# American Institute of Mathematical Sciences

## 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

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

Received  August 2020 Revised  November 2020 Published  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.

Citation: Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Analysis of an age-structured model for HIV-TB co-infection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021037
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Estimated number of deaths from HIV/AIDS and TB in 2017. Deaths from TB among HIV-positive people are shown in grey
Flowchart of the TB-HIV co-infection
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
The distribution histogram of the basic reproduction number $\mathcal{R}_{T}$
The PRCC values
The effect of changes in $m_{5}$ on the number of new TB cases
The effect of changes in $m_{4}$ on the number of new TB cases
The effect of changes in $\beta_{T}$ on the number of new TB cases
The effect of changes in $m_{5}$, $\beta_{T}$ and $m_{4}$ on the number of new TB cases
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
 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
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
 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
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
 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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