.49584$ \times 10^{-9} $ 2.29066$ \times 10^{-11} $ [ 1.495393$ \times 10^{-9} $, 1.496291$ \times 10^{-9} $] MCMC
$ m_{1} $ 0.085041492 0.002759644 [0.084987398, 0.085095587] MCMC
$ \delta_{i} $ 0.921801960 0.036083393 [0.921094653, 0.922509267] MCMC
$ m_{2} $ 1.405989 $ \times 10^{-9} $ 9.99339$ \times 10^{-11} $ [1.404030$ \times 10^{-9} $, 1.407948$ \times 10^{-9} $] MCMC
$ m_{4} $ 0.025716455 0.000489538 [0.025706859, 0.025726051] MCMC
$ \sigma $ 0.010987723 0.000575755 [0.010976437, 0.010999009] MCMC
$ \mu_{T} $ 0.894619519 0.0263988 [0.89410205, 0.895136989] MCMC
$ \mu_{c} $ 0.000943614 0.00004225 [0.000942785, 0.000944442] MCMC
$ e(0) $ 154136723 566182 [154125625, 154147821] MCMC
$ i(0) $ 2566318 597 [2566306, 2566329] MCMC
$ t_{c}(0) $ 7433129 289907 [7427446, 7438811] MCMC
$ I_{t}(0) $ 14052275 30076 [14051685, 14052864] MCMC
$ I_{c}(0) $ 27800 1803 [27765, 27835] MCMC
$ t_{s}(0) $ 77090 4091 [77010, 77170] MCMC
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Parameter Mean Std 95% CI Source
$ \Lambda $ 16439333 - - [25]
$ \mu $ 1/74.7 - - [25]
$ S(0) $ 1307560000 - - [25]
$ \alpha $ 0.002489009 0.000296466 [0.002483197, 0.00249482] MCMC
$ m_{5} $ 0.073713359 0.002394865 [0.073666415, 0.073760303] MCMC
$ \delta $ 0.054719731 0.002310545 [0.05467444, 0.054765022] MCMC
$ m_{3} $ 3.07 $ \times 10^{-9} $ 7.63$ \times 10^{-11} $ [3.070478$ \times 10^{-9} $, 3.073469$ \times 10^{-9} $ ] MCMC
$ \beta $ 1.591747005 0.02403578 [1.591275856, 1.592218155] MCMC
$ \beta_{T} $ 1.49584$ \times 10^{-9} $ 2.29066$ \times 10^{-11} $ [ 1.495393$ \times 10^{-9} $, 1.496291$ \times 10^{-9} $] MCMC
$ m_{1} $ 0.085041492 0.002759644 [0.084987398, 0.085095587] MCMC
$ \delta_{i} $ 0.921801960 0.036083393 [0.921094653, 0.922509267] MCMC
$ m_{2} $ 1.405989 $ \times 10^{-9} $ 9.99339$ \times 10^{-11} $ [1.404030$ \times 10^{-9} $, 1.407948$ \times 10^{-9} $] MCMC
$ m_{4} $ 0.025716455 0.000489538 [0.025706859, 0.025726051] MCMC
$ \sigma $ 0.010987723 0.000575755 [0.010976437, 0.010999009] MCMC
$ \mu_{T} $ 0.894619519 0.0263988 [0.89410205, 0.895136989] MCMC
$ \mu_{c} $ 0.000943614 0.00004225 [0.000942785, 0.000944442] MCMC
$ e(0) $ 154136723 566182 [154125625, 154147821] MCMC
$ i(0) $ 2566318 597 [2566306, 2566329] MCMC
$ t_{c}(0) $ 7433129 289907 [7427446, 7438811] MCMC
$ I_{t}(0) $ 14052275 30076 [14051685, 14052864] MCMC
$ I_{c}(0) $ 27800 1803 [27765, 27835] MCMC
$ t_{s}(0) $ 77090 4091 [77010, 77170] MCMC
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