An Epidemiological Model for Tuberculosis Considering Environmental Transmission and Reinfection

Abstract

As tuberculosis (TB) patients do not have lifetime immunity, environmental transmission is one of the key reasons why TB has not been entirely eradicated. In this study, an SVEIRB model of recurrent TB considering environmental transmission was developed to explore the transmission kinetics of recurrent TB in the setting of environmental transmission, exogenous infection, and prophylaxis. A more thorough explanation of the effect of environmental transmission on recurrent TB can be found in the model’s underlying regeneration numbers. The global stability of disease-free and local equilibrium points can be discussed by looking at the relevant characteristic equations. The Lyapunov functions and the LaSalle invariance principle are used to show that the local equilibrium point is globally stable, and TB will persist if the basic reproduction number is larger. Conversely, the disease will disappear if the basic reproduction number is less than one. The impact of environmental transmission on the spread of tuberculosis was further demonstrated by numerical simulations, which also demonstrated that vaccination and reducing the presence of the virus in the environment are both efficient approaches to control the disease’s spread.

1. Introduction

Tuberculosis (TB) can manifest in any area of the body, including the brain, intestines, kidneys, bones, endometrium, and more. In medicine, TB infection refers to the process by which TB bacilli infiltrate the body—only when the body’s defenses are compromised or when an excessive number of bacteria enter the body do people become ill. Because more than 95% of primary TB cases occur in the lungs, tuberculosis is the most prevalent TB condition. One of the main causes of death in the global total, TB is an infectious disease spurred on by Mycobacterium tuberculosis (MTB) infection [1]. In 2022, according to WHO data, the number of deaths due to tuberculosis worldwide has now reached 7.8 million. Even though TB eradication has become possible in some developed countries with low incidence, mortality due to TB remains high in developing countries such as Central Africa, Algeria, Botswana, Vietnam, North Korea, and South Africa [2]. Therefore, it is crucial to act on several fronts to stop the spread of TB in addition to making sure that all TB patients have access to treatment to lower the prevalence of TB on a worldwide scale.

TB is a transmissible illness that predominantly spreads through the air. When a person with TB coughs, sneezes, sings, laughs, or engages in other behaviors that include airborne routes of transmission, MTB is discharged into the air and is transmitted from one person to another in minute microscopic droplets [3]. Primary infections [4], endogenous recurrence [5], exogenous reinfection [6], and recurring infections are the most common tuberculosis infections [7]. MTB’s pathophysiology is defined as an infection that either does not progress or does so for an extended period. Either it advances directly to active TB disease, or the illness manifests right away [8]. The protracted amount of time required for the disease to transition from latent to active is the crucial distinction between TB and other infectious diseases. Mathematicians and biologists have conducted several theoretical investigations on various facets of nodule propagation dynamics over the last few decades [9,10,11,12]. Samuel Bowong [13] created an SEI model that outlines the TB transmission pathway in terms of typical contact rates. With the creation of the Lyapunov function and application of the LaSalle principle, the hypothesized model’s comprehensive dynamics are investigated. Feng et al. [8] took into account the possibility of exogenous secondary infection, including the onset of exogenous reinfection in latent individuals and endogenous onset in individuals, as well as the potential for relapse in recovered individuals due to exogenous reinfection. They came to conclude that abdicating the disease requires minimizing contact between the incubator and the patient. Additionally, numerous models have been enhanced by taking exogenous infections into account to account for factors including illness recurrence [12], epidemic models that account for immunological conditions [14,15], nonlinear infection rates [10], slow and rapid advancement [16], relating to incomplete immunity [17,18,19], infection–age structuring [20], coinfection [21,22], and medication resistance [23]. Fatima Sulayman [24] revealed that a substantial influence on TB transmission is triggered by immunity loss following several years of adopting TB control techniques through vaccination and treatment of infectious diseases, which are insufficient to eradicate TB. The spread and control of tuberculosis are largely influenced by all of the aforementioned factors, although tuberculosis is still widespread worldwide and cannot be eliminated.

MTB has been discovered to survive and be present in multiple natural and built environments for months to years, possibly after being contaminated by human sources, according to experimental and epidemiological evidence [25]. Recent research on the survivability of MTB in the environment and its detection utilizing cutting-edge techniques has produced consistent results. For a deeper understanding of Mycobacterium tuberculosis transmission, it is essential to conduct additional research to determine whether environmental MTB provides an infectious threat to humans. This possibility may have been overlooked in previous studies. It is clear from this discussion that neglecting the environment in the modeling process when taking recurrent TB into account may result in inaccurate and unclear forecasts. Although the environmental transmission barrier to disease prevention is not taken into account in the mathematical modeling studies of pulmonary tuberculosis that are currently in use, other epidemiology modeling processes have been successfully conducted that way. Meili Li et al. [26] constructed and examined the network epidemic SIWR model of illnesses, such as cholera, that can be spread through the environment, and they highlighted the impact of environmental transmission on epidemic transmission. Chinwendu E. Madubueze et al. [27] developed a deterministic model of the environmental transmission dynamics of monkeypox (MPX) in isolation and immunization, indicating that environmental transmission parameters are the primary drivers of infection. Preety Kumari et al. [28] developed a mathematical model of COVID-19 transmission that took into account asymptomatic people, quarantined employees, and environmental transmission factors, with an emphasis on the role of environmental factors on the transmission of the COVID-19 epidemic. Emmanuel Addai et al. [29] considered the effects of vaccination and environmental transmission on monkeypox virus dynamics taken into consideration using the Caputo operator. A sensitivity study demonstrates that environmental transmission increases the monkeypox viral infection rate. Therefore, the total eradication of infectious diseases depends greatly on environmental transmission. As a result, a significant development in the research of pulmonary tuberculosis transmission is the modeling of pulmonary tuberculosis dynamics taking environmental transmission into account.

The inclusion of environmental transmission parameters in the model has some research relevance in light of the tuberculosis transmission mechanism and the body of existing literature. The SVEIRE model is used in this work to analyze the effects of imperfect vaccines and other exogenous factors on the treatment and prevention of tuberculosis (TB), which was inspired by Fatima Sulayman et al. [24]. This essay will concentrate on the part environmental transmission plays in tuberculosis prevention in this paradigm. First, this research suggests a six-dimensional SVEIRB model address this issue. In contrast to other infectious diseases, the mathematical model takes into account secondary infection in light of the peculiarities of pulmonary tuberculosis patients without lifetime immunity. Second, this essay provides the prerequisites for the presence of endemic equilibrium points as well as evidence of both their worldwide stability and the existence of disease-free equilibrium points. Fatima Sulayman et al. [24] did not take into account the global stability of local equilibrium points, simply the existence and local stability of disease-free equilibrium points and endemic equilibrium points. We are aware that the key to deciding whether an outbreak is eradicated is global stability. As a result, a certain theoretical foundation for the eradication of diseases is provided by our analysis of the stability of the SVEIRB model.

The structure of this essay is as follows: Part 2 develops the SVEIRB model with environmental transmission in mind. In Part 3, the critical components of the model are derived. Section 4 resolves the actual in-growth number, existence, and stability of the disease-free equilibrium point. Section 5 investigates the existence and stability of the endemic equilibrium point. In Section 6, numerical simulations and discussions are carried out. Finally, the summary is laid out in Section 7.

2. Model Building

According to the pathogenesis of tuberculosis [11,30], the population is separated into the five groups below: susceptible, vaccinated, latent, infected, and recovered. We use $S\left(t\right),V\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right),B\left(t\right)$ to represent the number of susceptible, vaccinated, latent, infected, and recovered, $B\left(t\right)$ for the number of TB viruses in the time$t$environment, where $N\left(t\right)$represents the total population at the time and $N\left(t\right)=S\left(t\right)+V\left(t\right)+E\left(t\right)+I\left(t\right)+R\left(t\right)$. Class $S$ does not carry MTB but contact can be infected. Class $V$ is vaccinated newborns. Class$E$has been infected but not contagious. Class$I$is infected and infectious people. Class$R$for treatment recovered people but do not have lifelong immunity.

In this paper, all newborns were considered susceptible, and a flowchart of disease transmission was established, as shown in Figure 1.

Considering vaccination does not completely ensure that a population will have lifelong immunity [11,30,31], it is crucial to take environmental transmission into account when planning TB prevention and treatment. In this paper, a six-dimensional model of infectious disease dynamics was developed as follows:

$$\{\begin{array}{l}\frac{\mathrm{d}S}{\mathrm{d}t}=A+pV-\left(\mu +\omega \right)S-\beta SI-\delta SB\\ \frac{\mathrm{d}V}{\mathrm{d}t}=\omega S-\left(p+\mu \right)V-k\left(\beta VI+\delta VB\right)\\ \frac{\mathrm{d}E}{\mathrm{d}t}=\beta SI+\delta SB+k\left(\beta VI+\delta VB\right)+h\left(\beta RI+\delta RB\right)-\left(\mu +\epsilon \right)E-\alpha \beta EI\\ \frac{\mathrm{d}I}{\mathrm{d}t}=\epsilon E+\alpha \beta EI-\left(\gamma +\mu +{\mu }_1\right)I\\ \frac{\mathrm{d}R}{\mathrm{d}t}=\gamma I-\mu R-h\left(\beta RI+\delta RB\right)\\ \frac{\mathrm{d}B}{\mathrm{d}t}=mI-\tau B\end{array}.$$

(1)

where

$$\begin{gathered} S\left(0\right)=S_0\ge 0,V\left(0\right)=V_0\ge 0,E\left(0\right)=E_0\ge 0, \\ I\left(0\right)=I_0\ge 0,R\left(0\right)=R_0\ge 0,B\left(0\right)=B_0\ge 0 \end{gathered}$$

(2)

In this epidemiological system, the susceptible compartment is increased through the recruitment of individuals, either by immigration or birth, into the population at a constant rate $A$, while each compartment is diminished at the rate $\mu$. Administration of vaccines to the susceptible population moves individuals to the vaccinated compartment at the rate $\omega$; however, the protection provided by the vaccine wanes over time at the rate of $p$. The susceptible individuals become infected at a rate $\beta$. In the vaccinated compartment, individuals acquire infection through contact with exposed individuals and residual viruses in the environment. The indirect contact transmission rate $\delta$ of humans infected through environmental exposure to MTB as the vaccine offered to individuals is thought to be imperfect (i.e., it does not offer 100% assurance against the TB disease). The vaccinated individuals may become infected again at a rate $k$; but with a lower level than those in the susceptible compartment (i.e., unvaccinated individuals). The exposed compartment becomes infectious at the rate of $\epsilon$ and progresses to an actively infected state. Exogenous reinfection can enable individuals that are already infected to develop active tuberculosis. This happens when they contract new infections from infectious individuals at a constant rate $\alpha$. Infected individuals recovered with the rate $r$ and decreased due to TB-induced death at a rate ${\mu }_1$. The rate $m$ of the release of MTB into the environment by infected persons. Finally, the recovered individuals may return to the exposed compartment, due to low immunity, at the rate of $h$.

3. Mathematical Model Analysis

3.1. Positivity of Solutions

It must be demonstrated that all variables are positive for the constructed TB model to have value in real life.

Lemma 1.

Assume that $\left\{\left(S, V,E,I,R,B\right)\ge 0\right\}\in \Gamma$that for any$t>0$to have solution set $\left\{S\left(t\right),V\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right),B\left(t\right)\right\}$.

Proof. 

From the first equation of the TB model (1), we have

$$\frac{\mathrm{d}S}{\mathrm{d}t}=A+pV-\left(\mu +\omega \right)S-\beta SI-\delta SB,$$

(3)

then

$$\frac{\mathrm{d}S}{\mathrm{d}t} \ge -\left(\mu +\omega +\beta I+\delta B\right)S.$$

(4)

Using the separation of variables method yields

$$\int \frac{\mathrm{d}S}{S}\ge \int -\left(\mu +\omega +\beta I+\delta B\right)\mathrm{d}t,$$

(5)

$$\ln S \ge -\left(\mu +\omega +\beta I+\delta B\right)t,$$

(6)

then

$$S\left(t\right)\ge S\left(0\right)e^{-\left(\mu +\omega +\beta I+\delta B\right)t},$$

(7)

therefore

$$S\left(0\right)\ge 0.$$

(8)

The same can be obtained

$$V\left(t\right)\ge 0,E\left(t\right)\ge 0,I\left(t\right)\ge 0,R\left(t\right)\ge 0,B\left(t\right)\ge 0.$$

(9)

Therefore, for every solution of the system is positive, the disease is always present. □

3.2. Boundedness

Lemma 2.

For the initial condition (2), the solution of the model (1) is contained in the$\Gamma =\left\{S\left(t\right),V\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right),B\left(t\right)\right\}\in {\mathcal{R}}_{+}^6$region, then

$$\Gamma =\left\{ S\left(t\right), V\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right),B\left(t\right) \in {\mathcal{R}}_{+}^6: 0 \le N\left(t\right) \le \frac{A}{\mu }; 0<B\left(t\right) \le \frac{mA}{\tau \mu }\right\}.$$

(10)

Proof. 

Consider that model (1) is a nonlinear system and $N=S+V+E+I+R$.

Summing the first five equations of the system, we obtain

$$\frac{\mathrm{d}N}{\mathrm{d}t}\le A-\mu N-{\mu }_{1}I.$$

(11)

then

$$N\left(t\right)\le \frac{A}{\mu }-\left(\frac{A}{\mu }-N_0\right)e^{-\mu t},N\left(0\right)=N_0,$$

(12)

if

$$N\left(0\right)\le \frac{A}{\mu },$$

(13)

then $t\to \infty$ when

$$N \to \frac{A}{\mu }.$$

(14)

Since the $N$ is the total number of people, all feasible solutions of the model converge to the set $\mathsf{\Gamma }$. □

4. Disease-Free Equilibrium Point and Basic Regeneration Number [32,33]

4.1. Existence of Disease-Free Equilibrium Points

The absence of sick people results in a stable solution to model (1) at the disease-free equilibrium point. Solving for the disease-free equilibrium point$P_0$, the absence of infected individuals and MTB in each compartment, then $E=I=R=B=0$.

Bringing in the model (1), it is obtained that

$$\left\{\begin{array}{l}A+pV-\left(\mu +\omega \right)S=0\\ \omega S-\left(p+\mu \right)V=0\end{array}\right..$$

(15)

The solution gives:

$$S=\frac{A\left(p+\mu \right)}{\mu \left(p+\mu +\omega \right)},V=\frac{A\omega }{\mu \left(p+\mu +\omega \right)}.$$

(16)

Therefore, no disease equilibrium point

$$P_0=\left(\frac{A\left(p+\mu \right)}{\mu \left(p+\mu +\omega \right)},\frac{A\omega }{\mu \left(p+\mu +\omega \right)},0,0,0,0\right).$$

(17)

4.2. Basic Regeneration Number

The basic regeneration number $R_0$ gives an average number of secondary infections spread by a single infectious individual in completely susceptible individuals. To find $R_0$, we follow the method of the next-generation matrix, illustrated by RuiXu, Zhili Wang, and Fengqin Zhang.

Let us consider $X=\left(E,I,B\right)$ and rewrite the model (1) as

$$\frac{\mathrm{d}X}{\mathrm{d}t}=\overrightarrow{F}-\overrightarrow{V}.$$

(18)

where, $\overrightarrow{F}$ is the rate at which new infections take place whereas $\overrightarrow{V}$ is all other flows within and out of each compartment. Therefore, we have

$$\overrightarrow{F}=\left(\begin{array}{c}\beta SI+\delta SB+k\left(\beta VI+\delta VB\right)+h\left(\beta RI+\delta RB\right)\\ 0\\ 0\end{array}\right),$$

(19)

$$\overrightarrow{V}=\left(\begin{array}{c}\left(\mu +\epsilon \right)E+\alpha \beta EI\\ -\epsilon E-\alpha \beta EI+\left(\gamma +\mu +{\mu }_1\right)I\\ -mI+\tau B\end{array}\right).$$

(20)

The model (1) always has an infection-free steady state. Then the Jacobian matrix of $F$ and $V$ at $P_0$ is given by

$$F\left(P_0\right)=\left(\begin{array}{ccc}0& \beta S+k\beta V& \delta S+k\delta V\\ 0& 0& 0\\ 0& 0& 0\end{array}\right), V\left(P_0\right)=\left(\begin{array}{ccc}\epsilon +\mu & 0& 0\\ -\epsilon & \gamma +\mu +{\mu }_1& 0\\ 0& -m& \tau \end{array}\right).$$

(21)

Now the spectral radius $\rho$ of matrix $F{V}^{-1}$, gives the basic reproduction number $P_0$ of the model (1). After some algebraic manipulations, we obtain

$$R_0=\rho \left(F\left(P_0\right)V^{-1}\left(P_0\right)\right)=\frac{A\epsilon \left(p+\mu +k\omega \right)\left(\tau \beta +m\delta \right)}{\mu \tau \left(\epsilon +\mu \right)+\left(\gamma +\mu +{\mu }_1\right)\left(p+\mu +\omega \right)}.$$

(22)

When environmental transmission and vaccination are not considered, the

$$R_0^I=\frac{A\epsilon \beta }{\mu \tau \left(\epsilon +\mu \right)\left(\gamma +\mu +{\mu }_1\right)},$$

(23)

where

$$p=0,\omega =0,\delta =0,\tau =0.$$

(24)

When considering environmental transmission but not vaccination

$$R_0^B=\frac{A\epsilon \left(\tau \beta +m\delta \right)}{\mu \tau \left(\epsilon +\mu \right)\left(\gamma +\mu +{\mu }_1\right)},$$

(25)

where

$$p=0,\omega =0.$$

(26)

When vaccination is considered but not environmental transmission

$$R_0^V=\frac{A\epsilon \left(p+\mu +k\omega \right)}{\mu \tau \left(\epsilon +\mu \right)\left(\gamma +\mu +{\mu }_1\right)\left(p+\mu +\omega \right)},$$

(27)

where

$$\delta =0,\tau =0.$$

(28)

and

$$R_0^I<R_0^B, R_0^V<R_0.$$

(29)

4.3. Stability of Disease-Free Equilibrium Points

4.3.1. Local Asymptotic Stability of Disease-Free Equilibrium Points

The initial population size of affected people has an impact on the local stability of the disease-free equilibrium point. In addition, the stability of the disease-free balance point will also alter as the starting value of an infected person changes, which could result in the disease not being completely eradicated.

Theorem 1.

Disease-free equilibrium point$P_0$is locally asymptotically stable when$R_0<1$, the disease-free equilibrium point is unstable when$R_0>1$ [34].

Proof. 

Model (1) at the disease-free equilibrium$P_0$, the Jacobi matrix at$J\left(P_0\right)$, is shown below:

$$J\left(P_0\right)=\left(\begin{array}{cccccc}-\left(\mu +\omega \right)& p& 0& -\beta S& 0& -\delta S\\ \omega & -\left(p+\mu \right)& 0& -k\beta V& 0& -k\delta V\\ 0& 0& -\left(\mu +\epsilon \right)& \beta S+k\beta V& 0& \delta S+k\delta V\\ 0& 0& \epsilon & -\left(\gamma +\mu +{\mu }_1\right)& 0& 0\\ 0& 0& 0& \gamma & -\mu & 0\\ 0& 0& 0& m& 0& -\tau \end{array}\right).$$

(30)

From the Jacobi matrix $J\left(P_0\right)$ it is known that the eigenvalues are ${\lambda }_1=-\mu , {\lambda }_2=-\left(\mu +\omega \right),{\lambda }_3=-\left(p+\mu \right)$. The remaining three eigenvalues of the Jacobi matrix satisfy

$$J_1\left(P_0\right)=\left(\begin{array}{ccc}-\left(\mu +\epsilon \right)& \beta M_2& \delta M_2\\ \epsilon & -\left(\gamma +\mu +{\mu }_1\right)& 0\\ 0& m& -\tau \end{array}\right).$$

(31)

where

$$M_1=\frac{A}{\mu \left(p+\mu +\omega \right)}, M_2=\frac{A\left(k\omega +p+\mu \right)}{\mu \left(p+\mu +\omega \right)}$$

(32)

$$S+kV=M_2.$$

(33)

Matrix $J_1\left(P_0\right)$ The characteristic polynomial is

$$f\left(\lambda \right)={\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0.$$

(34)

where

$$a_2=2\mu +{\mu }_1+\epsilon +\tau +\gamma ,$$

(35)

$$a_1=\tau \left(\mu +\epsilon \right)+\left(1-R_0^I\right),$$

(36)

$$a_0=\tau \left(\epsilon +\mu \right)+\left(\gamma +\mu +{\mu }_1\right)\left(1-R_0\right).$$

(37)

Obviously

$$a_2>0, a_1>0, a_0>0,$$

(38)

$$a_1{a}_2-a_0=\left(\tau \epsilon +\tau \mu +3\mu +2{\mu }_1+\epsilon +\tau +2\gamma \right)+\left(\gamma +\mu +{\mu }_1\right)\left(1-R_0\right)>0.$$

(39)

if $R_0<1$.

According to Hurwitz criterion, it is obtained that at $R_0<1$ time, the disease-free equilibrium point is locally asymptotically stable. □

4.3.2. Global Stability of Disease-Free Equilibrium Points

The initial patient count will shift, but the disease will eventually go away if the disease-free equilibrium point exhibits global stability.

Theorem 2.

Disease-free equilibrium point $P_0$ globally asymptotically stable if  $R_0<1$; the disease-free equilibrium point is unstable if  $R_0>1$.

Proof. 

Constructing Lyapunov functions [35]

$$L=E+c_{0}I+c_{1}B.$$

(40)

$$\begin{gathered} \frac{\mathrm{d}L}{\mathrm{d}t}=\beta SI+\delta SB+k\left(\beta VI+\delta VB\right)+h\left(\beta RI+\delta RB\right)-\left(\mu +\epsilon \right)E \\ -\alpha \beta EI+c_0\epsilon E+c_0\alpha \beta EI-c_0\left(\gamma +\mu +{\mu }_1\right)I+c_1\left(mI-\tau B\right), \end{gathered}$$

(41)

$$\begin{gathered} \frac{\mathrm{d}L}{\mathrm{d}t}=\left[c_0\epsilon -\left(\mu +\epsilon \right)E\right]++\left(\delta M_2-c_1\tau \right)B \\ +\left[c_{1}m-c_0\left(\gamma +\mu +{\mu }_1\right)+\beta M_2\right]I \end{gathered}$$

(42)

Assuming that

$$\delta M_2-c_1\tau =0,$$

(43)

$$c_{1}m-c_0\left(\gamma +\mu +{\mu }_1\right)+\beta M_2=0,$$

(44)

The solution gives

$$c_0=\frac{\delta }{\tau } M_2,$$

(45)

$$c_1=\frac{M_2\left(\tau \beta +m\delta \right)}{\tau \left(\gamma +\mu +{\mu }_1\right)}.$$

(46)

then we obtain

$$L=E+\frac{\delta }{\tau }{M}_{2}I+\frac{M_2\left(\tau \beta +m\delta \right)}{\tau \left(\gamma +\mu +{\mu }_1\right)}B,$$

(47)

$$\frac{\mathrm{d}L}{\mathrm{d}t}=c_0\epsilon -\left(\mu +\epsilon \right)=\left(\mu +\epsilon \right)\left(R_0-1\right)\frac{\left(\gamma +\mu +{\mu }_1\right)I}{\epsilon +\alpha \beta I}.$$

(48)

When $R_0<1, \frac{\mathrm{d}L}{\mathrm{d}t} \le 0$, when and only when $I=0, \frac{\mathrm{d}L}{\mathrm{d}t}=0$, that is $P_0$ is the maximum invariant set of the model (1) in the set$\mathsf{\Gamma }$the set’s greatest invariant set, which is known by the LaSalle invariance principle [36], the disease-free equilibrium point $P_0$ in the set$\mathsf{\Gamma }$is globally stable. □

5. Existence and Global Stability of Endemic Equilibrium Points

5.1. Existence of Local Equilibrium Points

TB-infected people have always existed if there are endemic equilibrium points. Using Cartesian notation, we can determine the existence and number of roots of cubic polynomials.

Assuming the existence of a local equilibrium $P_1=\left(S_1,V_1,E_1,I_1,R_1,B_1\right)$. Letting the right-hand side of the equation of model (1) equal 0, we obtain

$$\{\begin{array}{l}A+p{V}_1-\left(\mu +\omega \right)S_1-\beta S_1{I}_1-\delta S_1{B}_1=0\\ \omega S_1-\left(p+\mu \right)V_1-k\left(\beta V_1{I}_1+\delta V_1{B}_1\right)=0\\ \beta S_1{I}_1+\delta S_1{B}_1+k\left(\beta V_1{I}_1+\delta V_1{B}_1\right)+h\left(\beta R{I}_1+\delta R_1{B}_1\right)-\left(\mu +\epsilon \right)E_1-\alpha \beta E_1{I}_1=0\\ \epsilon E_1+\alpha \beta E_{1}I-\left(\gamma +\mu +{\mu }_1\right)I_1=0 \\ \gamma I_1-\mu R_1-h\left(\beta R_1{I}_1+\delta R_1{B}_1\right)=0\\ m{I}_1-\tau B_1=0\end{array}.$$

(49)

From the first equation and the second equation of model (1), it follows that

$$S_1=\frac{A\left(k{D}_{3}I+\mu +p\right)}{D_3^2{I}^2+D_{2}I+D_1},$$

(50)

$$V_1=\frac{A\omega }{D_3^2{I}^2+D_{2}I+D_1},$$

(51)

From the fourth, fifth and sixth equations of model (1), it follows that

$$E_1=\frac{\left(\gamma +\mu +{\mu }_1\right)I}{\epsilon +\alpha \beta I},$$

(52)

$$R_1=\frac{\gamma I}{\mu +h{D}_{3}I},$$

(53)

$$B_1=\frac{m}{\tau }I.$$

(54)

Substituting$S_1,V_1,E_1,R_1 \mathrm{and} B_1$ into the third equation in model (1) yields

$$f\left(I\right)=b_1{I}^3+b_2{I}^2+b_{3}I+b_4.$$

(55)

where

$$b_1=kh{\beta }^3\left(\mu +\gamma \right),$$

(56)

$$\begin{gathered} b_2=\left(\left(\alpha +1\right)h+\alpha \right)k{\beta }^2{\mu }^2+\left(\left(\alpha +1\right)\delta +\alpha \omega +\epsilon +{\mu }_1\right)h \\ +\alpha \left({\mu }_1+\alpha \right)\tau {\beta }^{2}h\mu +\left(\epsilon {\mu }_1\beta +h\beta \left(-Ak\beta \omega +{\mu }_{1}p\right)\right)kh, \end{gathered}$$

(57)

$$\begin{gathered} b_3=(\left(h+\alpha +1+\tau \right)\beta {\mu }^3+\left(\left({\mu }_1+\omega +\alpha +\gamma \right)h+\left(\alpha +1\right){\mu }_1+\epsilon +\left(\omega +{\mu }_1\right)\alpha +\tau \right)\beta {\mu }^2 \\ -\left(Ak\alpha {\beta }^2+\left(\left(\epsilon +\omega \right){\mu }_1+\omega h\left(\epsilon +\tau \right)+\left(\alpha \omega +\epsilon \right)\left(\mu +\gamma \right)+h\beta \right)\mu +\epsilon \left(-A\beta \omega +{\mu }_1\omega \right)h\beta \right), \end{gathered}$$

(58)

$$b_4=\tau \left(\epsilon +\mu \right)+\left(\gamma +\mu +{\mu }_1\right)\left(\mu +p+\omega \right)\left(1-R_0\right).$$

(59)

It is evident that $I$ is given by the polynomial’s positive real roots [37]. The number of possible positive real roots of the cubic polynomial depends on the signs of $b_2,b_3,b_4$. We will explore this using Descarte’s rule of signs.

Table 1. The cubic polynomial equation’s potential positive roots.

Cases	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	${\mathit{b}}_4$	${\mathit{R}}_0$	Changes in Sign	Total Possible Positive Roots
1	$+$	$-$	$-$	$-$	$R_0>1$	1	1
2	$+$	$+$	$-$	$-$	$R_0>1$	1	1
3	$+$	$+$	$+$	$-$	$R_0>1$	1	1
4	$+$	$-$	$+$	$-$	$R_0>1$	3	1, 3
5	$+$	$-$	$-$	$+$	$R_0<1$	2	0, 2
6	$+$	$+$	$-$	$+$	$R_0<1$	2	0, 2
7	$+$	$-$	$+$	$+$	$R_0<1$	2	0, 2
8	$+$	$+$	$+$	$+$	$R_0<1$	0	0

lists several possible scenarios. As a conclusion, the following theorem may be applied to illustrate the results.

Theorem 3.

The TB model system (1) has the following:

1.

A particular endemic equilibrium when cases 1–3 and $R_0>1$ are met;

2.

One endemic equilibrium or many endemic equilibriums when $R_0>1$and instances 5–7 are met;

3.

No endemic equilibrium when $R_0<1$ and case 8 shows that all coefficients are positive.

5.2. Global Stability of Local Equilibrium Points

The worldwide stability of endemic equilibrium points can assist us in determining if the disease has always existed. Within the collection of endemic equilibrium points, local equilibrium points are globally asymptotically stable, indicating that the disease has always existed. Therefore, we must minimize the fundamental regeneration number to achieve disease eradication.

Theorem 4.

When the basic regeneration number $R_0>1$when the endemic equilibrium point $P_1=\left(S_1,V_1,E_1,I_1,R_1,B_1\right)$, in the collection$\Gamma$the global asymptotic stability within the set.

Proof. 

Assume that $P=\left(S,V,E,I,R,B\right)$ is an arbitrary positive solution of model (1), consider the Lyapunov function $L$ that

$$L=\left|S-S_1\right|+\left|V-V_1\right|+\left|E-E_1\right|+\left|I-I_1\right|+\left|R-R_1\right|+\frac{\mu }{2}\left|B-B_1\right|.$$

(60)

Obviously $L\left(P_1\right)=0$ when $P_1\ne P$, the $L\left(P_1\right)\ne 0$.

The solution estimate along model (1) $L$ the upper right inverse of:

$$\begin{array}{l}{D}^{+}L=\mathrm{sgn}\left(S-S_1\right)[A+pV-\left(\mu +\omega \right)S-\beta SI-\delta SB-(A+p{V}_1-\left(\mu +\omega \right)S_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\beta S_1{I}_1-\delta S_1{B}_1)]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathrm{sgn}\left(V-V_1\right)[\omega S-\left(p+\mu \right)V-k\left(\beta VI+\delta VB\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(\omega S_1-\left(p+\mu \right)V_1-k\left(\beta V_1{I}_1+\delta V_1{B}_1\right)\right)]+sgn(E\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-E_1)[\beta SI+\delta SB+k\left(\beta VI+\delta VB\right)+h\left(\beta RI+\delta RB\right)-(\mu \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\epsilon )E-\alpha \beta EI-(\beta S_1{I}_1+\delta S_1{B}_1+k\left(\beta V_1{I}_1+\delta V_1{B}_1\right)+h(\beta R{I}_1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\delta R_1{B}_1)-\left(\mu +\epsilon \right)E_1-\alpha \beta E_1{I}_1)\left]+\mathrm{sgn}\left(I-I_1\right)\right[\epsilon E+\alpha \beta EI\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(\gamma +\mu +{\mu }_1\right)I-\left(\epsilon E_1+\alpha \beta E_{1}I-\left(\gamma +\mu +{\mu }_1\right)I_1\right) ]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathrm{sgn}\left(R-R_1\right)[\gamma I-\mu R-h\left(\beta RI+\delta RB\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(\gamma I_1-\mu R_1-h\left(\beta R_1{I}_1+\delta R_1{B}_1\right)\right)]+\frac{\mu }{2m} \mathrm{sgn}\left(B-B_1\right)[mI\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\tau B-\left(m{I}_1-\tau B_1\right)].\end{array}$$

(61)

In the above equation, there are random combinations of $2^6$ cases where $S$ with $S_1$, $V$ with $V_1$,$E$ with $E_1$,$I$ with $I_1$,$R$ with $R_1$and $B$ with $B_1$. Now only for$S>S_1$, $V>V_1$, $E>E_1,I>I_1,R>R_1$ and $B>B_1$, the scenarios are discussed and analyzed by taking

$$q=min\left\{\frac{\mu }{2},\frac{\mu \tau }{2m}\right\}.$$

(62)

$$\begin{gathered} D^{+}L \le -\mu \left|S-S_1\right|-\mu \left|V-V_1\right|-\frac{\mu }{2}\left|E-E_1\right| \\ -\frac{\mu }{2}\left|I-I_1\right|-\frac{\mu }{2}\left|R-R_1\right|-\frac{\mu \tau }{2m}\left|B-B_1\right|, \end{gathered}$$

(63)

$$D^{+}L\le -qL.$$

(64)

Integrating both sides of the above equation yields

$$L\left(t\right)+q{{\displaystyle \int }}_{t_0}^{t}L\left(s\right)ds \le L\left(t_0\right) \le +\infty .$$

(65)

From the feasible domain $\mathsf{\Gamma }$ we know that $S, V,E,I,R,B$ are bounded, then their derivatives are also bounded, so $L$ is uniformly continuous. From Barbalat’s Lemma, we know that $\underset{t\to +\infty }{\lim }L\left(t\right)=0$, then the endemic equilibrium point $P_1=\left(S_1,V_1,E_1,I_1,R_1,B_1\right)$ in the set $\mathsf{\Gamma }$ is globally asymptotically stable. □

6. Numerical Simulation and Discussion

The vast majority of the parameter values taken from the influential literature on TB transmission models and provided in this research are biologically plausible [5,7,24,26,38]. Numerical simulations were carried out using MATLAB. By calculating the fundamental regeneration number $R_0$utilizing the group of parameter values displayed in

Table 2. The range of values of each parameter in the model (1).

Description	Parameters	Value	Unit	B
Recruitment of person through birth or immigration	$A$	1	year−1	-
Natural death rate	$\mu$	0.027	year−1	-
The virus lapse rate in the environment	$\tau$	0.6	days−1	-
Rate of virus release into the environment by infected individuals	$m$	0.45	-	-
Transmission rates between people and the environment	$\delta$	0–1, 0.36, 0.1	year−1	-
Transmission rate	$\mathsf{\beta }$	0–1	-	[3,8,11]
Exogenous reinfection	$\alpha$	0.3	-	[6]
Reinfection among the treated individuals	$h$	0–1, 0.26	year−1	[28]
Probability of endogenous morbidity	$\mathsf{\epsilon }$	0.002	year−1	[5,18]
Recovery rate	$\mathsf{\gamma }$	0.78	year−1	[12]
Vaccine viability	$k$	0.63	-	-
The rate of vaccine waning	$p$	0.23	-	-
The rate of vaccination of those who are vulnerable	$\omega$	0–1, 0.79	year−1	-

, we obtained $R_0=0.3793<1$. The initial model (1) further confirms that when the disease-free equilibrium point is less than 1, the equilibrium point is locally asymptotically stable because the susceptible and immunized newborns tend to stabilize at the disease-free equilibrium point, and the infected individuals vanish (See Figure 2).

The parameter values proposed in this paper are feasible within the range of changing the magnitude of the considered parameters when $R_0=7.6708>1$. According to model (1), the susceptible and the immunized human beings vanish while the infected, latent, and cured everybody tends to have stable values. This indicates that if $R_0=7.6708>1$, the prevalence of TB is still rising, and it has spawned endemic infections. Which validates that if $R_0>1$, elimination of the disease is not possible to achieve (See Figure 3).

We consider the parameters in the model (1) that affect the number of infected $\beta ,\delta ,\omega ,\gamma$. Figure 4 illustrates the effect of changing the direct contact rate’s size on the transmission rate of tuberculosis (TB) on tuberculosis. According to the law of infectious disease transmission, the fewer infected individuals there are, the lower the direct contact rate, the sooner TB will be eradicated. Considering the impact of environmental transmission on TB, it can be seen from Figure 5 that as the indirect contact rate of environmental transmission increases, the number of infected persons increases with it. However, the number of infected persons remains constant over time, which confirms that the endemic balance is globally asymptotically stable, indicating that the indirect contact rate of environmental transmission plays a crucial role in TB. As shown in Figure 6: when the vaccination rate increases from 0 to 1, the number of infected persons also decreases as the vaccination rate increases, indicating that vaccination is effective in preventing TB infection. At the same time, when the vaccination rate reaches 100%, the number of TB infections is not completely eliminated, indicating that the eradication of TB requires not only vaccination but also consideration of the influence of various external factors. As the cure rate rises, the number of infected persons rises as well (see Figure 7), and after a time of transmission, the infected person count stabilizes, resulting in endemic diseases. Therefore, a decline in infection rates can occasionally be accompanied by a rise in cure rates. Due to the risk of repeated infection, it is important to take responsibility for both the influence of linked factors and the likelihood of a cure when trying to eradicate the disease.

Finally, the 3D diagram is used to illustrate better the parameters affecting the total number of regenerations. Figure 8 considers the direct transmission rate and environmental transmission coefficient on the influence of basic; as the direct transmission rate increases, the basic regeneration number rises, increasing the environmental transmission coefficient, which in turn increases the basic regeneration number; this confirms that the direct transmission rate and environmental transmission coefficient is an important factor. The number of basic regenerations increases when the influence of vaccination rate and vaccine loss efficiency on the number of basic regenerations is taken into account (see Figure 9); as the vaccination rate increases, the number of basic regeneration rapidly declines. It is strongly correlated with the number of immunizations, which is extremely harmful to the eradication of tuberculosis. The number of basic regenerations can be effectively reduced and, to some extent, eliminated by increasing vaccination rates for the vaccine. In conclusion, taking into account environmental transmission and immunization is crucial for the prevention of TB transmission.

7. Summary

Although numerous research has highlighted the crucial role that environmental transmission plays in tuberculosis, so few have used this information to help mitigate its spread. In a bid to assess the effect of environmental transmission on the dynamics of recurrent tuberculosis infection transmission in the setting of vaccination, a deterministic numerical model was built. The next-generation matrix approach is used in this work to calculate the model’s actual regeneration number. The basic regeneration number impacts how quickly the trajectory of the whole model tends to converge. The six compartments that constitute the upper range have an ultimate asymptotically stable disease-free equilibrium point. In this study, the impact of vaccination and environmental transmission on the eradication of TB is also examined. As can be observed, manipulating the parameters of environmental transmission and vaccination has substantial consequences for the control of tuberculosis (TB). The environmental transmission would increase the difficulty of eliminating recurrent TB and have a negative epidemiological impact on the population for the SVEIRB model under study, resulting in a decline in the value of the environmental transmission parameters and a positive trend in disease eradication. We also noticed that vaccination, in some ways, is a hindrance to the eradication of tuberculosis because of the recurrent character of the disease. The principal findings of this survey’s conclusion point to a substantial role of environmental transmission in the fight against recurrent tuberculosis. The research findings indicate that by keeping the total regeneration number below 1, it may be feasible to make some effort toward the eradication of tuberculosis.

In striving to eradicate the source of TB transmission at the source, public policymakers should consider upgrading the living conditions. However, remnant Mycobacterium tuberculosis in the environment must still be effectively and fully eradicated to prevent the spread of tuberculosis (TB) since environmental transmission affects the basic regeneration figure in this paper’s analysis.

Continued studies can modify the model to take into account parameters such as swift and slow processes [16], drug resistance [23], nonlinear transmission rates [10], and cross-infection [21] into the many dynamics that affect the spread of TB infection. To reduce or eliminate TB infection, for example, certain model assumptions can be reduced or alternative indicators substituted. Data collected can also be applied to forecast TB transmission and generate stronger TB prevention [39].