Rich Dynamics of a Brucellosis Model with Transport

Brucellosis is one of the major infectious diseases in China. In this study, we consider an SI model of animal brucellosis with transport. )e basic reproduction numberR0 is obtained, and the stable state of the equilibria is analyzed. Numerical simulation shows that different initial values have a great influence on results of the model. In addition, the sensitivity analysis of R0 with respect to different parameters is analyzed. )e results reveal that the transport has dual effects. Specifically, transport can lead to increase in the number of infected animals; besides, transport can also reduce the number of infected animals in a certain range. )e analysis shows that the number of infected animals can be controlled if animals are transported reasonably.


Introduction
Brucellosis is a zoonosis. It is one of the animal diseases, especially domesticated livestock. Brucellosis mainly attacks sheep, cattle, horses, dogs, pigs, humans, and so on. is epidemic is highly infectious and can be transmitted directly or indirectly through brucella carriers. Humans and animals have certain immunity after suffering from the disease, but they can be infected again. Its incidence is higher in spring and summer. Humans are generally susceptible to the disease and can acquire certain immunity. ere is cross immunity among different Brucella species, and the reinfection rate is 2-7. Some scholars have studied the infectious diseases and obtained some results [1][2][3][4][5][6][7][8].
Many articles have studied brucellosis through dynamic models [9][10][11][12]. Hou et al. [9] proposed a model with general incidences and analyzed the dynamics of this model. Sun and Zhang [11] gave a brucellosis model which incorporated proportional birth and studied the global stability. Yang et al. [13] proposed to combine age structured brucellosis disease model with spatial diffusion infection firstly (R 0 was obtained by mathematical analysis), analyzed the sensitivity to parameters, and gave the prevention and control measures of the disease. Yang et al. [14] studied a nonlinear model including spatial and seasonal variations to study the transmission dynamics of brucellosis and proved the importance of spatial and seasonal heterogeneity in disease control. e mathematical model of sheep brucellosis and the effect of slaughtering policy have also been studied [15].
e Chinese government has taken many measures to prevent diseases: vaccination, disinfection and elimination, and so on. If the disease is found, culling measure is taken immediately. However, in fact, some breeders do not take these measures in order to reduce economic losses, which can cause the spread of disease; besides, because the disease has a latent period, breeders do not discover brucellosis in time. In these cases, they transport animals which include the infected.
ere has been some studies on the impact of transport on disease transmission [16][17][18]. A delay periodic patch model with transmission related infection was studied and the authors discussed the difference between the periodic model and the corresponding autonomous model [19]. Liu and Stechlinski [20] proposed a periodic SIS epidemic model with time delay and transport related infection in patchy environment.
ey discussed the dependence of R 0 on transport related infection parameters and fluctuation amplitude. is paper studies the disease transmission of the remaining animals after the animals are transported away, which will be helpful for disease prevention and control.
In this paper, we study an SI model of animal brucellosis with transport. We show that transport has dual effects. In Section 2, we study the existence of the equilibrium. In Section 3, we study the dynamic behavior of the equilibrium. In Section 4, the numerical simulation is given. Finally, some conclusion are summarized and discussed.

Equilibria
We study a two-dimensional model: densities of the susceptible (S) and densities of the infected (I), in this paper. It is assumed that the transport rate B is a nonzero constant when the number of infected is less than a certain value; when the number of infected is greater than a certain value, B equals 0. e detailed relation between susceptible and infected is given in Figure 1. en, the model is expressed in the following form: We assume that all parameters are nonnegative in this model.
A is the recruitment rate of the animal populations, β is the infection coefficient, B is transport proportion, 0 < C < 1 is the rate of susceptible in this transport, d is the natural death rate of the animal populations, and μ is the diseaserelated death rate.
When I ≤ I 0 , the steady state of (1) can be written as follows: When I > I 0 , the steady state of (1) can be written as follows: (1) has no disease-free equilibrium. Based on Ref. [21], we can obtain the basic reproduction number R 0 : System (4) has a positive equilibrium E * � (S * , Next, we calculate the positive solution of system (3). rough the first equation of system (3), we obtain S � (A − BC)/βI + d. en, we substitute it into the other equation of (3), so we have Consequently, we have the following expression: Δ ≥ 0 is equivalent to b < 0 is equivalent to . (11) b < 0 and Δ ≥ 0 if and only if (10) holds. If (10) holds, there are two solutions of (3): If the condition I 1 ≤ I 0 holds, then we have 2 Complexity By calculating, (14) is equivalent to As a conclusion, when R 0 ≥ max(p 1 , p 2 ) or R 0 ≤ p 1 , It is equivalent to an inequality as follows: We have that (19) is equivalent to According to the discussions above, we can obtain two theorems as follows.
So, we have the following.
According to eorems 1 and 2, the changes in the number of equilibrium with the basic reproduction number are shown in Figure 2. When p 0 < R 0 < p 2 , there are three endemic equilibria; when p 2 < R 0 < p 1 , two endemic equilibria exist. We can see that the number of infected is proportional to R 0 if there is no transport. On the contrary, the number of infected does not always increase with R 0 when there is transport. Consequently, the proper transport might reduce the chance of infection in this area.
Next, we give the relationship between the transport rate B and I in Figure 3. As seen from the Figure 3, model (1) has two endemic equilibria when there is transport. When B < B * or B > B * and I < I * , the number of the infected always increases with the increase of B; when B > B * and I > I * , transport is good for disease control. So, the transport has dual effects on the number of I; it can not only promote the growth of brucellosis but also inhibit the growth of the disease. Consequently, the transport rate B plays a very important role.

Theorem 3. E * is locally asymptotically when
is conclusion can be easily obtained. e dynamic behavior of the stable node indicates that susceptible and infected animal populations can coexist for a long time.
e Jacobian matrix of system (1) with respect to E i is If detJ < 0, then E 1 is a saddle. Note that I < I 0 , and we have the following results [25][26][27].

Numerical Results
In an epidemic model, the basic reproduction number R 0 is an important parameter which can predict the dynamics of the disease. If R 0 < 1, it indicates that the disease can be controlled and can even disappear. Otherwise, the disease will outbreak when the basic reproduction number R 0 > 1. In this paper, when the endemic equilibrium exists, the basic reproduction number R 0 > 1. Next, we analyze the spread of the disease over time. Figure 4 represents time series of the infected animal populations with different initial values. We take B � 50, I 0 � 110, d � 0.9, μ � 0.3, β � 0.01, c � 0.5, and A � 441.99 with different initial values: (a) S 0 � 200, I 0 � 130; (b) S 0 � 100, I 0 � 80; (c) S 0 � 300, I 0 � 180; (d) S 0 � 300, I 0 � 100. One can conclude from this figure that there is great distinction between different initial values. As seen from Figure 4(b), the number of infected decreases over time. On the contrary, in the other three figures, the number of infected increases in the beginning. erefore, initial values play an important role in the control of the disease.
Next, we give sensitivity analysis of R 0 in order to find better control strategies. Different parameters are substituted into the expression R 0 � βA/ (d(d + u)). We use the normalized forward sensitivity index and derive an analytical expression [33]: Table 1 shows the sensitivity index of R 0 with respect to different parameters. We can conclude from Table 1 that R 0 is more sensitive to the recruitment rate A and the infection coefficient β, which means that the main influencing factors of R 0 are A and β. Since A and β are proportional to R 0 , the most effective way to reduce R 0 is to reduce the recruitment rate and the infection coefficient.

Conclusion and Discussion
Brucellosis has always been a research focus in infectious diseases. Measures to control the spread of brucellosis are vaccinating susceptible and culling infected animal populations. However, in fact, due to some practical factors, such as the incubation period of the disease and economic factor, the breeders will transport the animals containing the infected and fail to take effective measures in time. is case is tremendously overlooked despite its real existence in China. In order to study the influence of transport on the spread of brucellosis, we present an SI model with transport.
rough mathematical analysis, we obtain the basic reproduction number R 0 and the positive equilibria. e existence and global stability of these equilibria are analyzed. Our results reveal that initial values have important effects on the spread of disease.
Our work demonstrates that the transport has dual effects. Specifically, transport can lead to increase in the number of the infected; besides, transport can also reduce the number of the infected animals in a certain range. is is an interesting result. is indicates that transport can help control the disease for a time. Besides, sensitivity of the basic reproduction number R 0 is calculated. e results show that the main influencing factors are the recruitment rate and the infection coefficient. Some measures should be taken, such as vaccination and disinfection. We suggest that breeders  can increase animal vaccination coverage rate and enhance the awareness of disease prevention. e analysis shows that the number of infected animals can be controlled if animals are transported reasonably. However, we do not analyze how the disease spreads in the transported animals; this may not be good at disease control. Consequently, we will analyze the dynamic behavior of the animals which have been transported and give a better strategy to control brucellosis. Besides, infectious diseases spread in space, and spatial infectious disease models can describe the dynamics of disease transmission. erefore, we will integrate the diffusion term into the disease model in the future study [34][35][36].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.