Prevention and Control of Tuberculosis Relying on a Tuberculosis Dynamic Model Based on the Cases of American

Background: Tuberculosis (TB) which is a preventable and curable disease, is claimed as the second largest number of fatalities and there are 9,025 cases reported in the United State in 2018. Many researches have done a lot of research and achieved remarkable results, but TB is also a serious problem for human being.The study is a further exploration. Methods: In the paper, we propose a new dynamic model to study the transmission dynamics of TB, then use global diﬀerential evolution and local sequential quadratic programming (DESQP) optimization algorithm to estimate parameters of the model. Finally, we use Latin hypercube sampling (LHS) and partial rank correlation coeﬃcients (PRCC) to analyze the inﬂuence of parameters on the basic reproduction number ( R 0 ) and the total infectious (including the diagnosed, undiagnosed and incomplete treatment infectious), respectively. Results: By the research, the basic reproduction is computed as 2.3597 which means TB will be epidemic in US. The diagnosed rate is 0.6082 which means the undiagnosed will be diagnosed after 1.6442 years. The diagnosed will be recovered with an average of 1.9912 years, especially, some diagnosed will end the treatment after 1.7550 years, for some reasons. By the study, it’s shown that there are 2.4% of the recovered will be reactivated and 13.88% of the newborn will be vaccination. However, the immunity will be lose after about 19.6078 years. Conclusion: Through the results of this study, we give some suggestions to help prevent and control the TB epidemic in the United State, such as prolonging the protection period of the vaccine by developing new and more eﬀective vaccines to prevent TB, using the chemoprophylaxis for incubation patients to prevent their conversion into active TB; raising people’s awareness of prevention and control of TB and treatment after illness; isolation treatment for the infected to reduce the spread of TB. According to the latest report, in the announcement came at the ﬁrst WHO Global Ministerial Conference on Ending Tuberculosis in the Sustainable Development Era, we predict that it’s diﬃcult to control TB in 2030.


Introduction
To date, tuberculosis (TB) is seen as one of the world's deadliest disease caused from a single infectious agent, second only to acquired immunodeficiency syndrome (AIDS) caused by human immunodeficiency virus (HIV) [1,2]. TB is caused by Mycobacterium tuberculosis (MTB) which can spread through the air when the infected people cough, talk, sneeze or sing [3]. In most situation, MTB generally affects the lungs of the infected individuals. Tuberculosis is so highly contagious that the susceptible are likely to develop tuberculosis when they inhaled tiny particles containing MTB. The MTB are very small and can stay in the air for a long time and keep moving. The immune system is an important line of defense that limits the growth and spread of MTB. If the immune system cannot suppress their growth, they will most likely spread throughout the body [4]. In fact, not all people infected with TB will be sick at once. Some people's incubation period may be 1 year to forever. Nowadays, reliable TB tests are inexistence [5], which cause there are many the undiagnosed infectious and increase the diffcult to control TB.
It is reported in Global Tuberculosis Report 2018 that there are 1.3 million deaths caused by TB [2]. People may be infected with TB in all countries and age groups, but overall 90% were adults (aged ≥ 15 years), 9% were people living with HIV (72% in Africa). And most cases happened in following eight countries: India (27%), China (9%), Indonesia (8%), the Philippines (6%), Pakistan (5%), Bangladesh (4%), Nigeria (4%), South Africa (3%). All of these countries and 22 other countries are listed by WHO as the top 30 countries with the high burden of TB, and they account for 87% of the world's cases. On the contrary, the global cases of the WHO European Region (3%) and the WHO American Region (3%) account for only 6% [2].
Although the number of people infected with tuberculosis has been declining, there are about 10 to 20 thousands cases of TB every year in the last 20 years (see Table 1 ) [6] and the death rate is from 0.05 to 0.07 in American. Therefore, it is essential for researchers to explore the factors related to the infection, outbreak and epidemic of tuberculosis, then we could take more measures to protect people from TB.
From 1945 to 1955, the widespread use of antibiotics reduced tuberculosis mortality in the United States by 70%, but the United States still has a serious TB epidemic [4,7]. Based on decades of technology and experience, most active and latent tuberculosis can be effectively treated, and latent tuberculosis can be treated with isoniazid, but treatment can be effective only if the course of cures last at least 6 months [4]. Active tuberculosis can be eliminated with a complex treatment regimen and treated with multiple drugs (isoniazid, rifampicin, pyrazinamide) for nine months to achieve the course of treatment [4,7].
In the United States, people who contract TB and receive treatment fall into two categories [8,9]: one type of people are who complete the treatment and eventually recover; the other are not complete treatment, these people may be because of adverse reactions to the drug, or lost, or refused treatment or other reasons. It is not excluded that it may be due to the high cost of tuberculosis treatment and a heavy burden relative to the general population. When the treatment is unfinished, the drug-resistant strains will reproduce which may seriously increase the difficulty of treatment [4,10,11]. According to the latest treatment outcome data for new cases in 2016, 82% people who are able to be successfully healed. In contrast, this is another reduction from 86% in 2013 to 83% in 2015 [2].
Bacillus Calmette-Guerin (BCG) is a vaccine which have been used to prevent TB for a long time. The protection period of BCG vaccine usually varies from 10 to 20 years [12]. BCG prevents about 20 percent of children from getting infected, while the vaccine protects about half of those already infected from getting worse [13]. The study of Fjallbrant et al. [14] suggested that primary vaccination and revaccination of negative tuberculin skin test (TST) young adults with BCG cause a significant increase in the T-helper 1 (Th1) response against mycobacterial antigens, suggesting a protective effect against TB. These give support for the policy of primary vaccination as well as of revaccination in this setting and age group. In some areas where tuberculosis is highly prevalent, primary BCG vaccination is important for tuberculosis control, but should not be the main measure for tuberculosis control. BCG is widely used in countries with high burden of TB, but for small burden countries may be different. In the United States, BCG is often used to special people [15] but not widely used to ordinary people and is replaced by chemoprophylaxis [16,17] which is cheap and easy to take and can prevent latent infectious from becoming active tuberculosis. For some country with little burden for TB, chemoprophylaxis is a good chance to control TB [18].
On the dynamic model study, many researchers have devoted big efforts for the research of the epidemic law and transmission dynamics of tuberculosis. In 1962, Waaler et al. established the first dynamic model of tuberculosis which is based on a susceptible-infected-recovered (SIR) model [19]. From then on, many models that take into account multiple influencing factors have been established, just like reinfection [20][21][22], vaccination [23][24][25],interactions with HIV [26,27], reactivated [28,29], chemoprophylaxis [18] and so on [30][31][32][33]. Revelle et al. considered prophylaxis, cure and BCG vaccination to research the optimal strategy to fight against TB, which was then extensively used to study the epidemic model of transmission for infectious disease in 1967 [34]. Buonomo et al. studied the global behavior of a non-linear susceptible-infectious-removed (SIR)-like epidemic model with a nonbilinear feedback mechanism [35]. The SEI model proposed by Bowong et al. exhibits the traditional threshold behavior [33]. Whang et al. use a SEIR model with the time-dependent parameters to develop a dynamic model for tuberculosis (TB) transmission in South Korea [36]. A mathematical model was proposed to understand the spread of tuberculosis disease in human population for both pulmonary and drug-resistant subjects by Mishra et al. [37]. Three control factors must be considered simultaneously to decrease the threat of TB by Gao et al., as the following: a preventive measure in the form of vaccination and two treatment measures aiming at the susceptible and individuals infected TB in the active stage and latent stage [38].
In developing countries, the increase of TB cases by a high level of undiagnosed infectious population and incomplete treated population is one of the greatest challenges to control TB. These people are more possible to develop multi-drug resistance relative than the diagnosed infectious population [20,39]. According the actual situation of tuberculosis in the United States, we considered some factors: slow-fast process [40][41][42], vaccination [23][24][25], reinfection [20][21][22], reactivated [28,29] and undiagnosed infection [20,43]and referenced the modeling thought of the D.P. Moualeu et al. [20,44,45] and Liu et al. [23], then established our model. Compared with their model, the biggest difference of our model is that the vaccination and recovery to the susceptible population [20,23,44,45].
The aim of our study is to analyze the factors affecting tuberculosis based on the dynamics model, give some measures to control and prevent the TB, and predict the epidemic trend in American. The structure of this paper is as follows. In the Section 2, we introduce our tuberculosis model expressed by ordinary differential equations (ODE) and give the definition of parameters. Then describe the model assumptions and modeling ideas in detail, and the disease-free equilibrium and basic reproduction number is given. In Section 3, the model is simulated by global differential evolution and local sequential quadratic programming (DESQP) [46,47] optimization algorithm based on the US cases. We analyze the fitting effect by the root mean square percentage error (RMSPE) and the mean absolute percentage error (MAPE). In Section 4, we make the uncertainty and sensitivity analysis of the parameters for our model by Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC). We have analyzed the sensitivity of each parameter on the basic reproduction number and the total infectious, respectively. In Section 5, we analyzed the results and made some suggestions to prevent tuberculosis, and carried out simulation experiments, then we discussed the deficiency of our study. Finally, we summarize our research.

Methods
In this section, we introduce our new mathematical model and shortly explain the structure of our model, then analysed the basic reproduction number.

Model Instruction
The total population is denoted by N (t) which is sub-divided into the following seven sub-populations: V (t) vaccinated: healthy people vaccinated with TB, S(t) susceptible: healthy people not exposed to TB, E(t) exposed: exposed to TB but not infectious, I(t) diagnosed infectious: infected with TB and diagnosed in hospital, J(t) undiagnosed infectious: infected with TB but undiagnosed in hospital, L(t) incomplete treated: have been diagnosed with active TB and begun their treatment in hospital or home, but quitted before the end, R(t) recovered:, recovered from TB after treatment.
Considering slow-fast process [40][41][42], vaccination [23][24][25], reinfection [20][21][22], reactivated [28,29] and undiagnosed infection [20,43]and referencing the modeling thought of the D.P. Moualeu et al. [20,44,45] and Liu et al. [23], we display our dynamic model with a flow diagram shown in Fig. 1 and introduce the model in detail as the following: For the dynamic system, there will be a certain recruitment to join with an average scale Λ. Considering the vaccination, a proportion χ will be vaccinated (i.e., primary vaccination) and become the vaccinated class, and the remainder (1 − χ) will be not vaccinated and become the susceptible class.
In fact, BCG will be ineffective for some newborn, and they will be infected by contacting with the infectious, we assume the fraction as ε . Consider the vaccine protection period which are 10 to 20 years [12], there will be a rate ψ losing the immunity and being the susceptible.
Considering the fast-slow process, the susceptible will be the exposed by contacting at a slow process (1 − p) and be the active TB at a fast p. Because of the test of TB is not sensitive, there will be a fraction f diagnosed and (1 − f ) undiagnosed. Here, we set p 1 = pf and p 2 = p(1 − f ). We not rule out that there will be a rate φ who will revaccinated and become the vaccinated. For the exposed, a proportion r will use chemoprophylaxis to prevent being active TB, and the remainder (1 − r) will be the active TB at a rate k. For the active TB, there will be a fraction h will be diagnosed and healed and (1 − h) will be undiagnosed.
In the diagnosed infectious, some will be recovered at a rate g. Some will be the uncomplete treatment at a rate δ. For the undiagnosed, some will be diagnosed and go to hospital at a rate θ and some will be the exposed by the immunity at a rate ρ.
For the uncomplete treatment, some will be conscious of themselves illness and go to hospital at a rate α. While, some will be recovered by self-immunity at a rate ω.
Considering the re-infected and reactivity, the recovered will lost the immunity at a rate γ. A fraction q 1 will be reactivity and become the diagnosed infectious, a fraction q 2 will be re-infected but not be the active, the remainder 1 − q 1 − q 2 will become the susceptible.
In the last, considering natural death, it is inevitable that there will be a certain number of people in each part of the system. In our article, the main feature of our model is to consider the transmission route and mechanism of tuberculosis in the context of reality in a comprehensive way, which has a strong practical significance. Therefore, the conclusion will be closer to reality.

Model Introducing
We given the definition and range of the parameters in Table 2, and proposed a mathematical model to state the transmission dynamics and epidemic of TB which is represented by the following system of ordinary differential equations: where The susceptible and vaccination are infected with tuberculosis through contacting with individuals with active TB at a rate v(I, J, L) given by: where β i , i = 1, 2, 3 are the rates of which the diagnosed, undiagnosed infectious and incomplete treated sufficiently and effectively transmit TB to the susceptible or the vaccinated [1].

Basic Reproduction Number
The basic reproduction number R 0 represents the number of infected during the initial patient's infectious (not sick) period [48]. Our model is a biological system model, so it must meet the biological conditions. Therefore, we only study the dynamic state of the solution of system (1) in the following feasible region: which can be confirmed as positively invariant. Therefore, we restrict our attention to the dynamics of model (1) in Ω.
For the threshold system, when R 0 < 1, the model will stabilize to the disease-free equilibrium, and the disease will be controlled and eventually become extinct. When R 0 > 1, the model will stabilize to the endemic equilibrium, and the disease will develop into an endemic disease. For other complex systems, this conclusion may not be valid, for example, backward bifurcation, multistable system and other complex dynamic behaviors may occur. Therefore, in order to control and prevent TB, the smaller R 0 is, the easier to control TB [1,49]. Here, we use the next-generation matrix approach to calculate the basic reproduction R 0 which were proposed by Van den Driessche. etc [49], and the detail calculation is given in Appendix.

Simulation
Based on the reported data from 1984 to 2018 by WHO [6] and the model (1), an global differential evolution and local sequential quadratic programming (DESQP) optimization algorithm was conducted to estimate the undetermined parameters [47,50]. DESQP which combine differential evolution (DE) [46] and local sequential quadratic programming (SQP) [51,52] is a method used to search for optimal solution of DE. In the method, DE is used as a base level search and SQP is used as a local search. DE is first applied to short term of problem to find best solution. This best solution is given to SQP as an initial condition to fine tune the solution to reach the global optimum or near global optimum.
We get the estimated value, standard deviation, confidence interval, p-value and t-statistic of the parameters which are listed in Table 4. Based on the estimated results, the basic reproduction number R 0 can be calculated R 0 = 2.3597.
The real data and the model results are shown in the following Fig. 2. We evaluate the fitting effect of our established model through the root mean square percent-age error (RM SP E) and the mean absolute percentage error (M AP E) which are significant evaluation indicators. The RM SP E and the M AP E are defined as: where I(t) * is the real value at time t and I(t) is its fitting value and n is the number of data used for prediction. The criteria of M AP E and RM SP E are shown in Table 3 [53,54]. We use model (1) to simulate the number of the infected, where M AP E = 4.7245% and RM SP E = 5.7676% which means the fitting effect is very well and our system have strong prediction ability and high prediction accuracy.

Sensitivity Analysis
In model (1), the precise estimation of parameter values is one of the greatest challenge. Direct measurements of specific biological parameters are rare, and many parameter's values are estimated within a large range of identified by fitting the model with limited experimental data [55]. Therefore, parameters estimation is always associated with uncertainty analysis (UA) and sensitivity analysis (SA). It is vital to study the influence of the uncertainty of these parameters on the model, which will help us to successfully use the mathematical and computational models of biological systems as prediction tools and to comprehend the functions of biological systems. Some factors are which we cannot control in model (1). Generally, we used to choose the factors which we can control to analysis the sensitivity, so that we can find good measures to eliminate TB. Because we can't control the death rate d 1 , d 2 and d 3 . So in our analysis, we only did sensitivity analysis for 22 parameters which we can control to eliminate TB. In order to find how the parameters impact on the outcome, a general and better way is to do the sensitivity analysis for each parameter. The ideal method is to use Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) to study the dependence of model parameters on the basic reproduction number and total infectious [56,57].

Latin Hypercube Sampling (LHS)
Generally, the input factors of most mathematical and computational model consist of initial conditions and parameters which are indepent and dependent model variables. Thanks to natural variation, lack of current techniques, measurement error and so on, the parameters are not always known with enough certainty [57]. The purpose of uncertainty analysis (UA) is to solve these problems. UA can quantify the degree of confidence in the experimental data and the estimation value of parameters [57].
In the article, the most popular and efficient Latin hypercube sampling-LHS which belongs to Monte Carlo (MC) class of sampling methods and was introduced by Mckay et al. was used to perform UA [57]. MC method is a common algorithm to solve various computational problems, and can realize the evaluation of multiple models and the results can not only be used to perform SA, but also to determine the uncertainty of model inputs. LHS can unbiased estimate the average output of the model, and fewer samples are required to achieve the same accuracy as simple random sampling [57].
The remaining 22 parameters have been chosen to do uncertainty analysis. We assume each parameter to be a random variable with normal distribution to analyze the uncertainty in the value of these parameters. Normal distribution for all parameters with the mean (i.e., estimated value) and variance value (i.e., square of standard deviation) are given in the Table 4. Latin hypercube sampling has been used to sample for these parameters which are considered for sensitivity analysis. Here, we set the sample size N=2000. Using Latin hypercube sampling method, probability density function for each parameter is stratified into 2000 equiprobable (1/2000) serial intervals. Then a single value is chosen randomly from each interval. This produces 2000 sets of values for each parameters, and we can compute 2000 sets of values for R 0 from 2000 sets of different parameters values mixed randomly and get the distribution hist of R 0 which is shown in Fig. 3.

Partial Rank Correlation Coefficient (PRCC)
Sensitivity analysis (SA) is a quantitative way to analysis the effects of the parameter uncertainty on the model's outputs. Based on the parameters, we are able to raise presumptions about the biological system that actuate the system behavior which can be measured by conducting experiments [58]. Local SA techniques, one class of SA, research the effects of small variations in individual parameters around some nominal point and have been applied to a number of signal transduction and metabolic pathway models [59,60]. Since the most influential parameters are determined, the predictability of the model can be greatly enhanced.
In the section, we compute PRCC to analyze the sensitivity of the parameters to the R 0 and the total infectious to identify which parameters have great effect on the variability in the outcome and how the parameters affect R 0 and the total infectious. Here we compute the PRCC of R 0 and the total infectious based on the LHS matrix, the result can be seen from Fig. 4, Table 5. In our experiment, we assume that the parameters have a significant effect when P-value< 0.01.
From Fig. 4, we can easily see that different parameters have different degree and different effect on R 0 and the total infectious which may be complex to take proper measures to control TB respectively. In order to better control TB, we put emphasis on analyzing the parameters whose PRCC > 0.2. And we assume these parameters have high degree and significant effect on R 0 and the total infectious. Expect for the uncontrollable factor (which we cannot take relative measure to control TB) from Fig. 4, we can easily see that r, ω, g and φ have significant and positive affect on the R 0 and the total infectious, and ε, k, ψ, δ and θ have significant and negative affect on the R 0 and the total infectious.

Results and Discussion
In this section, we present the results of simulation for the model and discuss the sensitivity of parameters to the R 0 and the total infectious. Furthermore, we give some measures to prevent TB.

Results
In generally, parameter estimation is an iterative process, which we use the current parameter values as the initial values of the next iteration [1]. All the parameter values of the first iterative process are set to be their initial guess values which are estimated with the lowest sub-condition. Then parameters estimation is carried out with a limited list of parameters which are previously non-identifiable. Finally, repeat the estimation process and check all the estimated parameters to see whether the new value of the previously unrecognized parameter affects the value of the identifiable parameter. We use the data of tuberculosis cases (i.e., diagnosed infectious) in American from 1984 to 2018 (see Table 1) published by Centers for Disease Control and Prevention (CDC) to estimate the parameters of the model (1).
In our model, some parameters have been estimated by WHO and some parameters have been estimated by the TB researchers and others are uncertainty. In the following we specify some parameter values.
(1) The natural mortality µ: It is assumed to be equal to the inverse of the life expectancy at birth and 1/µ = 79.30 is the average lifespan of human, then µ = 0.0126 [61].
(2) Progress rate of the exposed to infectious individuals (including diagnosed and undiagnosed infectious) k: Based on the parameter estimation k = 0.0421, the incubation period of tuberculosis 1/k = 23.7529 years. In fact, the latent period of TB is 1 year to forever in generally [1,18,62].
(3) Recovery rate of the diagnosed infectious g: In our simulation, g = 0.5022, and the course of recovery for the diagnosed infection is estimated 1/g = 1.9912 years. By 2010, the course of treatment of the tuberculosis patient that first time is commonly treated in 6 months, and the course of treatment of the tuberculosis patient that is reinfected is commonly 18-24 months [63,64].
(4) Diagnosis rate θ: θ = 0.6082 per year which means the undiagnosed individual will be diagnosed with active TB after 1/θ = 1.6442 years, generally, some people with active TB is difficult to be diagnosed.
(5) Progression rate at which diagnosed infectious people become incomplete treated δ: It has been estimated as δ = 0.5698 per years which means the diagnosed people may give up treatment after 1/δ = 1.7550 years. Generally, the average convalescence period of tuberculosis is about 1 year which means when people are treated 1 years, they will think they have recovered, but not [49].
(6) Reactivated ratio q 1 and reinfected ratio q 2 : It has been estimated as q 1 = 2.40%, q 2 = 54.25%, this shows that relapse for most people is slow progress, but re-infected is fast progress. (1 − q 1 − q 2 ) = 43.35%, which means the people will lost the immunity and become the susceptible.
(7) Progress rate at which incomplete treated people become diagnosed infectious α: It has been estimated as α = 0.1002 which means that the incomplete treated may be treated again after 1/α = 9.9800 years. (8) The natural vaccination rate of the newborn babies χ: It has been estimated as χ = 13.88%. In American, only a few people get BCG vaccination [15][16][17]. The United States and other western countries with low TB burden do not necessarily require people to be vaccinated against BCG for newborns, some Americans will be vaccinated at the doctor's advice [15][16][17].
(9) Progress rate that the vaccinated become the susceptible ψ. It has been estimated as ψ = 0.0510 which means the vaccination may be invalid after 1/ψ = 19.6078 years. BCG vaccine duration varies widely, ranging from 10 to 20 years [12].
(10) Progress rate which the recovered individuals lose the immunity γ. It is estimated as γ = 0.1444 per year which means the recovery individuals may be lose the immunity after 1/γ = 6.9252 years.
(11) Progression rate at which the undiagnosed become the exposed ρ. It has been estimated as ρ = 0.1993 which means it takes about 1/ρ = 5.0176 years for the undiagnosed to become a non-infectious individuals (exposed).
(12) Detection rate of active TB h. It is estimated as h = 39.98%, This shows that 60.02% of tuberculosis patients will not be diagnosed or will not be diagnosed for a short time.
(13) Vaccination coverage φ: It has been estimated as φ = 0.0500 which means after an average of 1/φ = 20.0000 years, people will lose antibodies to tuberculosis and be vaccinated again. Generally, BCG is not widely used in American, and the adult also will not choose to vaccinate if it's unnecessary [15,16]. (14) Chemoprophylaxis rate r: It has been estimated as r = 0.9219. Dye et al. have estimated r = 0.7 [65,66].
(15) Recovery rate of the incomplete treated ω: It has been estimated as ω = 0.1986 which means some incomplete treated will naturally recovery after 1/ω = 5.0352 years. Bacaër et al. have estimated that the natural recovery for HIV-negative TB and HIV-positive TB cases as 0.1390 and 0.2400 per year, respectively, HIVnegative TB and HIV-positive TB will be recovered after 7.1900 years and 4.1700 years without treatment [67]. (16) The rate of the susceptible become the diagnosed and undiagnosed infectious p 1 , p 2 : It is estimated that p 1 = 2.5% and p 2 = 34.14%, which means 2.5% + 34.14% = 36.54% of the people will be sick at once after being infected TB, while 2.5% have serious symptom and 34.14% have mild symptom which do not go to diagnosing. The remained (1 − p 1 − p 2 ) = 63.46% who enter a slow progression of TB infection become the exposed.

Discussion
TB is a high-prevalence infectious disease in the world and the infectious are widespread worldwide. It is vital to seize the main cause and find the best measures to prevent and control the disease. In the article, we have constructed a TB model to study the transmission dynamic and provide some measures to control and prevent TB in US. To find more ways to prevent TB, we analysis many factors which may have effect on R 0 and the total infectious. The sensitivity analysis of the parameters with R 0 and the total infectious have been done (see Fig. 4, Table 5). When we take measures to control TB, the result is shown in Fig. 5. In generally, we find we can control the factors with α, g, ψ, r, β i , i = 1, 2, 3, φ and δ. From the result in Fig. 5, we can clearly find that r has the greatest effect on the total infected, then the g has the secondly greatest effect, and others have the similar effect.
Strategy 1: We can find r is significantly negatively correlated with R 0 which means it's wise to use chemoprophylaxis to control TB. And with our control, the TB can be greatly control from Fig. 5. In order to prevent the TB outbreak, we can encourage people to have chemoprophylaxis by the media, and research new and more efficient chemoprophylaxis to improve the effect and reduce the harm to human body, improve the therapeutic effect [68]. Strategy 2: The parameter g have a negative effect on the R 0 , and with our control, it has clear reduction on the total infectious. By doing these, we should take much money and energy to research new medicine and therapy to reduce the period of treatment [69]. Nowadays, even though there has been a big improvement, but not enough, in the treatment of tuberculosis, still TB is the second leading cause of death in the world. So we still have a long way to go to end TB. Strategy 3: From Fig. 4, we can see β i , i = 1, 2, 3 have positive effect on R 0 . With our control, we can greatly reduce the total infectious by reducing the transmission rate. Despite the emphasis we have placed on treating TB patients in isolation, the majority of people were infected TB by contacting each year [70]. We must strictly monitor and take protective measures to avoid tuberculosis. Examine outsiders to avoid contact with TB patients [71]. If we decrease half of β i , i = 1, 2, 3, we can find the total infectious will decrease (see Fig. 5).
Strategy 4: It is also clearly shown that ψ is positively correlated with R 0 and the total infectious which means the longer the vaccine lasts, the easier it is to control TB. We make 0.8 × ψ, and find the total infectious decreased greatly which means we can decrease it to prevent TB. We can delay the duration of the BCG by researching new and better vaccination to prevent infecting TB. Currently, one of the main health interventions that can be used to prevent tuberculosis is to vaccinate children.
Strategy 5: The parameter δ have positive effect on the R 0 , and α have negative effect on the R 0 . From one hand, we can encourage people finish the treatment to cure completely by reducing the cost [72]. The state can make more medical insurance to relieve the financial burden and help people heal. For another, educate people to know more about TB treatment and to follow the doctor's plan, let they know health is more important than everything. Strategy 6: It is clearly shown that φ is negatively correlated with R 0 and the total infectious which means increase the people of vaccination per year can protect people from TB and control TB. Even the state have little burden of TB, BCG also can help to prevent TB.

Deficiency
In this paper, the data used for fitting is annual data. Due to the large time scale, the accuracy of the model will be reduced. In the later research, we will choose monthly data. Although our model took into account factors such as slow-fast process [40][41][42], vaccination [23][24][25], reinfection [20][21][22], reactivated [28,29] and undiagnosed infection [20,43], however, there were not enough and many factors which didn't take into account. We did not take into account factors such as interactions with HIV [26,27], immigration [73,74] and drug-resistant TB bacilli [65,75,76]. In addition, we did not take into account residents' medical expenditure and awareness of disease control to discuss the prevention and control measures in this study. As for the deficiencies in the study, we will analyze and discuss them in the follow-up study.

Conclusion
In generally, from the analysis, it is clearly that the prevalence of TB in the United States is heavily influenced by exposure, vaccination and treatment effectiveness which are similar to TB studies elsewhere. In our study, we also found that chemoprophylaxis affects the prevalence of TB more than other factors. However, each coin have two sides, chemoprophylaxis has certain harm to human body [68], so we should research new better prevent and control measures for TB. Based on the analysis, we give some strategies to control and end TB in two ways: prevent and treatment. Especially, chemoprophylaxis can greatly control TB with some side effect, so we should do much research to find better chemoprophylaxis.
In fact, because of the recovered infectious may be relapse, the difficulty of diagnosing TB and treatment, TB is hard to control. When all the control measures are implemented together (see Fig. 5, black solid line), the basic reproduction number of model (1) is 0.6915 < 1, but the case does not disappear, and the system is stable to the endemic equilibrium. For the measures we proposed, these may cannot eliminate TB, but these are critical useful for the epidemic of Tuberculosis. According to the latest report, in the announcement came at the first WHO Global Ministerial Conference on Ending Tuberculosis in the Sustainable Development Era, there are 75 ministers agreed to take urgent measures to end tuberculosis (TB) by 2030 [77]. From our analysis, it is difficult to end the tuberculosis till to 2030 under the existing conditions (Fig. 5). So we should find more and efficient methods to end TB. It will be difficult to eliminate TB in a short period of time, but we believe that in the future, with advanced technologies, TB can be eliminated completely.    show the PRCC of parameters with the total infected. Here,we assume that when P-value < 0.01, the parameters have significant effect R 0 and the total infectious. In order to better control TB, we put emphasis on analyzing the parameters whose PRCC > 0.2. Figure 5 control.eps Simulation of the total infectious with parameters 1.03 × r = 0.9496, 1.5 × α = 0.1503, 1.5 × g = 0.7533, 1.5 × φ = 0.0750, 0.9 × δ = 0.5128, 0.99 × ψ = 0.0505, 0.7 × β 1 = 3.1067, 0.7 × β 2 = 0.2096 and 0.7 × β 3 = 4.1063, when one parameters takes a specific value, others take the value of the first column in Table 4. Differently, we synthesize the effects of three contact rates β i , i = 1, 2, 3 into one contact rate β effect. 'With all control' means that we let all parameters specific values simultaneously. 'Without control' is the situation which we take no measures. We can find ψ has a mild effect on the total infected, with its line overlapping with 'without control' approximately.  Year Number(×10 9 ) N umber (11) Rate (11) N umber (12) Rate (12) N umber (21) Rate (21) N umber (22) Rate (22) Table 2 The definition and value range of parameters for model (1).     The data of population from [6] and others from [78]. Table 2 -The definition and value range of parameters for model (1). Table 3 -The criteria of M AP E and RM SP E. Table 4 -The t-statistic, P-value, CI Bound, Standard deviation and estimated value of parameters and initial condition of each compartment of model (1). Table 5 -The value of PRCC between each parameter and R0 and the total infected.     Here,we assume that when P-value < 0.01, the parameters have significant effect R0 and the total infectious. In order to better control TB, we put emphasis on analyzing the parameters whose PRCC > 0.2. Figure 5 -Simulation of the total infectious with parameters 1.04 × r = 0.9588, 1.9 × α = 0.1904, 1.6 × g = 0.8035, 1.7 × φ = 0.0850, 0.8 × δ = 0.4558, 0.8 × ψ = 0.0408, 0.7 × β1 = 3.1067, 0.7 × β2 = 0.2096 and 0.7 × β3 = 4.1063, when one parameters takes a specific value, others take the value of the first column in Table 4. Differently, we synthesize the effects of three contact rates βi, i = 1, 2, 3 into one contact rate β effect. 'With all control' means that we let all parameters specific values simultaneously. 'Without control' is the situation which we take no measures. We can find ψ has a mild effect on the total infected, with its line overlapping with 'without control' approximately.