Research Article | Open Access
Wei Zhang, Juan Zhang, Yong-Ping Wu, Li Li, "Dynamical Analysis of the SEIB Model for Brucellosis Transmission to the Dairy Cows with Immunological Threshold", Complexity, vol. 2019, Article ID 6526589, 13 pages, 2019. https://doi.org/10.1155/2019/6526589
Dynamical Analysis of the SEIB Model for Brucellosis Transmission to the Dairy Cows with Immunological Threshold
As we all know, bacteria is different from virus which with certain types can be killed by the immune cells in the body. The brucellosis, a bacterial disease, can invade the body by indirect transmission from environment, which has not been researched by combining with immune cells. Considering the effects of immune cells, we put a minimum infection dose of brucellosis invading into the dairy cows as an immunological threshold and get a switch model. In this paper, we accomplish a thorough dynamics analysis of a switch model. On the one hand, we can get a disease-free and bacteria-free steady state and up to three endemic steady states which may be thoroughly analyzed in different cases of a minimum infection dose in a switch model. On the other hand, we calculate the basic reproduction number and know that the disease-free and bacteria-free steady state is a global stability when , and the one of the endemic steady state is a conditionally global stability when . We find that different amounts of may lead to different steady states of brucellosis, and considering the effects of immunology is more serious in mathematics and biology.
Brucellosis is the zoonotic sex contagion which is named as Mediterranean relax heat, wave heat, or wave form heat [1–3]. It is characterized by extensive host, strong infectivity, and difficulty in radical treatment after infection [4, 5]. Besides, it has a serious harm to economy, society, and public health. Therefore, brucellosis is listed as one of the communicable diseases that must be notified in the World Organization for Animal Health and is classified as second kind of animal diseases in China. Meanwhile, it is as the first zoonotic disease to be controlled in the National Medium and Long Term Plan for Animal Disease Prevention and Control 2011-2020 formulated by the ministry of agriculture. Brucellosis is acute or chronic infectious disease caused by Brucella which is a group of small bulbous gram-negative bacteria [6, 7]. In 1985, the World Health Organization (WHO) divided Brucella into six species and nineteen biological types . They are more popular with Br. melitensis, Br. brovis, and Br. suis in China, among which the most popular Brucella is Br. melitensis and the second is Br. brovis. The most noticeable symptom of brucellosis is miscarriage in cows and orchitis in bulls. In addition, it can cause arthritis in the knee and wrist frequently [2, 3]. The route of spread is through direct contact with diseased animals and indirect infection with bacteria in the environment [8, 9]. And it can be divided into three ways. The first transmitted way is contacting with skin, such as direct contact with droppings, vaginal secretions, and vaginal delivery content of sick animals. It also can be indirectly exposed to the environment and objects contaminated by sick animals. The second infected way is via the digestive tract, like eating food, water, or milk contaminated with pathogens. And the last way of transmission is through the respiratory tract.
When bacteria enter the body, mainly in the liver, spleen, bone marrow, lymphatic tissue, and other cells, those will grow, multiply, and produce endotoxin which can cause damaged tissues. In the meantime, it often releases bacteria and toxins into the blood causing systemic bacteremia and allergic reactions. But the Brucella does not take effect immediately when it enters the body. The bacteria with a small number will be killed by immune cells and only amounts of bacteria over a certain threshold called a minimum infection dose (MID) in environment can do harm to body [10, 11]. And different bacteria have variant thresholds which should be tested by experiment. Some bacteria threshold can be seen in . Also, there were many authors researching the brucellosis, who had been unconscious of the impact on immunization [13–19]. By referring to [20, 21], we will put the pathogeny of brucellosis and MID into this paper. The authors [12, 20, 21] incorporated a MID, , into the incidence term , in a cholera transmission model, which was a piecewise continuous function which was zero under the MID and was a response curve over that. where the parameter represents MID, is half-saturation pathogen density, and the description of parameter is maximum rate of infection. One of typical mathematical models given in  is as follows:where all parameters are positive and the description of parameters can be found in Table 1.
In , the authors gave a detailed demonstration of the basic reproduction number , which can estimate the occurrence of the epidemic, and made brief dynamical analysis on the global stability.
Basing on model (2) and considering the immunology of Brucella in the environment, we get a switch system with MID . If the number of Brucella in environment is less than or equal to the MID, then we can get a globally asymptotically stable disease-free and bacteria-free steady state while the basic reproduction number is less than or equal to 1, an unstable disease-free and bacteria-free equilibrium and a conditionally globally stable endemic steady state when the basic reproduction number is more than 1. Else if , it is going to be a little bit complicated. Combining with the relationship between parameter and (which is the formula received from the system), we can attain the three different switch systems. One of them can attain four equilibria and can produce a backward bifurcation [23, 24] and might be two steady states at most under certain parameter conditions.
2. Dynamical Analysis of the Model
2.1. The Dynamical Model
We classify the dairy cows into three compartments: the susceptible compartment , the exposed compartment , and the infectious compartment in Figure 1. And the Brucella in environment is denoted by .
One of the key differences of (2) is the incidence term of indirect transmission of Brucella to susceptible cow , where is the pathogen density dependent component. Unlike [12, 20] using response curve when the bacteria density is over the MID, a simple linear form is used in this paper. There are chiefly the following reasons. Firstly, the half-saturation pathogen density () is difficult to be determined in actual application. Secondly, the Brucella which is discharged to the nature by dairy cows and other livestock is hard to reach saturation. Lastly, that can get more practical conclusions with immunological threshold than . The expression of in this paper is defined as
If the number of bacteria is more than MID, then the Brucella of environment can bring the indirect influence to the dairy cows. At this time, is representative of maximum rate of indirect infection, and is denoted MID of Brucella, is representative of bacteria amount which can enter the body and take effect. If the number is less than or equal to MID, then the bacteria of environment have no effect on the susceptible and dairy cows were infected just by the direct transmission from infected to susceptible cow.
Remark 1. In this paper, we also use the condition like .
2.2. Forward Invariance
The first equation of (4) gives us that , when . In the same way, we can get and . Hence, we know that , , and for . As , we can get by using the character of . Cause is a linear function . In the same way, if , then . Thus, the inequality for is true. If , then for any . To sum up, we get the following theorem.
Theorem 2 (feasible region). The setdefines a forward invariant region of system (2).
2.3. Equilibria of System (4)
2.3.1. Equilibria When Bacteria Are Less Than or Equal to MID
When , then and the model is
We solve the equation and can get , a disease-free and bacteria-free equilibrium where , and a endemic equilibrium , where
Remark 3. , , and .
According to the spectral radius theory [25, 26], we can directly obtain the basic reproduction number based on  when . That is Then we can attain the existence of equilibria: if , then there is only one equilibrium ; if , then there are disease-free and bacteria-free equilibrium and endemic equilibrium .
2.3.2. Equilibria When Bacteria Are More Than MID
When , the equation isObviously, there is no disease-free and bacteria-free equilibria because of . Thus, system (9) implies that
The equation of about can be written as . Define . From the expression, we can know that is a quadratic function, in which the first and third coefficients are positive. Therefore, the existence of solution for depends on the second coefficient and . If the second coefficient is nonnegative, no matter is, then has no positive solution. If the second coefficient is negative, the positive solution depends on the sign of : when , there is no solution; when , there is one positive solution; and when , there are two positive solutions.
Now we simplify them and can get the new second coefficient :where and the new about :
We can know is a liner function about and is still a quadratic function which the first and third coefficients are positive and the second coefficient is less than zero. Then we calculate the and can know that the sign of is always positive. So we can solve and get , , where
By the analysis with the feature of function, we can get the conclusion as follows.
Proposition 4. The existence of positive solutions for :
(i) When : if , then does not get any positive solution; if , then has one positive solution ; if , then has two positive solutions , ().
(ii) When : if , then does not get any positive solution; if or , then has one positive solution (); if or , then has two positive solutions , .
It just discusses the existence of positive solutions, but we need consider the value of , where only internal equilibria exist while positive solutions are more than . Hence, we will get conclusions as follows.
Theorem 5. The existence of internal equilibria about :(1)If :(i)when , there is an internal equilibrium ;(ii)when , there is an internal equilibrium ;(iii)when , there are two internal equilibria , ;(iv)when , there is an internal equilibrium ;(2)If and , then there is an internal equilibrium . where
Proof. We have known the positive value of ; now we need combine with the inequality . The proof uses character of function .
Firstly, we should know the value of . When , we can get , and the relationship between and is translated into the relationship between and : (1)When : if , then , if , then , and if , then .(2)When , we can get .Secondly, we need to know position of and symmetry axis of . We put them into the formula In the same way, we translate them into the relationship between and . It means that when : if , then ( is on the left of axis of symmetry); if , then ( is the axis of symmetry); and if , then ( is on the right of axis of symmetry).
Thirdly, we calculate the relationship of , , , . Then we can get the relationships among them as follows: (1)The relationships of and are always true.(2)When , there is . When , there is . When , there is Finally, we use the above-mentioned relational expressions and combine with Proposition 4; then, we can obtain the final chart as shown in Table 4.
Because positive solutions of is equal to existences of internal equilibria of system. Thus, the existence of internal equilibria about is proved.
Remark 6. is the criterion when .
After knowing internal equilibria when , we want to get the relationship between internal equilibria and , so we need to change the form above all. We had known the . Thus, we change to , to , and to . From the expression of these, we get when and when .
Theorem 7 (existence of equilibria in Figures 2 and 3). The equilibrium always exists in . Here are the existences of internal equilibria: (1)If :(i)If : when , there only exists ; when , there exist , ; when , there exist , , ; when , there exist , ; when , there exists .(ii)If : when , there exists ; when , there exist , , ; when , there exist , ; when , there exists .(iii)If : when , there exists ; when , there exist , ; when , there exist , , ; when , there exist , ; when , there exists .(2)If : when , there exists ; when , there exists .(3)If (in this case, .): when , there exists .
Theorem 7 describes the existence of equilibria under the different parameters condition (Table 5), where disease-free and bacteria-free equilibrium always exists. Besides, we give the stability of the equilibria, which will be proved in next part. We just give the stability of and ; others are too difficult to give the strict mathematical proofs because of lacking exact expression. Therefore, the local stability of and will be demonstrated numerically in Section 4.
3. Stability of Equilibria
3.1. Local Stability of and
We calculate the Jacobian to analyze the local stability of each of the equilibria. When , the Jacobian isNow considering , we use for eigenvalues. We can computeNote . We can find that , . Now we just consider the eigenvalues of . The coefficients of the areIf , then ; if , then . Thus the eigenvalues of are negative when . So the disease-free and bacteria-free equilibria are locally asymptotically stable while . If , then is a saddle-point equilibrium.
Now considering . The characteristic polynomial of Jacobian of is where The coefficients of the are where and , when .
As for the condition , we have the following expression: where
Now, we magnify them:
We know that , so and then . Hence, the