A Dynamic Model of Human and Livestock Tuberculosis Spread and Control in Urumqi, Xinjiang, China

We establish a dynamical model for tuberculosis of humans and cows. For the model, we firstly give the basic reproduction number R 0. Furthermore, we discuss the dynamical behaviors of the model. By epidemiological investigation of tuberculosis among humans and livestock from 2007 to 2014 in Urumqi, Xinjiang, China, we estimate the parameters of the model and study the transmission trend of the disease in Urumqi, Xinjiang, China. The reproduction number in Urumqi for the model is estimated to be 0.1811 (95% confidence interval: 0.123–0.281). Finally, we perform some sensitivity analysis of several model parameters and give some useful comments on controlling the transmission of tuberculosis.


Introduction
Tuberculosis (TB) is a worldwide public health problem that is chronic infectious disease of respiratory tract as the main route of transmission. In 1993, WHO declared a state of the global TB in emergency. Even if we already know how to effectively prevent and cure TB through the half a century of development and progress, there are still more than 1.6 million people who died of TB. In 2014, TB killed 1.5 million people (1.1 million HIV-negative and 0.4 million HIV-positive). The toll comprised 890 000 men, 480 000 women, and 140 000 children. India, Indonesia, and China had the largest number of cases: 23%, 10%, and 10% of the global total, respectively [1]. TB is caused by Mycobacterium tuberculosis and spread via air-borne droplets from a cough or sneeze. The majority of infected individuals never develop TB, and only few people would induce active TB.
Bovine tuberculosis (BTB) is zoonotic infectious disease that is by the OIE (Office International des Epizooties) classified as class B animal epidemics. Infected cattle can act as the primary source of infection in other animals and humans. The main route of transmission is the respiratory and digestive tract. Healthy people and animals will be infected by contacting infected animals or drinking their raw milk [2,3].
BTB is caused by Mycobacterium bovis and Mycobacterium tuberculosis. BTB is a major infection of work cattle and cows. Most of high-yielding dairy cows and young cattle are infected by BTB [4]. After being infected with TB, cows will decrease milk production and working cattle become emaciated; finally, most infected cattle died of heart failure. BTB not only restricts the development of the livestock industry but also threatens people's health. It has become a worldwide public health problem [5,6].
BTB has a very long history in Xinjiang, and it has a wide popularity and has a serious impact on the animal husbandry in Xinjiang. According to Xinjiang related archives and records, livestock infection of TB existed in Xinjiang before the founding of China [7]. After the founding of China, Xinjiang quarantined bovine tuberculosis in the early 1950s. Quarantine was via conjunctival sac or use the intradermal allergic reaction method. All positive cows detected in the country documents or policy specific requirements are to be slaughtered. From 1990 to 2007, a total of 1098651 head of cattle were quarantined in Xinjiang; positive rate was 0.88%. But because of the shortage of the subsidy funds, only a part of positive cattle was slaughtered [6].
Mathematical model is the important tool to measure control strategies against various infectious diseases [8]. Mathematical models have played a significant role in understanding the complexity of TB transmission dynamics. The original mathematical models for TB were developed by Blower et al. in 1995 [8]. They established a simple model and a complex model to explain the spread of TB in the population. They demonstrated that it takes one to several hundred years for a TB epidemic to rise, fall, and reach a stable endemic level. Since then, a large number of mathematical models have been created for tuberculosis [9][10][11][12][13][14][15][16][17][18]. Blower et al. introduced chemoprophylaxis and treatment in previous models due to drug sensitivity and drug resistance expansion [9]. They concluded that, in order to control TB, treatment failure rates must be lower in developing countries than in developed countries.
Although many studies of dynamical TB models spread between humans have been reported, little work has been performed on such models spread between humans and animals up to now. The purpose of this paper is to propose a TB model between humans and cows to investigate the BTB epidemic situation and analyze the effect of current control strategies in Urumqi. In this paper, based on the reference of the literature, exploring TB transmission mechanism between humans and cows, the dynamic model is established.
The paper is organized as follows. In Section 2, we introduce the data sources. The model establishing and analysis were shown in Section 3, including the calculation of the basic reproductive number and the discussion of positive equilibrium points. The parameter estimation and sensitivity analysis of the model were carried out in Section 4. A discussion is given in Section 5.

Data Sources
This paper used data from the human and livestock TB epidemiological investigation in Urumqi [19]. The epidemiological investigation was to find the rules and characteristics between human and livestock TB in Urumqi and control the spread of TB better.  Table 1). Results: the result shows that there are 333 positive cows in quarantined cows, so positive rate is 0.40%. For 14 large-scale dairy farms, 35634 cows were quarantined, and the positive rate was 0.51%. Large-scale dairy farm from 2007 to 2014 TB positive rates were 0.37%, 0.64%, 0.55%, 0.52%, 1.55%, 0.09%, 0.15%, and 0.18% (see Table 2). For eight counties in Urumqi, in the cows of scattered households, 46637 cows were quarantined, and the positive rate was 0.32%. The positive rates were 0.06%, 0.63%, 0.17%, 0.32%, 0.60%, 0.23%, 0.14%, and 0% (see Table 3). The positive rate of cows of scattered households was lower than large-scale dairy farm.    In 2011, Sanlu milk powder caused damage to a lot of people because of toxic ingredients melamine. Dairy industry had a great adverse impact after this point. As a consequence of the not acquired raw milk, farmers sold and slaughtered a large number of cows so that the large number of cows declined sharply.

Object and
Therefore, we can get the point estimate and interval estimation of TB positive cows in Urumqi city in 2007-2014 (see Table 4).

Model Formulation.
We use a TB model to study the transmission of TB in Urumqi, Xinjiang, China [6,13,19]. Model consists of two parts; cow TB model captures the temporal dynamics of three groups: susceptible cows ( ), cows infected with Mycobacterium tuberculosis ( ), and cows that are removed after infection with Mycobacterium tuberculosis ( ) (including quarantined and slaughtered cows); human TB model captures the temporal dynamics of four groups: susceptible individuals ( ℎ ), latently infected individuals ( ℎ ), active infectious TB cases ( ℎ ), and recovered ( ℎ ). The transmission flow among humans and cows is illustrated in Figure 1. The model is described by the following system of seven ordinary differential equations: The parameters of the model are explained below: is recruiting of susceptible cows; is natural death rate of cows; 1 is the rate of cows infected TB via cows; 2 is the rate of cows infected TB via humans; is mortality rate due to TB of cows; is the slaughter rate to infected cows; is the quarantine rate to infected cows; ℎ is recruiting of susceptible humans; ℎ is the removal rate of livestock workers in dairy farm; 3 is the rate of humans infected TB via cows; 4 is the rate of humans infected TB via humans; is the progression rate to TB; ℎ is mortality rate due to TB of humans; is the cure rate to TB; is the rate of relapse to active TB.

Model Analysis.
Notice that is independent of the first six equations, and we start by considering the first six equations: Simple algebraic calculation shows that model (2) always has a unique disease-free equilibrium 0 ( / , 0, ℎ / ℎ , 0, 0, 0). According to the concepts of next generation matrix and reproduction number presented in [24,25], we define Noting that the disease-free equilibrium of model (2) Hence, the next generation matrix is Computational and Mathematical Methods in Medicine The basic reproduction number is given by ( −1 ) and According to the conclusions of the literature [24,25], the following results are obtained. Theorem 1. When 0 < 1, 0 is local stable; when 0 > 1, 0 is unstable.
Using a similar argument as in the proof of proposition 3.3 in [26], we can show that when 0 > 1, model (2) has at least one endemic equilibrium * . On the stability of the endemic equilibrium, one has the following theorem.

Computational and Mathematical Methods in Medicine
Through the same calculation, we obtain Similarly, it is easy that Now, construct the following Lyapunov function: Then, It can be verified that the largest invariant set where = 0 is singleton * . Therefore, by LaSalle's invariance principle, * is globally asymptotically stable.
Computational and Mathematical Methods in Medicine 7

Parameter Estimation.
The values of parameters for model (1) are listed in Table 5. According to the national policy, the positive livestock infected TB should be slaughtered, however, due to the lack of funds and the nontimely payment of the slaughter of livestock, resulting in the fact that TB positive livestock are not completely slaughtered. So we choose = 0.85 and = 0.12.
The parameters 1 , 2 , and 3 are obtained by fitting the model to data. We ignored humans infected TB via humans; hence, we make 4 = 0. By least-square fitting and Bootstrap method, we can obtain the point estimation and confidence interval for transmission coefficient which are listed in Table 6, respectively.
Based on Table 6, we obtained the basic reproduction number 0 ≈ 0.1811. The result shows that disease will not break out under current situation by Theorem 1. We give a histogram of 0 obtained by using the Bootstrap method (see Figure 2). In 2011, some of the large-scale dairy farm owners changed and the new buy cows from other places, so the rate of TB positive cows is very high. We regard this point as outlier. We discard this point, estimate the number of TB positive cows in 2007-2014, and draw the 95% confidence interval (see Figure 3). The result shows that the fitting effect is good; in fact, we estimate the number of TB positive cows in 2007-2014 and provide the confidence belt by all of the data (see Figure 4). We can predict the general tendency of the epidemic according to the current situation, which is presented in Figure 5. The prediction shows that disease will vanish around 2020 (see Figure 5).

Sensitivity Analysis.
For the sensitivity analysis, Latin hypercube sampling was used to sample parameters that appear in the derived expression for basic reproduction number 0 . Uncertainty and sensitivity analysis based on Latin hypercube sampling has been previously applied to disease transmission models. Thus, in order to examine the sensitivity of our results to parameter variations, we use Latin   We choose sample size = 2000, parameters of interest as the input variables, and the value of 0 as the output variable. The PRCC values of ten parameters are listed in Table 7 and shown in Figure 6. The ordering of these PRCCs corresponds to the level of statistical influence the parameter has on the variability for the basic reproduction number 0 . The larger the PRCCs in absolute value, the more important the parameter in responding to the change in 0 . Plus sign or minus sign means the influence is positive or negative, respectively. Figure 6 shows that 1 and 3 have positive impact upon 0 , whilst , , , and have negative impact. We also know that 0 is not sensitive to parameters , , , and 2 . Table 7 shows that the slaughter rate to infected cows (|PRCC| = 0.9008) has the greatest impact on 0 . Then, the quarantine rate (|PRCC| = 0.7349) to infected cows has the greater impact on 0 . Hence, from sensitivity and mathematical analysis, we conclude that the most effective approach to reduce the TB infection is to increase parameters and .
Computational and Mathematical Methods in Medicine In the following, we focus on parameters and . The influence of parameters and on the number of cows TB positive cases is shown in Figure 7. We can see from Figure 7 that, with the increase in slaughter rate, the positive rate of TB in dairy cows will be greatly reduced. Similarly, this phenomenon is also reflected in the effect of quarantine rate on the number of TB positive cows; appropriate increase of the quarantine rate of TB positive cows can also be a good control of the spread of TB.
It is very significant to investigate the effect of slaughter rate and quarantine rate on basic reproduction number 0 . Due to the lack of funds and the nontimely payment of the slaughter of cattle, resulting in the fact that TB positive cattle are not completely slaughtered, when the slaughter rate can not reach a high proportion of cases, appropriate improvement to the quarantine of sick cattle can also control the epidemic of BTB.

Discussion
TB infection exists widely in the world. In Xinjiang, TB is one of the major infectious diseases that seriously endanger the health of people. Xinjiang is one of the large pastoral areas in China. The prevalence of BTB not only restricts the development of the livestock industry in Xinjiang but also threatens people's health. To investigate the prevalence of BTB in Urumqi, a total of 82271 cows in Urumqi areas from 14 large-scale dairy farms and 8 counties of grazed cows were quarantined [19]. We establish a dynamical model for TB of humans and cows. We get the disease-free equilibrium point, discuss the positive equilibrium point, estimate the parameters, and conduct the sensitivity analysis. The sensitivity coefficients (PRCCs) of the parameters with respect to the basic reproduction number are shown in Figure 6. The results indicate that the slaughter rate and quarantine rate are the main factors affecting the spread of BTB, so the standard slaughter and quarantine management of the TB positive cows will inhibit the spread of BTB effectively. The simulation results reveal the main trend of BTB epidemic in Urumqi and also a prediction for the trend of the BTB infection. In 2011, some of the large-scale dairy farm owners changed and the new brought cows from other places, so the rate of TB positive cows was very high. This point has a little impact on our fitting effect. Finally, we predict the number of TB positive cows in Urumqi from 2014 to 2024. Figure 5 shows that the number of TB positive cows will be close to zero in our model. According to recent epidemiological investigation, BTB effective control had been obtained in Urumqi. The result shows that the current control measures are effective.