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

Zhong-Kai Guo; Hai-Feng Huo; Hong Xiang

doi:10.3934/dcdsb.2021037

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2022, Volume 27, Issue 1: 199-228. Doi: 10.3934/dcdsb.2021037

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

Received: July 31, 2020

Revised: October 31, 2020

Early access: January 2021

Published: January 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 37N25, 92B05; Secondary: 37B25.

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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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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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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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