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Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system
Analysis of an age-structured model for HIV-TB co-infection
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 |
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.
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/. |









Parameters | Description |
the recruitment rate of the susceptible class | |
the natural death rate of the population | |
the transmission coefficient of HIV class to susceptible class | |
the transmission coefficient of active TB class to susceptible class | |
the death rate due to HIV | |
the rate at which latent class progress into infectious class | |
the death rate due to TB | |
the death rate due to co-infection | |
the rate at which treatment infectious individuals progress into | |
susceptible class | |
the rate at which treatment latent individuals progress into | |
susceptible class | |
the transmission coefficient of TB infectious class to HIV class |
Parameters | Description |
the recruitment rate of the susceptible class | |
the natural death rate of the population | |
the transmission coefficient of HIV class to susceptible class | |
the transmission coefficient of active TB class to susceptible class | |
the death rate due to HIV | |
the rate at which latent class progress into infectious class | |
the death rate due to TB | |
the death rate due to co-infection | |
the rate at which treatment infectious individuals progress into | |
susceptible class | |
the rate at which treatment latent individuals progress into | |
susceptible class | |
the transmission coefficient of TB infectious class to HIV class |
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 |
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 |
Parameter | Mean | Std | 95% CI | Source |
16439333 | - | - | [25] | |
1/74.7 | - | - | [25] | |
1307560000 | - | - | [25] | |
0.002489009 | 0.000296466 | [0.002483197, 0.00249482] | MCMC | |
0.073713359 | 0.002394865 | [0.073666415, 0.073760303] | MCMC | |
0.054719731 | 0.002310545 | [0.05467444, 0.054765022] | MCMC | |
3.07 |
7.63 |
[3.070478 |
MCMC | |
1.591747005 | 0.02403578 | [1.591275856, 1.592218155] | MCMC | |
1.49584 |
2.29066 |
[ 1.495393 |
MCMC | |
0.085041492 | 0.002759644 | [0.084987398, 0.085095587] | MCMC | |
0.921801960 | 0.036083393 | [0.921094653, 0.922509267] | MCMC | |
1.405989 |
9.99339 |
[1.404030 |
MCMC | |
0.025716455 | 0.000489538 | [0.025706859, 0.025726051] | MCMC | |
0.010987723 | 0.000575755 | [0.010976437, 0.010999009] | MCMC | |
0.894619519 | 0.0263988 | [0.89410205, 0.895136989] | MCMC | |
0.000943614 | 0.00004225 | [0.000942785, 0.000944442] | MCMC | |
154136723 | 566182 | [154125625, 154147821] | MCMC | |
2566318 | 597 | [2566306, 2566329] | MCMC | |
7433129 | 289907 | [7427446, 7438811] | MCMC | |
14052275 | 30076 | [14051685, 14052864] | MCMC | |
27800 | 1803 | [27765, 27835] | MCMC | |
77090 | 4091 | [77010, 77170] | MCMC |
Parameter | Mean | Std | 95% CI | Source |
16439333 | - | - | [25] | |
1/74.7 | - | - | [25] | |
1307560000 | - | - | [25] | |
0.002489009 | 0.000296466 | [0.002483197, 0.00249482] | MCMC | |
0.073713359 | 0.002394865 | [0.073666415, 0.073760303] | MCMC | |
0.054719731 | 0.002310545 | [0.05467444, 0.054765022] | MCMC | |
3.07 |
7.63 |
[3.070478 |
MCMC | |
1.591747005 | 0.02403578 | [1.591275856, 1.592218155] | MCMC | |
1.49584 |
2.29066 |
[ 1.495393 |
MCMC | |
0.085041492 | 0.002759644 | [0.084987398, 0.085095587] | MCMC | |
0.921801960 | 0.036083393 | [0.921094653, 0.922509267] | MCMC | |
1.405989 |
9.99339 |
[1.404030 |
MCMC | |
0.025716455 | 0.000489538 | [0.025706859, 0.025726051] | MCMC | |
0.010987723 | 0.000575755 | [0.010976437, 0.010999009] | MCMC | |
0.894619519 | 0.0263988 | [0.89410205, 0.895136989] | MCMC | |
0.000943614 | 0.00004225 | [0.000942785, 0.000944442] | MCMC | |
154136723 | 566182 | [154125625, 154147821] | MCMC | |
2566318 | 597 | [2566306, 2566329] | MCMC | |
7433129 | 289907 | [7427446, 7438811] | MCMC | |
14052275 | 30076 | [14051685, 14052864] | MCMC | |
27800 | 1803 | [27765, 27835] | MCMC | |
77090 | 4091 | [77010, 77170] | MCMC |
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