The influence of ambient air pollution on the transmission of tuberculosis in Jiangsu, China

Abstract

In this paper, based on the statistical data, we investigate the effects of long-term exposure to ambient particulate air pollution on the transmission dynamics of tuberculosis (TB) in Jiangsu, China by studying the threshold dynamics of the TB epidemic model via the statistical data analytically and numerically. The basic reproduction number ${\mathcal{R}}_0>1$ reveals that TB in Jiangsu, China is an endemic disease and will persist for a long time. And the numerical results show that, in order to control the TB in Jiangsu effectively, we must decrease the depuration coefficient of PM₁₀ in the body, the proportion of TB symptomatic infectious by direct transmission, the reactivation rate of the pre-symptomatic infectious and the effect coefficient of PM₁₀ and MTB inhaled of TB transmission, and increase the uptake coefficient, the recovery rate of the symptomatic/pre-symptomatic infectious and the influence coefficient of PM₁₀ on the body of mortality. Our study shows that PM₁₀ is closely related to the incidence of TB, and the effective control efforts are suggested to focus on increasing close-contact distance and wearing protective mask to decrease the influence of PM₁₀ on the TB transmission, which may shed a new light on understanding the environmental drivers to TB.

1. Introduction

Tuberculosis (TB) is a communicable disease caused by Mycobacterium tuberculosis (MTB) and several related species (Blower, Small, & Hopewell, 1996). MTB can spread through the air while people with active TB in their lung cough, spit, speak or sneeze (Dye & Williams, 2000). TB is usually regarded as a major public health problem throughout the world, one of the top 10 causes of death worldwide and the leading cause of death from a single infectious agent (ranking above HIV/AIDS) (The World Health Organization, 2022). Globally, 10.1 million new cases of TB were estimated in 2020 – an increase from 10.0 million in 2019. And there were approximately 95% of the estimated 10.6 million (range 9.9–11.0 million) new cases of TB occurring in 2021 (The World Health Organization, 2022).

In recent years, the Chinese government has increased its investment in public health, which provides a severe basic guarantee for effectively controlling major infectious diseases. Particularly, the incidence of TB has decreased significantly. In 2019, China had 775,764 incident TB (died 2990); in 2020, 679,538 (died 1919); and in 2021, 639,548 (died 1763) (National Health Commission of the People’ s Republic of China, 2018). Obviously, there is a reduction trend of TB, but TB is still an important public health issue in China (Hu & Sun, 2013).

In China, 80% of the reported number of people diagnosed with TB exists in rural areas, particularly in north and north-western regions with low socioeconomic status (Hu & Sun, 2013). However, in the past 20 years in Jiangsu province, China, one of the most economically developed regions in China, TB ranked first in the number of notifiable B infectious diseases (Jiangsu Commision of Health, 2020). In 2019, the reported number of people diagnosed with TB in Jiangsu, China was 25,159 (died 31); in 2020, 22,922 (died 56); and in 2021, 21,729 (died 69). Although the incidence of TB has decreased significantly, the situation of prevention and control of TB in Jiangsu, China is still very serious.

The World Health Organization (WHO) has declared that air pollutants are one of the most dangerous environmental carcinogens on the earth. Particulate matter (PM), which includes respirable particles with 10 μm (μm) or less (PM₁₀) in diameter and fine particles with 2.5 μm or less (PM_2.5) in diameter, can contribute to a reduction of visibility, arouses public health concerns because of its toxicity and the widespread human exposure to this pollutant (Liu et al., 2019a). There is no doubt that air pollution, sometimes called “fog and haze”, has negatively affected transportation, economy and tourism, hindered the improvement in people's living standards, caused a lot of inconvenience and, worst of all, is severely hazardous to human health (Chen, Liu, Chen, Wang, & Fu, 2018; He et al., 2018, 2019; Ruckerl, Schneider, Breitner, Cyrys, & Peters, 2011; Tang et al., 2018; Wang, Jin, & Wang, 2018; Xia et al., 2017). Both short-term and long-term exposure to pollutants will reduce respiratory functions, leading to increases in hospital respiratory admissions, in medication use by asthmatic subjects, and in death, particularly for sensitive groups such as children, the older people and those with chronic respiratory illnesses (He et al., 2018; Huang et al., 2016; Raaschou-Nielsen et al., 2013). Particularly, it is reported that long-term exposure to PM_2.5/PM₁₀ increases the risk of death from TB and other diseases among TB patients (Brunekreef & Holgate, 2002; Liu et al., 2019b; Peng et al., 2017). In addition, Cai et al. (Cai et al., 2021) studied the effects of the contaminated environment on the TB transmission dynamics in Jiangsu, China, and claimed that TB in Jiangsu, China will persist for a long time, and in order to control the TB in Jiangsu efficiently, we must decrease the virus shedding rate, and increase the recovery rates and the environmental clearance rate.

This study will focus on the effects of long-term exposure to ambient particulate air pollution on the transmission dynamics of TB in Jiangsu, China through studying the threshold dynamics of the TB epidemic model and provide the effective control strategies to curb the spread of TB in Jiangsu, China.

2. Model derivation

TB has slow intrinsic dynamics, and the slow progression of TB at the individual level leads to long-term outcomes of TB at the population level. Long-term effects of TB can be examined using epidemiological models consisting of compartments which represent sets of individuals grouped by disease status (Ozcaglar et al., 2012), since the pioneering work of Waaler et al. (Waaler, Geser, & Andersen, 1962). And a number of theoretical studies have been carried out on the mathematical modeling of TB transmission dynamics (Dye et al., 1998; Feng, Castillo-Chavez, & Capurro, 2000; Guo & Li, 2011; Knight, McQuaid, Dodd, & Houben, 2019; Liu, Zhao, & Zhou, 2010; Ozcaglar et al., 2012; Porco & Blower, 1998; Suzanne et al., 2005; Ziv, Daley, & Sally, 2004). Based on the models in the references above, we divide the host population into four compartments: the susceptible (individuals not yet infected) S(t), the pre-symptomatic infectious T_e(t), the infectious with symptoms T(t) and the recovered R(t). And then we can construct the following STeTRS model:

$$\{\begin{array}{l}\frac{\mathrm{d}S}{\mathrm{d}t}=\mathrm{\Lambda }-\mu S-\beta (t)ST+\theta R,\\ \frac{\mathrm{d}{T}_e}{\mathrm{d}t}=(1-q)\beta (t)ST-(\mu +\nu +{\gamma }_1)T_e,\\ \frac{\mathrm{d}T}{\mathrm{d}t}=\nu T_e+q\beta (t)ST-(\mu +\delta +{\gamma }_2)T,\\ \frac{\mathrm{d}R}{\mathrm{d}t}={\gamma }_1{T}_e+{\gamma }_{2}T-(\mu +\theta )R,\\ \end{array}$$ (2.1)

where N = S + T_e + T + R, all the parameters are positive constants, Λ represents the recruitment rate of susceptible, μ the per capita natural mortality rate, q the proportion of TB symptomatic infectious by direct transmission, δ the TB-induced death rate, ν the reactivation rate of the pre-symptomatic infectious, θ the recovered rate, γ₁ and γ₂ are the recovery rate of the pre-symptomatic/symptomatic infectious, respectively. And β(t) is the transmission rate coefficient, which is a continuous, positive ω − periodic function.

Thanks for the insightful work in (Huang et al., 2013, 2015; Saha & Samanta, 2019; Wang et al., 2018) for modelling the environmental toxin in the population/epidemic models, in the present paper, we take care of those models to formulate the PM₁₀ into the basic TB model (2.1). A schematic diagram is provided in Fig. 1 for clear understanding of the model system. Then we obtain the following epidemic model to study the effect of PM₁₀ on the TB transmission dynamics:

$$\left\{\begin{array}{l}\frac{\mathrm{d}S}{\mathrm{d}t}=\mathrm{\Lambda }-\left(\mu +\frac{kC}{N}\right)S-\left(1+\frac{mC}{bN+C}\right)\beta (t)ST+\theta R,\\ \frac{\mathrm{d}{T}_e}{\mathrm{d}t}=(1-q)\left(1+\frac{mC}{bN+C}\right)\beta (t)ST-\left(\mu +\frac{kC}{N}+\nu +{\gamma }_1\right)T_e,\\ \frac{\mathrm{d}T}{\mathrm{d}t}=\nu T_e+q\left(1+\frac{mC}{bN+C}\right)\beta (t)ST-\left(\mu +\frac{kC}{N}+\delta +{\gamma }_2\right)T,\\ \frac{\mathrm{d}R}{\mathrm{d}t}={\gamma }_1{T}_e+{\gamma }_{2}T-\left(\mu +\frac{kC}{N}+\theta \right)R,\\ \frac{\mathrm{d}C}{\mathrm{d}t}=\sigma EN-\xi C,\\ \frac{\mathrm{d}E}{\mathrm{d}t}=a(t)-\eta (t)E-\sigma EN,\end{array}\right)$$ (2.2)

where C(t) represents the concentration of PM₁₀ in the body of total population at time t, m the effect coefficient of PM₁₀ and MTB inhaled of TB transmission, k the effect coefficient of PM₁₀ on the body of mortality. E(t) denotes the concentration of PM₁₀ at time t. Based on the law of mass action (Huang et al., 2013, 2015; Wang et al., 2018), σEN is the population uptake PM₁₀ from the environment and σ is the uptake coefficient, ξ the depuration coefficient of toxin in the body due to the metabolic processes, a(t) the exogenous input of PM₁₀ into the environment, and η(t) the depuration rate of PM₁₀ in the environment. And assume that a(t) and η(t) are ω-periodic functions.

Fig. 1.

Flow diagram representing TB transmission routes.

Consider the sixth equation in (2.2), the amount of the PM₁₀ at time t is mainly determined by the exogenous input, and the depuration of PM₁₀ and the consumption of PM₁₀ by the population. But the amount of PM₁₀ consumed by the population is usually very small in quantity and hence can be neglected. Then the sixth equation in (2.2) for E can be simplified as:

$$\frac{\mathrm{d}E}{\mathrm{d}t}=a(t)-\eta (t)E.$$ (2.3)

Then the research object of this paper is as follows:

$$\left\{\begin{array}{l}\frac{\mathrm{d}S}{\mathrm{d}t}=\mathrm{\Lambda }-\left(\mu +\frac{kC}{N}\right)S-\left(1+\frac{mC}{bN+C}\right)\beta (t)ST+\theta R,\\ \frac{\mathrm{d}{T}_e}{\mathrm{d}t}=(1-q)\left(1+\frac{mC}{bN+C}\right)\beta (t)ST-\left(\mu +\frac{kC}{N}+\nu +{\gamma }_1\right)T_e,\\ \frac{\mathrm{d}T}{\mathrm{d}t}=\nu T_e+q\left(1+\frac{mC}{bN+C}\right)\beta (t)ST-\left(\mu +\frac{kC}{N}+\delta +{\gamma }_2\right)T,\\ \frac{\mathrm{d}R}{\mathrm{d}t}={\gamma }_1{T}_e+{\gamma }_{2}T-\left(\mu +\frac{kC}{N}+\theta \right)R,\\ \frac{\mathrm{d}C}{\mathrm{d}t}=\sigma EN-\xi C,\\ \frac{\mathrm{d}E}{\mathrm{d}t}=a(t)-\eta (t)E,\\ S(0)=S_0>0,T_e(0)=T_{e0}>0,T(0)=T_0>0,R(0)=R_0\ge 0,\\ C(0)=C_0>0,E(0)=E_0>0.\end{array}\right)$$ (2.4)

3. The basic reproduction number and the threshold dynamics

Easy to know that, for model

$$\left\{\begin{array}{ll}\frac{\mathrm{d}E}{\mathrm{d}t}=a(t)-\eta (t)E(t),& t>0,\\ E(0)=E_0,\end{array}\right)$$ (3.1)

there is a unique continuous periodic solution E∗(t) which is globally asymptotically stable, where

$$E^{\ast }(t)=\exp \left\{-{\int }_0^t\eta (s)\mathrm{d}s\right\}\left(E_0+{\int }_0^{t}a(s)\exp \left\{{\int }_0^s\eta (\tau )\mathrm{d}\tau \right\}\mathrm{d}s\right).$$ (3.2)

The study of the dynamics of model (2.4) requires the introduction of the following set:

$$\mathrm{\Gamma }=\left\{\left(S,T_e,T,R,C,E\right)\in {\mathbb{R}}_{+}^6:0\le N\le \frac{\mathrm{\Lambda }}{\mu },0<C\le \widehat{C},0<E\le \widehat{E}\right\},$$

where $\widehat{E}={\sup }_{t\in [0,\omega )}{E}^{\ast }(t),\widehat{C}=\frac{\sigma \mathrm{\Lambda }\widehat{E}}{\xi \mu }$. And Γ is positive invariant set for model (2.4).

When the disease dies out, that is, only S survives, T_e, T and R go to extinction, we can get the following system from model (2.4):

$$\{\begin{array}{l}\frac{\mathrm{d}S}{\mathrm{d}t}=\mathrm{\Lambda }-kC-\mu S,\\ \frac{\mathrm{d}C}{\mathrm{d}t}=\sigma E^{\ast }(t)S-\xi C,\\ \end{array}$$ (3.3)

which is equivalent to

$$\frac{\mathrm{d}u}{\mathrm{d}t}=A(t)u+f(t),$$

where u = (S,C)^T and $A(t)=\left(\begin{array}{cc}-\mu & -k\\ \sigma E^{\ast }(t)& -\xi \end{array}\right)$,  f(t) = (Λ,0)^T, and T denotes the matrix/vector transposition.

Following (Aronsson & Kellogg, 1978; Wang, Xiao, & Cheke, 2016), we know that model (3.3) with initial condition (S₀, C₀) has a unique positive ω-periodic solution u∗(t) = (S_∗(t), C_∗(t)), which is globally asymptotically stable.

Let $x(t)={(T_e(t),T(t),C(t),S(t),R(t),E(t))}^{\mathbf{T}}$, and then model (2.4) admits a unique disease-free ω-periodic solution

$$x_{\ast }(t)={(0,0,C_{\ast }(t),S_{\ast }(t),0,E^{\ast }(t))}^{\mathbf{T}},$$

where E∗(t) is defined as (3.2) and C_∗(t), S_∗(t) satisfy (3.3).

According to the work of (Diekmann, Heesterbeek, & Metz, 1990; Diekmann & Heesterbeek, 2000; Van den Driessche & Watmough, 2002; Wang & Zhao, 2012), we can determine the basic reproduction number ${\mathcal{R}}_0$ of model (2.4). Let $\mathcal{F}(t,x)$ be the input rate of newly infected individuals and $\mathcal{V}(t,x)$ be the rate of transfer of individuals, then

$$\mathcal{F}(t,x)=\left(\begin{array}{c}(1-q)\left(1+\frac{mC}{bN+C}\right)\beta (t)ST\\ q\left(1+\frac{mC}{bN+C}\right)\beta (t)ST\\ 0\\ 0\\ 0\\ 0\end{array}\right)$$

and

$$\mathcal{V}(t,x)=\left(\begin{array}{c}\left(\mu +\frac{kC}{N}+\nu +{\gamma }_1\right)T_e\\ -\nu T_e+\left(\mu +\frac{kC}{N}+\delta +{\gamma }_2\right)T\\ -(\sigma EN-\xi C)\\ -\left(\mathrm{\Lambda }-\left(\mu +\frac{kC}{N}\right)S-\left(1+\frac{mC}{bN+C}\right)\beta (t)ST+\theta R\right)\\ -\left({\gamma }_1{T}_e+{\gamma }_{2}T-\left(\mu +\frac{kC}{N}+\theta \right)R\right)\\ -(a(t)-\eta (t)E)\end{array}\right),$$

then

$$D_x\mathcal{F}(t,x){|}_{x=x_{\ast }(t)}=\left(\begin{array}{cc}F(t)& 0\\ 0& 0\end{array}\right),D_x\mathcal{V}(t,x){|}_{x=x_{\ast }(t)}=\left(\begin{array}{cc}V(t)& 0\\ J(t)& -M(t)\end{array}\right),$$

where

$$\begin{array}{c}F(t)=\left(\begin{array}{cc}0& (1-q)\beta (t)S_{\ast }(t)\left(1+\frac{m{C}_{\ast }(t)}{b{S}_{\ast }(t)+C_{\ast }(t)}\right)\\ 0& q\beta (t)S_{\ast }(t)\left(1+\frac{m{C}_{\ast }(t)}{b{S}_{\ast }(t)+C_{\ast }(t)}\right)\end{array}\right),\\ V(t)=\left(\begin{array}{cc}\mu +\nu +{\gamma }_1+\frac{k{C}_{\ast }(t)}{S_{\ast }(t)}& 0\\ -\nu & \mu +\delta +{\gamma }_2+\frac{k{C}_{\ast }(t)}{S_{\ast }(t)}\end{array}\right),\end{array}$$

and

$$\begin{array}{c}J(t)=\left(\begin{array}{cc}-\sigma E^{\ast }(t)& -\sigma E^{\ast }(t)\\ -\frac{k{C}_{\ast }(t)}{S_{\ast }(t)}& -\frac{k{C}_{\ast }(t)}{S_{\ast }(t)}+\left(1+\frac{m{C}_{\ast }(t)}{b{S}_{\ast }(t)+C}\right)\beta S_{\ast }(t)\\ -{\gamma }_1& -{\gamma }_2\\ 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}M(t)=\left(\begin{array}{cccc}-\xi & \sigma E^{\ast }(t)& \sigma E^{\ast }(t)& \sigma S_{\ast }(t)\\ -k& -\mu & \frac{k{C}_{\ast }(t)}{S_{\ast }(t)}+\theta & 0\\ 0& 0& -\mu -\frac{k{C}_{\ast }(t)}{S_{\ast }(t)}-\theta & 0\\ 0& 0& 0& -\eta (t)\end{array}\right).\end{array}$$

Then F(t) is non-negative, and −V(t) is cooperative in the sense that the off-diagonal elements of −V(t) are non-negative. Let Φ_M(t) be the fundamental solution matrix of the linear ω − periodic system

$$\frac{\mathrm{d}z}{\mathrm{d}t}=M(t)z.$$

Then the spectral radius ρ(Φ_M(t)) of Φ_M(t) is less than unity.

We can see that the eigenvalues of −V are the diagonal elements and negative. So −V is stable, namely ρ(Φ_−V(ω)) < 1.

Let Y(t, s), t ≥ s be the evolution operator of the linear ω − periodic system

$$\frac{\mathrm{d}y}{\mathrm{d}t}=-V(t)y.$$ (3.4)

That is, for each $s\in \mathbb{R}$, the 2 × 2 matrix Y(t, x) satisfies

$$\left\{\begin{array}{ll}\frac{\mathrm{d}}{\mathrm{d}t}Y(t,s)=-V(t)Y(t,s),& \forall t\ge s,\\ Y(s,s)=\mathbf{I},\end{array}\right)$$

where I = diag(1, 1) is an identity matrix. Thus, the monodromy matrix Φ_−V(t) of (3.4) equals Y(t, 0), t ≥ 0.

Following (Wang & Zhao, 2008), let φ(s) be ω-periodic in s and the initial distribution of infectious individuals, so F(s)φ(s) is the rate of new infections produced by the infected individuals who are introduced at time s. When t ≥ s, Y(t, s)F(s)φ(s) gives the distribution of those infected individuals who are newly infected by φ(s) and remain in the infected compartments T_e and T at time t. Naturally,

$${\int }_{-\infty }^{t}Y(t,s)F(s)\phi (s)\mathrm{d}s={\int }_0^{\infty }Y(t,t-a)F(t-a)\phi (t-a)\mathrm{d}a$$

is the distribution of cumulative new infections at time t produced by all those infected individuals φ(s) introduced at previous time s to t.

Let C_ω be the ordered Banach space of all ω-periodic functions from $\mathbb{R}$ to ${\mathbb{R}}^2$, which is equipped with the maximum norm ‖ ⋅‖ and the positive cone $C_{\omega }^{+}≔\{\phi \in C_{\omega }:\phi (t)\ge 0,\forall t\in \mathbb{R}\}$. Then we define a linear operator L implies that

$$(L\phi )(t)={\int }_0^{\infty }Y(t,t-a)F(t-a)\phi (t-a)\mathrm{d}a,\forall t\in \mathbb{R},\phi \in C_{\omega },$$

which is called the next infection operator, and the spectral radius of L is defined as the basic reproduction number

$${\mathcal{R}}_0≔\rho (L)$$ (3.5)

for the periodic epidemic model (2.4).

In a special case of β(t) = β, a(t) = a and η(t) = η, the basic reproduction number ${\mathcal{R}}_0$ of model (2.4) is ${\overline{\mathcal{R}}}_0=\rho (F{V}^{-1})$, i.e.,

$${\overline{\mathcal{R}}}_0=\frac{\mathrm{\Lambda }\beta {\eta }^2{\xi }^2\left(akq\sigma +\eta \mu q\xi +\eta q\xi {\gamma }_1+\nu \eta \xi \right)\left(am\sigma +b\eta \xi +a\sigma \right)}{\left(b\eta \xi +a\sigma \right)\left(ak\sigma +\eta \mu \xi \right)\left(ak\sigma +\eta \mu \xi +\nu \eta \xi +{\gamma }_1\eta \xi \right)\left(ak\sigma +\delta \eta \xi +\eta \mu \xi +{\gamma }_2\eta \xi \right).}$$ (3.6)

Furthermore, we can obtain the global threshold dynamics of model (2.4). If ${\mathcal{R}}_0<1$, model (2.4) admits a unique disease-free periodic solution x_∗(t) = (0, 0, C_∗(t), S_∗(t), 0, E∗(t)) which is globally asymptotically stable, while if ${\mathcal{R}}_0>1$, it is unstable. If ${\mathcal{R}}_0>1$, there exists a positive constant ζ > 0 such that for any initial values x(0) ∈ Γ, the solution of model (2.4) satisfies

$${\mathrm{lim\; inf}}_{t\to \infty }(T_e,T,C,S,R,E)\ge (\zeta ,\zeta ,\zeta ,\zeta ,\zeta ,\zeta ).$$

That is, model (2.4) is uniformly persistent.

4. Numerical results via TB disease dynamics

4.1. Data collection

The TB case data in Jiangsu Province during 2015–2019 (60 months) were derived from the Jiangsu Center for Disease Control and Prevention (Jiangsu Commision of Health, 2020), which included information of incidence case number and death number. And the demographic data were collected from Jiangsu Provincial Bureau of Statistics (Jiangsu Provincial Bureau of Statistics, 2022). The information regarding air pollution is the data of PM₁₀ which were gathered from the National Meteorological Information Center (China online air quality monitoring). See Fig. 2.

Fig. 2.

The monthly reported number of TB cases and the data of PM₁₀ in Jiangsu, China from 2015 to 2019.

4.2. Parameter estimation

We first use the function fminsearch with Nelder-Mead algorithm in Matlab to estimate the parameters a₁, a₂, a₃ in a(t) and c in η(t) via model (3.1). That is, we use algorithm fminsearch to find the variables a₁, a₂, a₃ and c so that the square of the difference between the solution of model (3.1) and the data of PM₁₀ gets the minimum.

Substituting the values of $a(t)=a_1(1+a_2\sin (\frac{\pi }{6}t+a_3))$ and η(t) = c(1 + t)^1/3 into model (2.4), similar to that in (Cai et al., 2021), we compute the i-th month cumulative TB cases Z_i as Z_i≔ ∫_{i-th month}κγ₂T dt, where κ ∈ (0, 1) is a constant which represents a combined effect of the TB symptomatic rate and the reporting rate, where we adopt κ = 0.01. Furthermore, we model the i-th monthly reported cumulative number of TB cases C_i as a hidden Markov model from the theoretical results Z_i in model (2.4). And we adopt the Poisson-distributed priors for C_i such that all C_i are assumed to follow Poisson distributions according to the theoretical outcomes Z_i. More precisely, we assume that the rate of Poisson distribution is a variable depending on the Z_i, and the reported cumulative number of TB cases C_i is a random sample from the Poisson distribution in model (2.4). Hence, C_i ∼Poisson (mean = Z_i). And model fitting is performed by Markov Chain Monte Carlo (MCMC) method (Guo et al., 2021, 2022; Luo et al., 2019; Zhao et al., 2018, 2019).

4.3. Fitting and projecting results

We show the parameters’ values used for the numerical simulation and sensitivity analysis for model (2.4) in Table 1.

Table 1.

The summary table of model parameters’ values.

parameter	value	unit	statu
Λ	60000	person	Jiangsu Provincial Bureau of Statistics (2022)
μ	0.00905	per month	Jiangsu Provincial Bureau of Statistics (2022)
q	0.05	unit-free	Cai et al. (2021)
γ1 = γ2	0.5	per month	Cai et al. (2021)
σ	1000	per case month	assumed
ξ	1.608	unit-free	assumed
θ	0.45	per month	fitted
ν	0.25	per month	fitted
δ	0.0003	per month	Jiangsu Commision of Health (2020)
m	0.005	unit-free	fitted
b	5.1	unit-free	fitted
k	10–8	unit-free	fitted
β(t) = b1 sin(b2t + b3)	b1 = 0.00031446
	b2 = 0.00000063	unit-free	fitted
	b3 = 0.00005273
$a(t)=a_1+a_2\sin (\frac{\pi }{6}t+a_3)$	a1 = −0.1845
	a2 = 13.6848	unit-free	fitted
	a3 = 8.5138
$\eta (t)=c\sin (\frac{\pi }{6}t)$	c = 0.0479	unit-free	fitted
N(0)	7987495	person	Jiangsu Provincial Bureau of Statistics (2022)
S(0)	7976000	person	Jiangsu Provincial Bureau of Statistics (2022)
T(0)	2495	person	Jiangsu Commision of Health (2020)
Te(0)	8000	person	assumed
R(0)	1000	person	assumed
C(0)	2000	μg/m3	assumed
E(0)	135	μg/m3	(China online air quality monitoring)

In numerical experiments, the monthly cumulative number of TB cases and the values of PM₁₀ of Jiangsu, China are used to fit the number of infected cases (see Fig. 3). Also, we find that a decreasing trend in the projection results for 2020–2022 (see Fig. 3). And to evaluate the accuracy of our model, following (Cui et al., 2020; Hu et al., 2022), we employ four statistical indices, the correlation coefficient (CC), absolute error (AE), rote mean square error (RMSE) and the distance between indices of simulation and observation (DISO), and the results are shown in Table 2. We can conclude that the model fits the reported cases in Jiangsu, China well generally.

Fig. 3.

Fitting/Projection model to the cumulative TB cases for years 2015–2019/2020–2022, respectively. The red dots are the reported number of people diagnosed with TB, and the black line is the model fitting result. The shading areas from the darkest to the lightest correspond to the 99%, 95%, 90% and 50% credible intervals [CrI].

Table 2.

Evaluation results of the simulation of weekly cumulative TB cases and the model values.

	CC	AE	RMSE	DISO
Cumulative TB cases	1.00	0.00	0.00	0.00
Model values	1.00	−160.90	1395.66	0.02

4.4. Sensitivity analysis

Consider the basic reproduction number ${\mathcal{R}}_0$ defined in (3.5) again, we agree that it should be a time-varying function in an autonomous setting that all model parameters are fixed. For example, we can see the term β in (3.6) is the annual average of time-varying parameter β(t). Thus, the basic reproduction number ${\overline{\mathcal{R}}}_0$ in (3.6) can be taken as the average of the time-varying basic reproduction number ${\mathcal{R}}_0$ defined in (3.5).

It is well known that the basic reproduction number ${\mathcal{R}}_0$ is dependent on almost all parameters in model (2.4) implicitly. In order to identify the impacts of these parameters on TB transmission and prevalence, we adopt the partial rank correlation coefficient (PRCC) to study the influence of the input variables (parameters in model) on the output variables (the basic reproduction number ${\overline{\mathcal{R}}}_0$ in the special autonomous case). The larger the absolute values of PRCC, the more important the parameters in responding to the change of new TB cases. In Fig. 4, we fix β(t) = 0.0000021, a(t) = 50.7419, η(t) = 0.0181, other parameters taken as those in Table 1, and show the values of PRCC on the outcome of ${\overline{\mathcal{R}}}_0$. We can see that parameters Λ, ξ, and β have a positive impact on ${\mathcal{R}}_0$, and conversely, μ, σ and γ₂ have a negative impact, which should be given priorities in controlling the TB epidemics.

Fig. 4.

The values of PRCC on the outcome of ${\mathcal{R}}_0$. ∗ represents significant test (P < 0.05).

In model (2.4), the parameters k, m, σ and ξ are closely related to the effect of PM₁₀ on the incidence of TB. For learning the effects of these parameters on ${\mathcal{R}}_0$ further, in Fig. 5, as an example, we fix β(t) = 0.0000021, a(t) = 50.7419, η(t) = 0.0181, other parameters taken as those in Table 1, and show the relations between ${\mathcal{R}}_0$ with k, σ and ξ, respectively. Obviously, ${\mathcal{R}}_0$ decreases with the increase of k and σ, and increases with the increase of ξ. More precisely, if k > 3.7529⁻⁸, ${\mathcal{R}}_0<1$, the TB will go into extinction, while if k < 3.7529⁻⁸, ${\mathcal{R}}_0>1$, the TB will persist (see Fig. 5(a)); if σ > 3727.0709, ${\mathcal{R}}_0<1$, the TB will be exterminated, while if σ < 3727.0709, ${\mathcal{R}}_0>1$, the TB will be protracted (see Fig. 5(b)); if ξ < 0.4314, ${\mathcal{R}}_0<1$, the TB will die out, while if ξ > 0.4314, ${\mathcal{R}}_0>1$, the TB will be an endemic disease (see Fig. 5(c)).

Fig. 5.

The relations between ${\mathcal{R}}_0$ with (a) k; (b) σ; (c) ξ.

4.5. Control the incidence of TB

It is worthy to note that the results in the subsection above characterize the long-term (i.e., t → ∞) dynamics of model (2.4). Next, we will focus on the short-term control strategies of TB in Jiangsu, China by conducting the numerical simulations to present the changing dynamics of the TB epidemics and the levels of PM₁₀ with changes in the epidemiological parameters, and the parameters taken as those in Table 1.

In order to study the effects of the parameters on the disease dynamics further, in Fig. 6, we show the relations between the solutions of model (2.4) with parameters k, m, σ and ξ which are closely related to the incidence of TB.

• •In Fig. 6(a), when k increases from 10⁻⁸ to 10⁻⁷ and 10⁻⁶, the solutions of model (2.4) decrease significantly.
• •In Fig. 6(b), when m decreases from 0.05 to 0.03 and 0.01, the solutions of model (2.4) decrease.
• •In Fig. 6(c), when σ decreases from 1000 to 500, the solutions of model (2.4) increase; while σ increases to 2000, the solutions of model (2.4) decrease.
• •In Fig. 6(d), when ξ decreases from 1.608 to 1.0 and 0.5, the solutions of model (2.4) decrease.

Fig. 6.

The relations between the solutions of model (2.4) with parameters (a) k; (b) m; (c) σ; (d) ξ.

In a nutshell, to control the spread of the TB, we must decrease m (see Fig. 6(a)), and ξ (see Fig. 6(c)), and increase k (see Fig. 6(b)) and σ (see Fig. 6(d)).

Furthermore, we study the effects of other parameters, such as q, ν, γ₁ and γ₂, on the TB disease dynamics of model (2.4) and the results are shown in Fig. 7.

• •In Fig. 7(a), when q decreases from 0.05 to 0.03 and 0.01, the solutions of model (2.4) decrease.
• •In Fig. 7(b), when ν decreases from 0.25 to 0.2 and 0.15, the solutions of model (2.4) decrease.
• •In Fig. 7(c) and (d), when γ₁ and γ₂ decrease from 0.5 to 0.45, respectively, the solutions of model (2.4) increase; while γ₁ and γ₂ increase to 0.55, respectively, the solutions of model (2.4) decrease.

Fig. 7.

The relations between the solutions of model (2.4) with parameters (a) q; (b) ν; (c) γ₁; (d) γ₂.

And hence, for control the incidence of TB in Jiangsu, China, we must decrease q (see Fig. 7(a)) and ν (see Fig. 7(b)), or increasing γ₁ (see Fig. 7(c)) and γ₂ (see Fig. 7(d)).

5. Conclusion and discussions

The main focus of this study is to investigate the effects of the PM₁₀ on the TB transmission dynamics. We define the basic reproduction number ${\mathcal{R}}_0$ (3.5) which can be used to govern the threshold dynamics of the model: if ${\mathcal{R}}_0<1$, the unique disease-free equuilibrium is globally asymptotic stable, while if ${\mathcal{R}}_0>1$, there is at least one positive periodic solution and TB will persist uniformly. Epidemiologically, we find that there is a reduction trend of the incidence of TB (see Fig. 3), but the average basic reproduction number ${\overline{\mathcal{R}}}_0>1$, then TB is an endemic disease and will persist in Jiangsu, China for a long time. This is the same result as in (Cai et al., 2021).

Sensitivity analysis is vital to identify key parameters and to find effective control strategies for combatting the spread of TB. The results of sensitivity analysis indicate that the recruitment rate of susceptible Λ, the depuration coefficient of PM₁₀ in the body due to the metabolic processes ξ, the transmission rate coefficient β, the per capita natural mortality rate μ, the uptake coefficient σ and the symptomatic infectious recover rate γ₂ are the more sensitive parameters to ${\overline{\mathcal{R}}}_0$ in the long-term than any other parameters (see Fig. 4).

The effective control efforts in the short-term are suggested to focus on decreasing the depuration coefficient of PM₁₀ in the body ξ (see Fig. 6(c)), the proportion of TB symptomatic infectious by direct transmission q (see Fig. 7(a)) and the reactivation rate of the pre-symptomatic infectious ν (see Fig. 7(b)), or on increasing the uptake coefficient σ (see Fig. 6(d)), the recovery rate of the pre-symptomatic infectious γ₁ (see Fig. 7(c)) and the recovery rate of the symptomatic infectious γ₂ (see Fig. 7(d)).

It is worthy to note that, increasing close-contact distance and wearing protective mask can efficiently decrease the effect coefficient m of PM₁₀ and MTB inhaled of TB transmission (see Fig. 6(a)) and increase the influence coefficient k of PM₁₀ on the body of mortality (see Fig. 6(b)), two parameters closely related to ambient particulate air pollution (e.g., PM₁₀), and therefore be useful to control the spread of TB in Jiangsu, China.

Declaration of competing interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Handling Editor: Dr. Jianhong Wu