# Exploring the Stochastic Host-Pathogen Tuberculosis Model with Adaptive Immune Response

Authors: S. P. Rajasekar; M. Pitchaimani; Quanxin Zhu; Kaibo Shi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this literature, we probe a stochastic host-pathogen tuberculosis model with the adaptive immune response of four states of epidemiological classification:Mycobacterium tuberculosis, uninfected macrophages, infected macrophages, and immune response CD4+ T cells. This model is pertinent to the latent stage of tuberculosis infection and active tuberculosis-infected individuals. The stochastic host-pathogen tuberculosis model in pathology is constituted based on the environmental influence on the Mycobacterium tuberculosis and macrophage population, elucidated by stochastic perturbations, and it is proportional to each state. We evince the existence and a unique global positive solution of a stochastic tuberculosis model. We attain sufficient conditions for the extinction of the tubercle bacillus. Moreover, we acquire the existence of the stationary distribution of the positive solutions by the Lyapunov function method. Eventually, numerical simulations validate analytical findings and the dynamics of the stochastic TB model.


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## Body

## 1. Introduction

In the annals of history, there have been many threats such as natural calamity, diseases, and wild animals for the existence of humans; among these, diseases seem to be the main cause of death. In the previous centuries, tuberculosis posed a great threat to the survival of human beings. It is believed that it was the cause of death of more than a third population [1]. And block death in Europe is the standing testimony for these facts. Fortunately, due to the development of medical science, tuberculosis is completely curable but still persists.Tuberculosis is traced back to 3400-2400 BC. It is as ancient as that of ancient countries such as Egypt, Greece, Rome, and India. The German Nobel laureate Robert Koch (1843–1910) discovered the tubercle bacillus (March 24, 1882) at the backdrop of one in seven who died in Europe due to mysterious disease.Infection of the lungs bacteria (Mycobacterium tuberculosis (MTB)) leads to tuberculosis (TB). The bacterial infection may also spread into the brain, kidneys, spine, etc., but it is fortunate that tuberculosis is curable and preventable [1]. Tuberculosis is contagious and spreads through the air. Even the inhalation of a few of these germs is potential enough to infect a person. The physical contact, the nearness, and the air around the infected person can also infect. Sharing the same space with the infected person, weak immunity, malnutrition, unemployment, poor lifestyle, poor food habits, lack of shelter, unhygienic environment, drug abuse, imperfect sexual attitude, ill-literacy, and smoking are vulnerable factors for the spread of tuberculosis [2].Infection is mainly occurred by the aerosol route like inhalation, wherein bacilli contained droplet nuclei sets in the alveoli. Here, alveolar macrophages [3] engulf the bacteria and ultimately bring out the infection to dendritic cells and all other cells as epithelial cells. These alveolar macrophages possess multiple microbicidal mechanisms which include phagolysosome fusion and respiratory burst to prevent postinfective mechanisms. For successful sustainability of infection, the tuberculosis bacilli must understand its encounter with the collection of activated macrophages (granuloma) [4, 5] and eventually gain access to the lymphocytes, neutrophils, fibroblasts, eosinophils, or the bloodstream [6, 7]. Macrophages with the potential to sterile and kill the bacteria are segmented apart in the tissue. In the case of adaptive immunity with the accumulation of effector CD4+ and CD8+ T cells setting in lungs, the multiplication of the bacterial population is contained and enumerable patients become asymptomatic and unable to shed bacteria, and they are then considered as vulnerable to latent TB infection (LTBI). Varied mechanisms cause the limited ability with adaptive immune responses to encounter and eradicate MTB, some of which are then categorized and characterized such as weakened histocompatibility complex (MHC) class II-mediated antigen presentation; adaptation of the anti-inflammatory mediator lipoxin A4; containing by regulatory T cells; monitoring bacterial antigen gene expression and hence incapable of reducing antigen-specific CD4+ T cells; and immune to the macrophages activating effector cytokines such as interferon-γ (INF-γ) and tumour necrosis factor-α (TNF-α) [4, 8].The latent stage of the tuberculosis infection is nonchallenged; though the germs are present in the body, they are inactive and harmless. Antibiotics such as isoniazid (INH), rifampin (Rifadin, Rimactane), and rifapentine can also be used to contain the aggravation of the germs at this stage. It is very easy to wipe out the germs from the body at this stage. Indications of active tuberculosis infection are coughing, sudden weight loss, loss of appetite, sweating during nights, fever, and chest pain.The potential roles for adjunctive immunotherapy in multidrug-resistant TB (MDR-TB) and extensively drug-resistant TB (XDR-TB) [9–11] consist in enhancing cure rates and reducing time spent to cultural shifts, reducing the tissue damage concerned with nonproductive postimmune reactions, and elevating overall health status by subdoing systematic impacts of prolonged chronic inflammation. In addition, enumerable powerfully useful cytokines and other immunomodulatory agents already exist in medical use for other conditions.A significant role in the immunity against pathogen agent is played by T-cell differentiation and memory and effector T cells [12–14]. Hence, an effective T-cell response decides if the infection decreases or develops into a medically evident disease. Studies prove that CD4+ T cells [13–15] do their part in the protection against MTB as strengthened by the evidence that CD4+ T cells such as T helper 1 (Th1), Th2, Th17, and regulatory T cells (Tregs) and all these subsets support one another to restrict infection. MTB-specific CD4+ Th1 cell response is recognized as possessing a protective role for the power to generate cytokines such as INF-γ and TNF-α that involve the processes of recruitment and activation of innate immune cells as monocytes and granulocytes. In this way, when other antigen-specific T cells such as CD8+ cells [14], natural killer (NK) cells, γδT cells, nonclassical (MHC) class I molecule CD1, controlled T cells are able to raise INF-γ during MTB infection and they are unable to equalize the lack of CD4+ T cells.Of late, studies have revealed that multifunctional/polyfunctional T cells (i.e., T cells possessed with multiple effector functions) [13, 14] can produce IFN-γ coupled with IL-2, and subsequently, a subset of cells is able to concurrently produce IFN-γ, TNF-α, IL-2, and IL-17 and was diagnosed in patients with active TB disease compared to LTBI, which rapidly decreased on anti-TB treatment. And also, the polyfunctional T cells were recognized for their potentiality to proliferate and to secrete very many cytokines, and these cells were discovered to act a protective role in antiviral immunity in chronic infections (when the antigen load is low).Memory T cells are a subset of infection-fighting T cells (T lymphocytes) that are highly proliferative and possess a strong immune response. Based on the role of L-selectin and CC-chemokine receptor 7, memory T cells can be segmented intoCD62LhiCCR7+ and CD62LloCCR7− [12–14]. In the case of in vitro stimulation, humanCD62LhiCCR7+ memory CD4+T cells caused the production of IL-2 but hardly produce interferon-γ, IL-4, or IL-5. But CD62LloCCR7− cells rapidly produced these effector cytokines but produced little IL-2. CCR7+ cells named as central memory (TCM) cells [13, 14] are potential enough to produce numerous IL-2 but low lying of effector cytokines and home to secondary lymphoid tissues. But CCR7− cells termed as effector memory (TEM) cells [14] are capable of producing high order of cytokines and home to peripheral tissues. Subsequently, there are various other subpopulations of memory T cells such as memory T stem cells (TSCM, multiple stem-like properties), T naive phenotype (TN), and transitional memory T cells (TTM) [13, 14], very many of which were secreted in the peripheral blood of healthy individuals.Regulatory T cells (Treg) [16] are inevitable for monitoring peripheral tolerance, preventing self-generating immunity, and reducing chronic inflammatory disease. In fact, Treg have varied mechanisms within their powers, and one such mechanism is the suppression by inhibitory cytokines which include IL-10, transforming (TGF-β), and latest identified IL-35 which are prime mediators of Treg− cell function. CD4+CD25+FOXP3+ circulating Treg play a primary role in restricting immune responses by downregulating CD4+ or CD8+ cell functions. Hectic suppression by Treg is important in regulating immune responses against MTB.In spite of the fact that the role of CD8 T cells [13] in TB is less clear than that of CD4 T cells, they are considered in general to impart optimal immunity and protection. CD8 T cells contain enumerable antimicrobial effector mechanisms that are less vital or cease to exist in CD4 Th1 and Th17 T cells. CD8 T cells emit cytokines or cytotoxic molecules, which cause apoptosis of target cells. Th1 CD4 T cells propel effector functions in macrophages that contain intracellular MTB, and their role has been monitored with protection. Besides, many studies have depicted that Th17 cells, which can produce IL-17, are implicated in immune protection against MTB, mainly due to the effect of this cytokine in incurring and vibrating neutrophils. Th 17 cells have been implicated in the protection against TB at initial levels, for their power to employ monocytes and Th1 lymphocytes to the space of granuloma formation [5]. Contrarily, very many studies have manifested that unrestricted Th17 stimulation confirms an exaggerated inflammation catalyzed by neutrophils and inflammatory monocytes that rush to the place of disease during the damage of tissue.MTB-specific CD4+ T-cell protective response is simply due to Th1 cells and is negotiated by IFN-γ and TNF-α that employ monocytes and granulocytes and enhance their antimicrobial attitudes. Subsequently, higher part of antigen-specific effector memory cells and diminished frequency of central memory CD4+ T cells have been discovered in patients with active TB in comparison with distribution witness in LTBI individuals. All the information converges on the increase of this complicated disease with the aim of improving diagnosis, prognosis, drug therapy, and vaccination.Zhang [17] acclaimed the following host tuberculosis model:(1)dBdt=δB1−BK+Miη1b+η2γTc+T−MuBζ+η3β,dMudt=sM−μMMu−βMuB,dMidt=βMuB−bMi−γMiTc+T,dTdt=sT+cMMiT1+emT+cBBT1+eBT−μTT.The biological meaning of all positive parameters and symbols in the deterministic TB model (1) are listed in Table 1. According to the theory in [17], the deterministic TB model (1) has the following properties:Table 1
Biological meaning of variables and parameters of the deterministic model (1).
SymbolInterpretationVariablesBtMycobacterium tuberculosis (MTB) population at time tMutPopulation of uninfected macrophages at timetMitPopulation of infected macrophages at timetTtPopulation of immune response CD4+ T cells at timetParametersδBacterium growth rateKCarrying capacityη1Death rate of infected macrophagesη2Extracellular bacteriaB rate by burstη3Engulfment rate of uninfected macrophagescHalf saturation rate in uninfected macrophagesζKilling rate of extracellular bacteriasMInflux rate of uninfected macrophagesμMDeath rates of uninfected macrophagesβInfection rates of uninfected macrophagesbKilling rate of infected macrophagesγCell-mediated adaptive immune response ratesTNatural recruitment rate of activated CD4+ T cellscBMaximum killing rates of extracellular bacteria macrophagescMMaximum killing rates of infected macrophageseBSaturating factor of extracellular bacteria macrophageseMSaturating factor of infected macrophagesμTDeath rate of activated CD4+ T cellsThe positive invariant set Γ is given by(2)Γ=B,Mu,Mi,T∈ℝ+4:Bt<ℬ,Mu+Mit≤sMminμM,b,Tt<1μTsT+cMeMℳ+cBeBℬ,whereℳ≔sMminμM,b.The disease-free equilibrium is given byE0=Mu0,T0,B0,Mi0=sM/μM,sT/μT,0,0.Ifℛ0≤1, then the disease-free equilibrium is globally asymptotically stable, where ℛ0=δμM/2η3β+ζsM+1/24βcμT+sTη1b+η2γsT/μTc+sT/η3β+ζbcμT+bsT+γsT+δ2μM2/η3β+ζ2sM2.The tuberculosis model (1) has a unique positive infected equilibrium E1 if and only if ℛ0>1 and it is globally asymptotically stable when it exists. The positive infected equilibrium E1=Mu1,T1,B1,Mi1, where Mu1=sM/βB1+μM,Mi1=βsMB1T1+c/βB1+μMbc+bT1+γT1, and(3)T1=−cβB1+μMδ+η1−η3β−ζsMK−δB1βB1+μMbγ+bβB1+μMδ+η1−η3b+γη2−η3β−ζγ+bsMK−δB1γ+bβB1+μM,andB1 is determined by(4)fB1=dTdt|Mu1,T1,Mi1=cBT1eMT1+1B1+−eMμTT12+cMMi1+eMsT−μTT1+sTeBT1+1=0.In reality, the environmental noise perverts through and medal with the population dynamics, ecology, environmental sciences, and mathematical biology. Stochasticity has its impact upon various biological [18–25] and other models [26–30]. Many researchers probed stochastic tuberculosis models in which the total host population can be divided into three, four, and five epidemiological classes, respectively, such as susceptible S, exposed E, and infected I [31]; susceptible S, latent L, infectious I, and treated T [32]; susceptible S, exposed E, infected I, and recovered R [33]; and susceptible S, vaccinated V, infected with TB in latent stage L, infected with TB in active stage I, and treated individuals infected with TB T [34, 35]. On the other hand, Khalid Hattaf et al. [36, 37] investigated specific functional response and temporary immunity and delayed viral infection models with adaptive immune response like cytotoxic T lymphocyte (CTL) cells which is in MHC class I molecules. In all the above models such as SEI,SLIT,SEIR,SVLIT, the researchers probed on host population, temporary immunity, and CTL cells. However, the authors have not examined the functions of macrophages and immune response CD4+ T cells which are in MHC class II molecules so far.The macrophages inundate the MTB and quarantine them in a cellular compartment where the bacteria are killed or cannot thrive. If the number of MTB is higher than that of macrophages, it will result in the death of the macrophages in the granuloma. Then the granuloma’s core liquidizes and is transmitted through airways to other people. During infection, the MTB cells set sporadically in the alveoli of an individual. The immunologic effects of both infected and uninfected macrophages vary randomly due to environmental sources of MTB. The adaptive immune responses in an individual start with very few number of cells and later vary greatly among individual immune systems. This variation is random, and on the other hand, individuals with smoking habit, weak immunity, malnutrition, etc., possess low immune response and have to be treated with more dosages of antigens to increase the immunity contingently.This study is challenging and useful in the sense that TB in its various dimensions and nature can be disclosed. Based on the above factors, we put forth the stochastic host-pathogen TB model with adaptive immune response. The stochastic host-pathogen TB model on the basis of the influence of the environment on theMycobacterium tuberculosis and macrophage population was manifested by stochastic perturbations and it is proportional to each state [38, 39].We establish the following stochastic host-pathogen TB model:(5)dBt=δB1−BK+Miη1b+η2γTc+T−MuBζ+η3βdt+σ1BdW1t,dMut=sM−μMMu−βMuBdt+σ2MudW2t,dMit=βMuB−bMi−γMiTc+Tdt+σ3MidW3t,dTt=sT+cMMiT1+emT+cBBT1+eBT−μTTdt+σ4TdW4t.LetWt=W1t,W2t,W3t,W4t be a 4-dimensional Wiener process defined on the given probability space. The components of Wt are supposed to be mutually independent. In the stochastic model (5), the nonnegative constants σ1,σ2,σ3, and σ4 denote the intensities of the environmental white noise.The intention of this literature is structured as follows. In Section2, the existence of the global and positive solutions to the stochastic host-pathogen TB model (5) is examined. In Section 3, sufficient conditions for the extinction of the tubercle bacillus are probed. In Section 4, we probe that there is a unique ergodic stationary distribution of the positive solutions of the stochastic TB model (5) under some conditions. In Section 5, we demonstrate some numerical simulations that validate our analytical findings. Eventually, Section 6 provides conclusions and future directions.


## 2. Global Positive Solutions of the Stochastic TB Model

SinceB,Mu,Mi, and T in the stochastic TB system (5) denotes the population of MTB, uninfected macrophages, infected macrophages, and immune response CD4+ T cells, respectively, they should be nonnegative. For a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are eventually required to satisfy the linear growth condition and local Lipschitz condition. Yet, the coefficients of equation (5) do not satisfy the linear growth condition; nevertheless, they are locally Lipschitz continuous, and thus, the solution of equation (5) may explode at a finite time. Hence, for further studies, we should set some preconditions under which stochastic system (5) has unique global positive solutions.Theorem 1.
For any given initial valueB0,Mu0,Mi0,T0∈ℝ+4, there exists unique positive solutions Bt,Mut,Mit,Tt of the stochastic TB model (5) on t≥0 and the solution will remain in ℝ+4 with probability one.Proof.
It is clear that the stochastic TB system (5) satisfies the local Lipschitz condition for any given initial value B0,Mu0,Mi0,T0∈ℝ+4. Then, there exist unique maximum local solutions Bt,Mut,Mit,Tt on t∈0,τ∗, where τ∗ is the explosion time [43]. To exhibit that the solution is universal, i.e., inevitable to affirm that τ∗=∞ a.s. Basically, we exhibit that Bt,Mut,Mit, and Tt do not explode to infinity in a finite time. Let κ0>0 be adequately large B0,Mu0,Mi0,T0∈1/κ0,κ0. For each integer κ≥κ0, define the stopping time [40]:(6)τκ=inft∈0,τ∗:minBt,Mut,Mit,Tt≤1κormaxBt,Mut,Mit,Tt≥κ,with the typical format infϕ=∞, where ϕ reflects the empty set. Evidently, τκ is increasing as κ⟶∞. We have τ∞=limκ⟶∞τκ; therefore, τ∞≤τ∗ a.s. If we evince that τ∞=∞ a.s., then τ∗=∞ and Bt,Mut,Mit,Tt∈ℝ+4 a.s. for all t≥0. Presume that τ∞<∞, then there is a pair of constants T˜>0 and ε∈0,1 such that(7)ℙτ∞≤T˜>ε.
Thus, there exists an integerκ1≥κ0 such that(8)ℙτκ≤T˜≥ε,for allκ≥κ1.
Define aC2-function V:ℝ+4⟶ℝ+ by(9)VB,Mu,Mi,T=φ2B−φ1−φ1lnBφ1+Mu−1−lnMu+Mi−1−lnMi+φ2T−1−lnT,where φ1>0 and φ2>0 will be determined later. The nonnegativity of VB,Mu,Mi,T function occurs from u−1−lnu≥0,∀u>0.
By Itô’s formula toV, we obtain(10)dV=ℒVdt+φ2σ1B−φ1dW1+σ2Mu−1dW2+σ3Mi−1dW3+φ2σ4T−1dW4,where ℒV:ℝ+4⟶ℝ+ is defined as(11)ℒV=φ21−φ1BδB1−BK+Miη1b+η2γTc+T−MuBζ+η3β+φ1φ2σ122+1−1MusM−μMMu−βMuB+σ222+1−1MiβMuB−bMi−γMiTc+T+σ322+φ21−1TsT+cMMiT1+emT+cBBT1+eBT−μTT+φ2σ422,ℒV=−φ2δKB2+φ2δφ1+KKB−φ1φ2δ+φ2Miη1b+η2γTc+T−φ2MuBζ+η3β−φ1φ2MiBη1b+η2γTc+T+φ1φ2Muζ+η3β+sM−μMMu−sMMu+μM+βB−bMi−γMiTc+T−1MiβMuB+b+γTc+T+φ2sT+φ2cMMiT1+emT+φ2cBBT1+eBT−φ2μTT−φ2sTT−φ2cMMi1+eMT−φ2cBB1+eBT+φ2μT+φ1φ2σ122+σ222+σ322+φ2σ422,ℒV≤maxB∈ℝ+−φ2δKB2+φ2δφ1+K+Kβ+Kφ2cBKB+φ1φ2ζ+η3β−μMMu+φ2η1b+η2γ+cM−bMi+sM+μM+b+γ+φ2sT+φ2μT+φ1φ2σ122+σ222+σ322+φ2σ422.
Chooseφ2=b/η1b+η2γ+cM such that φ2η1b+η2γ+cM−b=0 and let φ1=μM/φ2ζ+η3β such that φ1φ2ζ+η3β−μM=0, and then we have(12)ℒV≤maxB∈ℝ+−φ2δKB2+φ2δφ1+K+Kβ+Kφ2cBKB+sM+μM+b+γ+φ2sT+φ2μT+φ1φ2σ122+σ222+σ322+φ2σ422≔Λ,where Λ is a positive constant. Thus,(13)dV≤Λdt+φ2σ1B−φ1dW1+σ2Mu−1dW2+σ3Mi−1dW3+φ2σ4T−1dW4.
Integrate both sides of equation (13) from 0 to τκ∧T˜=minτκ,T˜,(14)∫0τκ∧T˜dVBτ,Muτ,Miτ,Tτ≤∫0τκ∧T˜Λdt+∫0τκ∧T˜φ2σ1B−φ1dW1+σ2Mu−1dW2+σ3Mi−1dW3+φ2σ4T−1dW4.
Taking expectations on both sides of (14) yields(15)EVBτκ∧T˜,Muτκ∧T˜,Miτκ∧T˜,Tτκ∧T˜≤VB0,Mu0,Mi0,T0+E∫0τκ∧TΛdτ,EVBτκ∧T˜,Muτκ∧T˜,Miτκ∧T˜,Tτκ∧T˜≤VB0,Mu0,Mi0,T0+ΛT˜.
LetΩκ=τκ≤T˜ for all κ≥κ1, and by (8), then ℙΩκ≥ε. Notice that for every ω∈Ωκ, there exists at least Bτκ,ω or Muτκ,ω or Miτκ,ω or Tτκ,ω which is equal to either κ or 1/κ.
IfBτκ,ω=κ or 1/κ, then(16)Bτκ,ω,Muτκ,ω,Miτκ,ω,Tτκ,ω≥φ2κ−φ1−φ1lnκφ1∧1κ−φ1−φ1ln1φ1κ=φ1φ2κφ1−1−lnκφ1∧1κφ1−1+lnφ1κ.
Similarly, ifMuτκ,ω=κ or 1/κ, then(17)Bτκ,ω,Muτκ,ω,Miτκ,ω,Tτκ,ω≥κ−1−lnκ∧1κ−1+lnκ.
IfMiτκ,ω=κ or 1/κ, then(18)Bτκ,ω,Muτκ,ω,Miτκ,ω,Tτκ,ω≥κ−1−lnκ∧1κ−1+lnκ.
IfTτκ,ω=κ or 1/κ, then(19)Bτκ,ω,Muτκ,ω,Miτκ,ω,Tτκ,ω≥φ2κ−1−lnκ∧1κ−1+lnκ.
Therefore,(20)Bτκ,ω,Muτκ,ω,Miτκ,ω,Tτκ,ω≥φ1φ2κφ1−1−lnκφ1∧1κφ1−1+lnφ1κ∧κ−1−lnκ∧1κ−1+lnκ∧φ2κ−1−lnκ∧1κ−1+lnκ.
Then, we attain(21)VB0,Mu0,Mi0,T0+ΛT˜≥E1ΩκωVBτκ∧T˜,Muτκ∧T˜,Miτκ∧T˜,Tτκ∧T˜,=E1ΩκωVBτκ,ω,Muτκ,ω,Miτκ∧T˜,Tτκ,ω,≥E1Ωκωφ1φ2κφ1−1−lnκφ1∧1κφ1−1+lnφ1κ∧κ−1−lnκ∧1κ−1+lnκ∧φ2κ−1−lnκ∧1κ−1+lnκ,VB0,Mu0,Mi0,T0+ΛT˜≥εφ1φ2κφ1−1−lnκφ1∧1κφ1−1+lnφ1κ∧κ−1−lnκ∧1κ−1+lnκ∧φ2κ−1−lnκ∧1κ−1+lnκ,where 1Ωκ is the indicator function of Ωκω. Letting κ⟶∞, we get ∞>VB0,Mu0,Mi0,T0+ΛT˜=∞. It is a contradiction, and hence, we obtain τ∞=∞ a.s., which completes the proof of Theorem 1.Remark 1.
Considering the regionℝ+4 is positively invariant, the unique solution of the stochastic TB model (5) exists for any given initial value in ℝ+4. Hence, it is sufficient to probe its dynamics of the host-pathogen stochastic TB model in the region ℝ+4.Remark 2.
As stated in the introduction, immunotherapy with LTBI and active TB individuals and adaptive immune responses are known to be quite effective against MTB. However, the prevailing opinion among public health and medical practitioners in the entire world, prior to 1944 [44], was a less effective treatment for TB patients. During the years between 2016 and 2035, the efforts at the global, regional, and national levels to ease and eradicate the burden of TB disease have the colossus objective “ceasing TB epidemic,” within the purview of the UN’s Agenda for Sustainable Development and based on the WHO’s End TB Strategy [1]. Hence, it is fruitful to study the eradication of the MTB among LTBI and active TB individuals. This is done as follows.


## 3. Extinction of the MTB

In this section, the extinction of the MTB is discussed. The death rate of uninfected macrophages and the intensities of the white noise in the stochastic system (5) are the consequences of the random fluctuations. The population dynamics show direct results on the qualitative outcome of the MTB dynamics in the system with regard to the factors that decide MTB eradication during the long course of time. Firstly, we shall present a lemma which will be used in our analysis.Lemma 1.
LetBt,Mut,Mit,Tt be the solution of the stochastic TB system (5) with any positive initial value B0,Mu0,Mi0,T0∈ℝ+4. Then(22)limt⟶∞Btt=0,limt⟶∞Mutt=0,limt⟶∞Mitt=0,limt⟶∞Ttt=0a.s,
Moreover, ifμM>σ12∨σ22∨σ32∨σ42/2, then(23)limt⟶∞∫0tBτdW1τt=0,limt⟶∞∫0tMuτdW2τt=0,limt⟶∞∫0tMiτdW3τt=0,limt⟶∞∫0tTτdW4τt=0.Define a parameter as follows:(24)ℛ0˜=4βsM+μMδ+bN1+γN2μMσ12∧σ32.Theorem 2.
For any given initial valueB0,Mu0,Mi0,T0∈ℝ+4, then the MTB of the stochastic TB system (5) will cease out if μM>σ12∨σ32/2 and ℛ0˜<1, i.e., limt⟶∞Bt=0,limt⟶∞Mit=0 a.s.Proof.
Consider the second and third equations of the stochastic TB system (5). Integrating these equations from 0 to t and dividing both sides by t, we get(25)Mut−Mu0t+Mit−Mi0t=sM−μMMu−bMi−γMiTc+T+σ2t∫0tMuτdW2τ+σ3t∫0tMiτdW3τ.
It follows that(26)Mu≤1μMsM−Mut−Mu0t−Mit−Mi0t+σ2t∫0tMuτdW2τ+σ3t∫0tMiτdW3τ
Consequently,(27)limt⟶∞Mu≤1μMlimt⟶∞sM−Mut−Mu0t−Mit−Mi0t+σ2t∫0tMuτdW2τ+σ3t∫0tMiτdW3τ.
By Lemma1, we have(28)limt⟶∞Mu≤sMμMa.s.
On the other hand, letPt=Bt+Mit.
Applying Itô’s formula, we obtain(29)dlnPt=1B+MiδB1−BK+Miη1b+η2γTc+T−MuBζ+η3β+βMuB−bMi−γMiTc+T−12B+Mi2σ12B2+σ32Mi2dt+σ1BB+MidW1t+σ3MiB+MidW3t,dlnPt≤1B+MiδB+Mibη1−1+MiγTc+Tη2−1+βMuB−12B+Mi2σ12B2+σ32Mi2dt+σ1BB+MidW1t+σ3MiB+MidW3t,≤δ+bN1+γN2+βMu−14σ12∧σ32dt+σ1BB+MidW1t+σ3MiB+MidW3t,where N1=η1−1>0 and N2=η2−1>0.
Integrating both sides of (28) from 0 to t, we attain(30)lnBt+Mit≤lnB0+Mi0+β∫0tMuτdτ+δ+bN1+γN2t−t4σ12∧σ32+∫0tσ1BB+MidW1τ+∫0tσ3MiB+MidW3τ.
Dividing both sides of byt and then taking the limit superior yield(31)limsupt⟶∞lnBt+Mitt≤βsMμM+δ+bN1+γN2−14σ12∧σ32,=14σ12∧σ324βsM+μMδ+bN1+γN2μMσ12∧σ32−1,=14σ12∧σ32ℛ0˜−1<0.
The conclusion is confirmed.Remark 3.
The immunomodulatory agents are compatible to immune responses such as IL-2 and IL-4 to effectively active T helper cells. Immunosuppressive agents such as TNF-α moderate dangerous inflammation and cytokine therapy INF-γ tends to accelerate the mycobacterial activity of effector immune cells [42]. All these facts exist in reality, which wipe out the MTB.Remark 4.
Patients have lack of information, lack of money for treatment, side effects, lack of commitment to a long course, social barriers, irregular treatment and do not take proper drugs and so they have a low level of immunity. These are the factors for the TB persistent and prevalence among population.


## 4. Ergodic Stationary Distribution

When considering epidemic dynamical systems, we are interested in when the disease will persist and prevail in a population. In this section, we present some theories about the stationary distribution (see Has’minskii [43]), and we show that there exists an ergodic stationary distribution, which reveals that the disease will persist. Let Xt be a homogeneous Markov process in ℝ+4, which is described by the following stochastic differential equation:(32)dXt=fXtdt+∑r=1kgrXdWrt.The diffusion matrix is defined as follows:(33)Ax=aijx,aijx=∑r=1kgrixgrjx.Lemma 2 (see [39]).
The Markov processXt has a unique ergodic stationary distribution π. if there exists a bounded domain D⊂ℝ+4 with regular boundary Γ, having the following properties:A1: there is a positive number M such that ∑i,j=14aijxξiξj≥Mξ2,forx∈D,ξ=ξ1,ξ2,ξ3,ξ4∈ℝ+4.A2: there exist a nonnegative C2-function Vx and a positive constant C^ such that ℒV≤−C^ for any x∈ℝ+4∖D. Then(34)ℙlimT⟶∞1T∫0TfXtdt=∫ℝ+4fxπdx=1,for all x∈ℝ+4, where f. is a function integrable with respect to the measure π.Define a parameter(35)ℛ0s=sMβϑ+σ12/2μM+σ22/2b+γ+σ32/2η1b+η2γsTsT+cμT+cσ42/2,whereϑ=ζ+η3βμMsM+ζ+η3βMu1+ζ+η3βμMMu1σ222.Theorem 3.
Assume thatℛ0s>1, then for any initial value B0,Mu0,Mi0,T0∈ℝ+4, the stochastic TB system (5) admits a unique stationary distribution π. and it has the ergodic property.Proof.
The diffusion matrix of the stochastic TB system (5) is given by(36)σ12B20000σ22Mu20000σ32Mi20000σ42T2.
SelectM˜=minB,Mu,Mi,T∈D˜K⊂ℝ+4σ12B2,σ22Mu2,σ32Mi2,σ42T2, and we have(37)∑i,j=14aijB,Mu,Mi,Tξiξj=σ1Bξ1σ2Muξ2σ3Miξ3σ4Tξ4σ1Bξ1σ2Muξ2σ3Miξ3σ4Tξ4,=σ1B2ξ12+σ2Mu2ξ22+σ3Mi2ξ32+σ4T2ξ42,≥M˜ξ2for anyB,Mu,Mi,T∈D˜k,ξ=ξ1,ξ2,ξ3,ξ4∈ℝ4,where D˜k=1/k,k×1/k,k×1/k,k×1/k,k and k>1 is an adequately larger integer, and then condition A1 in Lemma 2 holds.
Next, we tend to prove the conditionA2. Define(38)V1B,Mu,Mi=−lnB−k1lnMu−k2lnMi,where k1 and k2 are positive constants to be resolved later. By applying Itô’s formula to V1, we get(39)ℒV1=−1BδB1−BK+Miη1b+η2γTc+T−MuBζ+η3β+σ122−k1MusM−μMMu−βMuB+k1σ222−k2MiβMuB−bMi−γMiTc+T+k2σ322,=δBK−Miη1bB−Miη2γBTc+T+Muζ+η3β−δ+σ122−k1sMMu+k1μM+k1Bβ+k1σ222−k2βMuBMi+k2b+k2γTc+T+k2σ322,=−Miη1bB−k1sMMu−k2βMuBMi+δBK−Miη2γBTc+T+Muζ+η3β−δ+σ122+k1Bβ+k1μM+σ222+k2b+k2γTc+T+k2σ322,ℒV1≤−3η1bsMβk1k23+δBK−Miη2γBTc+T+Muζ+η3β+σ122+k1Bβ+k1μM+σ222+k2b+γ+σ322−δ.
Choose(40)k1=η1bsMβμM+σ22/22b+γ+σ32/2,k2=η1bsMβμM+σ22/2b+γ+σ32/22.
Then(41)ℒV1≤−η1bsMβμM+σ22/2b+γ+σ32/2+δBK−Miη2γBTc+T+Muζ+η3β+σ122+k1Bβ−δ.
Define(42)V2B,Mu,Mi,T=V1B,Mu,Mi−k3clnT−k4lnMu−k5lnMi,where k3,k4, and k5 are positive constants to be determined later. By calculating,(43)ℒV2≤−Miη2γBTc+T−η1bsMβμM+σ22/2b+γ+σ32/2+δBK+Muζ+η3β+σ122+k1Bβ−δ−k3cTsT+cMMiT1+eMT+cBBT1+eBT−μTT+k3cσ422−k4MusM−μMMu−βMuB+k4σ222−k5MiβMuB−bMi−γMiTc+T+k5σ322,≤−Miη2γBTc+T−k3c+TTsT−k4sMMu−k5βMuBMi−η1bsMβμM+σ22/2b+γ+σ32/2+δBK+Muζ+η3β+σ122+k1Bβ−δ+k3sT−k3ccMMi11+eMT−k3ccBB11+eBT+k3cμT+k3cσ422+k4μM+k4Bβ+k4σ222+k5b+k5γTc+T+k5σ322,ℒV2≤−4η2γsTsMβk3k4k54−η1bsMβμM+σ22/2b+γ+σ32/2+δBK+Muζ+η3β+σ122+k1Bβ+k4Bβ+k3sT+cμT+cσ422+k4μM+σ222+k5b+γ+σ322−δ.
Choose(44)k3=η2γsTsMβsT+cμT+cσ42/22μM+σ22/2b+γ+σ32/2,k4=η2γsTsMβsT+cμT+cσ42/2μM+σ22/22b+γ+σ32/2,k5=η2γsTsMβsT+cμT+cσ42/2μM+σ22/2b+γ+σ32/22.
Hence,(45)ℒV2≤−η2γsTsMβsT+cμT+cσ42/2μM+σ22/2b+γ+σ32/2−η1bsMβμM+σ22/2b+γ+σ32/2+Muζ+η3β+σ122+δK+k1β+k4βB−δ,ℒV2≤−sMβμM+σ22/2b+γ+σ32/2η1b+η2γsTsT+cμT+cσ42/2+Muζ+η3β+σ122+δK+k1β+k4βB−δ.
Define(46)U1B,Mu,Mi,T=V2B,Mu,Mi,T+ζ+η3βμMMu−Mu1−Mu1lnMuMu1,ℒU1≤−sMβμM+σ22/2b+γ+σ32/2η1b+η2γsTsT+cμT+cσ42/2+Muζ+η3β+σ122+δK+k1β+k4βB−δ+ζ+η3βμM1−Mu1MusM−μMMu−βMuB+ζ+η3βμMMu1σ222,≤−sMβμM+σ22/2b+γ+σ32/2η1b+η2γsTsT+cμT+cσ42/2+ζ+η3βμMsM+σ122+ζ+η3βMu1+ζ+η3βμMMu1σ222+δK+k1β+k4β++ζ+η3βμMβMu1B,ℒU1≤−sMβμM+σ22/2b+γ+σ32/2η1b+η2γsTsT+cμT+cσ42/2+ϑ+σ122+ϕB=−ϑ+σ122ℛ0s−1+ϕB=−λ+ϕB,where(47)λ≔−ϑ+σ122ℛ0s−1,ℛ0s≔sMβϑ+σ12/2μM+σ22/2b+γ+σ32/2η1b+η2γsTsT+cμT+cσ42/2,ϑ≔ζ+η3βμMsM+ζ+η3βMu1+ζ+η3βμMMu1σ222,ϕ≔δK+k1β+k4β++ζ+η3βμMβMu1.
Define aC2-function U:ℝ4⟶ℝ, as follows:(48)UB,Mu,Mi,T=MU1+U2+U3+U4+U5+U6,where U2=−lnB,U3=−lnMu,U4=−lnMi,U5=−lnT,U6=1/θ+1B/η1b+η2γ+Mu+Mi+T/4θ+1, in which θ is a sufficiently small constant satisfying 0<θ<minμM/σ22,b/1+cM/4+σ32,T/σ42;b>1+cM/4, and select a suitable constant M>0 which satisfies the following condition:(49)−Mλ+E≤−2,where(50)E=supB,Mu,Mi,T∈ℝ+4δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422.
Easily we can check that(51)liminfn→∞B,Mu,Mi,T∈ℝ+4\DnUB,Mu,Mi,T=+∞,where Dn=1/n,n×1/n,n×1/n,n×1/n,n. Furthermore, UB,Mu,Mi,T is not only continuous. Hence, UB,Mu,Mi,T has a minimum point B0˜,Mu0˜,Mi0˜,T0˜ in the interior of ℝ+4. Then, we define a C2-function U^B,Mu,Mi,T=UB,Mu,Mi,T−U˜B0˜,Mu0˜,Mi0˜,T0˜.
By applying Itô’s formula, we obtain(52)ℒU2=−δ1−BK−MiBη1b+η2γTc+T+Muζ+η3β+σ122≤−δ+δBK+Muζ+η3β+σ122,ℒU3=−sMMu+μM+βB+σ222,ℒU4=−βMuBMi+b+γTc+T+σ322≤−βMuBMi+b+γ+σ322,ℒU5=−sTT−cMMi11+eMT−cBB11+eBT+μT+σ422≤−sTT+μT+σ422,ℒU6=Bη1b+η2γ+Mu+Mi+T4θ1η1b+η2γδB1−BK+Miη1b+η2γTc+T−MuBς+η3β+sM−μMMu−bMi−γMiTc+T+sT4+cMMiT41+eMT+cBBT41+eBT−μTT4+θ2Bη1b+η2γMu+Mi+T4θ−1×σ1Bη1b+η2γ2+σ22Mu2+σ32Mi2+σ42T42,≤Bη1b+η2γ+Mu+Mi+T4θsM+sT4+δη1b+η2γ+cBB4B−δB2η1b+η2γK−μMMu−b−1+cM4Mi−μTT4+θ2Bη1b+η2γMu+Mi+T4θ−1×σ12Bη1b+η2γ2+σ22Mu2+σ32Mi2+σ42T42,≤sM+sT4Bη1b+η2γ+Mu+Mi+T4θ+δη1b+η2γ+cBB4×Bη1b+η2γ+Mu+Mi+T4θB−δBθ+2η1b+η2γK−μMMuθ+1−b−1+cM4Miθ+1−μTT4Tθ+1+θ2σ12Bη1b+η2γθ+1+σ22Muθ+1+σ32Miθ+1+σ42T4θ+1,ℒU6≤−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F,where(53)F=supB,Mu,Mi,T∈ℝ+4−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+sM+sT4Bη1b+η2γ+Mu+Mi+T4θ+δη1b+η2γ+cB4×Bη1b+η2γ+Mu+Mi+T4θB+θ2σ12Bη1b+η2γθ+1+σ22Muθ+1+σ32Miθ+1+σ42T4θ+1.
It follows that(54)ℒU^≤−Mλ+MϕB+δBK+Muζ+η3β−sMMu+βB−βMuBMi−sTT−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422.
Now we are in the portion to construct a compact subsetDϵ such that condition A2 in Lemma 2 holds. Define the following bounded closed set(55)Dϵ=B,Mu,Mi,T∈ℝ+4:ϵ≤B≤1ϵ,ϵ≤Mu≤1ϵ,ϵ3≤Mi≤1ϵ3,ϵ≤T≤1ϵ,where 0<ϵ<1 is a sufficiently small constant. In the set ℝ+4∖Dϵ, we can choose ϵ sufficiently small such that the following conditions hold:(56)−Mλ+Mϕϵ+E≤−1,(57)−sMϵ+G≤−1,(58)−βϵ+G≤−1,(59)−sTϵ+G≤−1,(60)−δ4η1b+η2γK1ϵθ+2+H≤−1,(61)−μM41ϵθ+1+J≤−1,(62)−b−1+cM/441ϵ3θ+3+N≤−1,(63)−μT161ϵθ+1+O≤−1,where E,G,H,J,N, and O are positive constants which can be found from the equations (50), (66), (70), (72), (74), and (76). Conveniently, we can divide ℝ+4∖Dϵ into the following eight domains:D1=B,Mu,Mi,T∈ℝ+4:0<B<ϵ,D2=B,Mu,Mi,T∈ℝ+4:0<Mu<ϵ,D3=B,Mu,Mi,T∈ℝ+4:0<Mi<ϵ3,B≥ϵ,Mu≥ϵ,D4=B,Mu,Mi,T∈ℝ+4:0<T<ϵ,D5=B,Mu,Mi,T∈ℝ+4:B>1/ϵ,D6=B,Mu,Mi,T∈ℝ+4:Mu>1/ϵ,D7=B,Mu,Mi,T∈ℝ+4:Mi>1/ϵ3,D8=B,Mu,Mi,T∈ℝ+4:T>1/ϵ.
Clearly,Dϵc=D1∪D2∪D3∪D4∪D5∪D6∪D7∪D8. Next, we prove that ℒU^≤−1 on Dϵc, which is equivalent to proving it on the above eight domains, respectively.Case 1.
IfB,Mu,Mi,T∈D1, we have(64)ℒU^≤−Mλ+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,≤−Mλ+MϕB+E≤−Mλ+Mϕϵ+E.
According to (56) and (64), we get that ℒU^≤−1 for any B,Mu,Mi,T∈D1.Case 2.
IfB,Mu,Mi,T∈D2, we attain(65)ℒU^≤−sMMu+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,≤−sMMu+G≤−sMϵ+G,where(66)G=supB,Mu,Mi,T∈ℝ+4MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422.
In view of (57), we obtain that ℒU^≤−1 for any B,Mu,Mi,T∈D2.Case 3.
IfB,Mu,Mi,T∈D3, we obtain(67)ℒU^≤−βMuBMi+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,≤−βMuBMi+G≤−βϵ+G.
By virtue of (58), we can conclude that ℒU^≤−1 for any B,Mu,Mi,T∈D3.Case 4.
IfB,Mu,Mi,T∈D4, we get(68)ℒU^≤−sTT+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,≤−sTT+G≤−sTϵ+G.
In view of (59), we can deduce that ℒU^≤−1 for any B,Mu,Mi,T∈D4.Case 5.
IfB,Mu,Mi,T∈D5, then one can see that(69)ℒU^≤−δBθ+24η1b+η2γK+MϕB+δBK+Muζ+η3β+βB−δBθ+24η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,≤−δBθ+24η1b+η2γK+H≤−δ4η1b+η2γK1ϵθ+2+H,where(70)H=supB,Mu,Mi,T∈ℝ+4MϕB+δBK+Muζ+η3β+βB−δBθ+24η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422.
By condition (60), we get that ℒU^≤−1 for any B,Mu,Mi,T∈D5.Case 6.
IfB,Mu,Mi,T∈D6, we get(71)ℒU^≤−μM4Muθ+1+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM4Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,ℒU^≤−μM4Muθ+1+J≤−μM41ϵθ+1+J,where(72)J=supB,Mu,Mi,T∈ℝ+4MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM4Muθ+1−b−1+cM/42Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422..
According to condition (62), we attain that ℒU^≤−1 for any B,Mu,Mi,T∈D6.Case 7.
IfB,Mu,Mi,T∈D7, we have(73)ℒU^≤−b−1+cM/44Miθ+1+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/44Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,≤−b−1+cM/44Miθ+1+N≤−b−1+cM/441ϵ3θ+3+N.where(74)N=supB,Mu,Mi,T∈ℝ+4MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/44Miθ+1−μT8Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422.
It leads toℒU^≤−1 for any B,Mu,Mi,T∈D7 if condition (62) is satisfied.Case 8.
IfB,Mu,Mi,T∈D8, we attain(75)ℒU^≤−μT16Tθ+1+MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT16Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422,ℒU^≤−μT16Tθ+1+O≤−μT161ϵθ+1+O,where(76)O=supB,Mu,Mi,T∈ℝ+4MϕB+δBK+Muζ+η3β+βB−δBθ+22η1b+η2γK−μM2Muθ+1−b−1+cM/42Miθ+1−μT16Tθ+1+F−δ+μM+b+γ+μT+σ122+σ222+σ322+σ422.
It follows from (63) that ℒU^≤−1 for any B,Mu,Mi,T∈D8.
Obviously, from (64), (65), (67)–(69), (71), (73), and (75), we attain that for a sufficiently small ϵ,ℒVB,Mu,Mi,T≤−1,∀B,Mu,Mi,T∈ℝ+4∖Dϵ. Hence, condition A2 of Lemma 2 holds. It follows from Lemma 2 that the stochastic system (5) is ergodic and has a unique stationary distribution π.. This completes the proof.Remark 5.
Notice that the expression ofℛ0s converges with the basic reproduction number ℛ0 for the deterministic model (1) if the environmental white noise is not taken into account. For example, we choose the real-life parameter values in Table 2, except δ=7.2×10−4,b=1.25,sM=1×109,μM=7,β=1.5×10−9,γ=2×10−2,sT=2×10−4,cM=1×10−4,eM=1×10−6,cB=5×10−6,eB=1×10−2,μT=50 with σ1=0=σ2=σ3=σ4. We have ℛ0=1.3507 and ℛ0s=1.3500.
On the other hand, we select the above parameters except the environmental white noisesσ1=0.3,σ2=0.2,σ3=0.4,andσ4=0.3. We have ℛ0s=1.2589, and the conditions of Theorem 2 are satisfied. Figure 1 represents the stochastic TB system (5) fluctuates around the endemic equilibrium E1=B1,Mu1,Mi1,T1 of deterministic TB system (1), where B1=4.5202×1010,Mu1=1.3368×107,Mi1=7.1372×108,andT1=1.4269×109.Table 2
List of parameters.
ParametersInterpretationValuesReferenceδBacterium growth rate5×10−4[17]KCarrying capacity1×108[17]η1Death rate of an infected macrophages50[17]η2Extracellular bacteria (B) rate by burst20[17]η3Engulfment rate of uninfected macrophages25[17]cHalf saturation rate in uninfected macrophages150[17]ζKilling rate of extracellular bacteria1.25×10−8[17]sMInflux rate of uninfected macrophages5000[17]μMDeath rates of uninfected macrophages25AssumedβInfection rates of uninfected macrophages2×10−7[17]bKilling rate of infected macrophages1.1×10−4AssumedγCell-mediated adaptive immune response rate5×10−8[17]sTNatural recruitment rate of activated CD4+ T cells6.6[17]cBMaximum killing rates of extracellular bacteria macrophages5×10−3[17]cMMaximum killing rates of infected macrophages1×10−3[17]eBSaturating factor of extracellular bacteria macrophages1×10−4[17]eMSaturating factor of infected macrophages1×10−4[17]μTDeath rate of activated CD4+ T cells0.33[17]Figure 1
The first segment of the figure represents the stochastic TB system (5) which fluctuates around the endemic equilibrium of deterministic TB system (1) if the environmental white noises are sufficiently small. The second segment of the figure reflects the probability distribution of the unique ergodic stationary distribution of the stochastic TB system (5).
(a)(b)(c)(d)(e)(f)(g)(h)(78)ℛ0˜=4βsM+μMδ+bN1+γN2μMσ12∧σ32≈0.59312.2500≈0.2636<1,


## 5. Numerical Simulations

In this section, we probe the extinction of the MTB, ergodic stationary distribution of the stochastic system (5), and the effect of varied environmental noise on model dynamic behavior based on the real-life parameters of theoretical outcomes by numerical simulations. Here, we utilize Milstein’s higher-order method [44] and the host-pathogen stochastic TB model can be written as the subsequent discretization equations:(77)Bk+1=Bk+δBk1−BkK+Mikη1b+η2γTkc+Tk−MukBkζ+η3βΔt+σ1BkΔtϑk+σ122BkΔtϑk2−Δt,Muk+1=Muk+sM−μMMuk−βMukBΔt+σ2MukΔtψk+σ222MukΔtψk2−Δt,Mik+1=Mik+βMukBk−bMik−γMikTkc+TkΔt+σ3MikΔtρk+σ322MikΔtρk2−Δt,Tk+1=Tk+sT+cMMikTk1+emTk+cBBkTk1+eBTk−μTTkΔt+σ4TkΔtνk+σ422TkΔtνk2−Δt,where ϑk,ψk,ρk, and νk,k=1,2,…,n, are the k th perception of four mutually independent Gaussian random variables with N0,1, σi,i=1,2,3,4 are the intensities of environmental noise, and Δt>0 is the increase of time. Here we choose time step Δt=0.02.Throughout the literature, the initial valuesB0=0.4,Mu0=0.2,Mi0=0.3,T0=0.1 and suppose that the unit of time is two hours in a day. We choose the parameter values of deterministic TB system (1) as given in Table 2.Note that,ℛ0=0.2816<1 and the deterministic TB model (1) has an infection-free equilibrium E0=Mu0,T0,B0,Mi0=200,20,0,0, which is globally asymptotically stable on Γ. We fix the parameter values in Table 2 and intensities of environmental noise σ1=0.4,σ2=0.4,σ3=0.3,σ4=0.3. By calculating, and 25=μM>σ12∨σ32/2=min0.08,0.045=0.045, the conditions of Theorem 2 are satisfied. Figure 2 exhibits that the population of MTB and population of infected macrophages (LTBI or active TB) of the stochastic TB system (5) to eradicate the tuberculosis bacilli germs are almost surely from the infected TB individuals at this stage.Figure 2
Simulations of the stochastic TB model (5) with initial conditions B0,Mu0,Mi0,T0=0.4,0.2,0.3,0.1. Parameter values used are as given in Table 1.
(a)(b)(c)(d)On the other hand, increasing the infected macrophage killing rateb from 1.1×10−4 to 1.25, increasing bacterium growth rate B, increasing the death rate of activated CD4+ T cells, and decreasing the natural recruitment rate of activated CD4+ T cells, etc., are vulnerable factors for tubercle bacillus which can grow from latency TB infection to oscillation and then reach as to active TB infection.Keep all the parameter values same as in Table2, except δ=5×10−3,sM=3×109,β=3×10−9,b=1.25,γ=0.4,sT=2×10−4,cB=5×10−6,cM=1×10−4,eB=1×10−2,eM=1×10−4,μT=50, with the intensities of environmental white noise as follows: σ1=0.2,σ2=0.5,σ3=0.2, and σ4=0.1. By computing ℛ0s=sMβ/ϑ+σ12/2μM+σ22/2b+γ+σ32/2η1b+η2γsT/sT+cμT+cσ42/2=1.0171>1, the conditions of Theorem 2 are fulfilled. Elseways, the deterministic TB model (1) has a unique positive infected equilibrium E1=B1,Mu1,Mi1,T1, where B1=2.4718×1010,Mu1=3.0256×107,Mi1=1.3598×109,andT1=2.7432×107, which is globally asymptotically stable on Γ. For the already stipulated small intensities of white noise, the stochastic TB system (5) oscillates around the positive infected equilibrium E1. Figure 3 evinces the existence of the unique ergodic stationary distribution of the stochastic TB model (5) if ℛ0s>1, which indicates that the disease will persist and prevail in an MTB population. The second segment of Figure 3 evaluates the probability density distribution functions of the stationary distribution.Figure 3
The first and second segments of the figures exhibit the existence of the unique stationary distribution of the stochastic TB model (5) and the probability distribution of the stationary distribution, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)We keep all the parameter values as in Figure3 with different intensities of environmental large white noise σ1=1.25,σ2=0.4,σ3=1.25, and σ4=0.2. Figure 4 reveals that the population of MTB and population of infected macrophages are swept away in the stochastic TB model (5) even while they are still persistent in the deterministic TB system (1). Hence, large intensity environmental white noise may be helpful for suppressing the disease outbreak and eradicating the MTB.Figure 4
The intensities of large environmental white noiseσ1=1.25,σ2=0.4,σ3=1.25 and σ4=0.2 cause extinction of the MTB.
(a)(b)(c)(d)


## 6. Conclusions and Future Directions

MTB is a leading communicable opportunistic disease in many countries and the main cause of chronic lung disease. In this literature, we studied the dynamical behavior of a stochastic host-pathogen tuberculosis model with adaptive immune response. Firstly, we exhibited that the stochastic TB model (5) has certain unique global positive solutions. We established sufficient conditions for the extinction of the tubercle bacillus. Also, we attained sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions to the stochastic TB model (5) by applying a suitable Lyapunov function. Finally, the numerical simulations are provided to validate analytical outcomes and the dynamics of the stochastic host-pathogen TB model.Certain interesting topics and practical realism earn further consideration. On the flip side, one may suggest some more realistic but complex models, such as considering the effects of impulsive perturbations on the stochastic TB system (5). The motivation for impulsive perturbations is that discontinuity is a common phenomenon and the real-life scenarios are generally discontinuous fashion like time and space. During the anti-TB treatment, pharmacokinetic variation and association of several drugs in a regimen are naturally nonlinear and occur only within some limited precise data spaces and often in a discontinuous manner [45]. Additionally, in the summer the incidence of TB is high, since it is speculated that the disease may have been obtained during the winter season. Due to air pollution, carbon monoxide stimulates bacillary reactivation and other factors such as poor water quality, soil pollution, and inadequate sanitation, which increase the risk of TB outbreaks. Owing to these sudden environmental changes, it is arousing curiosity to introduce the colored noise, for instance, continuous-time Markov chain into the stochastic TB system (5). Often, the switching among distinct environments is memoryless and the waiting time for the next switch is exponentially distributed [46, 47].Finally, the memory CD8+ T cells can swiftly obtain effector functions to destroy infected cells and/or secrete inflammatory cytokines such as IL-2, IL-4, and IL-5 prevent replication of the host-pathogen [14]. The predominant bestow factor is that the exhibition of MHC-I molecules is nearly omnipresent, whereas MHC-II molecules are revealed on a more limited set of cells. In this connection, the memory CD8+ T cells have more opportunities to encounter antigen [14]. Hence, in the stochastic TB model (5), another compartment has to be added like immune response CD8+ T cells which may be more effective and efficient to eradicate the tuberculosis bacilli. We leave these explorations for our future work.



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