# Rich Dynamics of a Brucellosis Model with Transport

Authors: Juan Liang; Zhirong Zhao; Can Li
Journal: Complexity
 (2020)

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## Abstract

Brucellosis is one of the major infectious diseases in China. In this study, we consider an SI model of animal brucellosis with transport. The 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. The 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. The analysis shows that the number of infected animals can be controlled if animals are transported reasonably.


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## Body

## 1. 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. This 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. There is cross immunity among different Brucella species, and the re-infection rate is 2–7. Some scholars have studied the infectious diseases and obtained some results [1–8].Many articles have studied brucellosis through dynamic models [9–12]. 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 (R0 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. The mathematical model of sheep brucellosis and the effect of slaughtering policy have also been studied [15].The 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.There has been some studies on the impact of transport on disease transmission [16–18]. 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. They discussed the dependence of R0 on transport related infection parameters and fluctuation amplitude. This 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 Section2, 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.


## 2. 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. The detailed relation between susceptible and infected is given in Figure 1. Then, the model is expressed in the following form:(1)S˙=A−BC−βSI−dS,I˙=−B1−C+βSI−dI−uI,where(2)B≠0,ifI≤I0,B=0,ifI>I0.Figure 1
Flow diagram of disease transmission. Here,S denotes susceptible, I denotes infected, B is transport proportion, and C is the proportion of susceptible in the transport.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 disease-related death rate.WhenI≤I0, the steady state of (1) can be written as follows:(3)A−BC−βSI−dS=0,−B1−C+βSI−dI−uI=0.WhenI>I0, the steady state of (1) can be written as follows:(4)A−βSI−dS=0,βSI−dI−uI=0.(1) has no disease-free equilibrium.Based on Ref. [21], we can obtain the basic reproduction number ℛ0:(5)ℱ=βSI,ν=d+uI,F=βS=βAd,V=d+u,ℛ0=ρFV−1=βAdd+u.System (4) has a positive equilibrium E∗=S∗,I∗=d+u/β,dℛ0−1/β.Evidently, we have(6)ℛ0>βI0d+1.Next, we calculate the positive solution of system (3). Through the first equation of system (3), we obtain S=A−BC/βI+d. Then, we substitute it into the other equation of (3), so we have(7)βd+μI2+μd+d2+Bβ−AβI+Bd1−C=0.Assumeb=μd+d2+Bβ−Aβ. Note that A−BC>0.Ifb≥0, it is easy to know that there is no positive equilibrium; then, we consider b<0, where(8)b=μd+d2+Bβ−Aβ=−ℛdd+u+ud+d2+Bβ.Consequently, we have the following expression:(9)Δ=−−ℛdd+u+ud+d2+Bβdd+u+ud+d2+Bβ2−4βd+μ1−CBd.Δ≥0 is equivalent to(10)ℛ0≤1+Bβdd+u−2βd+μ1−CBddd+μ,ℛ0≥1+Bβdd+u+2βd+μ1−CBddd+μ=Δp0.b<0 is equivalent to(11)ℛ0≥1+Bβdd+u.b<0 and Δ≥0 if and only if (10) holds. If (10) holds, there are two solutions of (3):(12)I1=−b−Δ2βd+μ,I2=−b+Δ2βd+μ.If the conditionI1≤I0 holds, then we have(13)−b−Δ<2βd+μI0.This is equivalent to(14)b+2βd+μI0≤0,b+2βd+μI02≤Δ,or(15)b+2βd+μI0≥0.By calculating, (14) is equivalent to(16)ℛ0≥1+Bβdd+u+2βI0d=Δp1,ℛ0≥1+Bβdd+u+βI0d+1−CBd+uI0=Δp2.(15) is equivalent to(17)ℛ0≤p1.As a conclusion, whenℛ0≥maxp1,p2 or ℛ0≤p1, I1<I0 holds.IfI2<I0 holds, we have(18)−b+Δ≤2βd+μI0.It is equivalent to an inequality as follows:(19)b+2βd+μI0≥0,b+2βd+μI02≥Δ.We have that (19) is equivalent to(20)ℛ0≤minp1,p2.According to the discussions above, we can obtain two theorems as follows.Theorem 1.
E∗ is the unique endemic equilibrium of (1) if and only if βI0/d+1<ℛ0<p0.
Note thatp1<p2 is equivalent to βd+μI02<1−CBd. Besides, if Ei exist, they must satisfy ℛ0≥p0 which is equivalent to βd+μI02>1−CBd. Therefore, we can obtain that p1<p2 is impossible. Note that p1>p2 is equivalent to βd+μI02>1−CBd. So, we have the following.Theorem 2.
(1)
Ifℛ0<p0, endemic equilibria E1 and E2 do not exist.(2)
Ifp0<ℛ0<p2 and βd+μI02>1−CBd, then E1S1,I1 and E2S2,I2 exist.(3)
Ifp2<ℛ0<p1 and βd+μI02>1−CBd, then E1S1,I1 exists.According to Theorems1 and 2, the changes in the number of equilibrium with the basic reproduction number are shown in Figure 2. When p0<ℛ0<p2, there are three endemic equilibria; when p2<ℛ0<p1, two endemic equilibria exist. We can see that the number of infected is proportional to ℛ0 if there is no transport. On the contrary, the number of infected does not always increase with ℛ0 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 rateB 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.Figure 2
The density of the infected animal populations as a function of basic reproduction numberℛ0. P0 indicates that the equilibrium will appear when there is no transport, P1 indicates that the equilibrium will disappear in the case of transport, and P2 indicates that there are three equilibria in two cases. Parameter values: B=50,I0=110,d=0.9,μ=0.3,β=0.01,andC=0.5.Figure 3
The relationship between the transport rateB and the infected animal populations I. B∗ indicates that there are two equilibria. Parameter values: I0=110,d=0.9,μ=0.3,β=0.01,C=0.5, and A=300.


## 3. Stability Analysis

J is the Jacobian matrix of (1) with respect to E∗:(21)J=−dℛ0−d+udℛ0−10,with trJ=−dℛ0<0, detJ=dd+uℛ0−1. Then, λ1<0,λ2<0, and we have the following result [22–24].Theorem 3.
E∗ is locally asymptotically when ℛ0>β/dI0+1.
This conclusion can be easily obtained. The dynamic behavior of the stable node indicates that susceptible and infected animal populations can coexist for a long time.The Jacobian matrix of system (1) with respect to Ei is(22)J=−dℛ0−d+uβIβA−BCβI+d−d−μ,with(23)detJ=−βA−BC+d+μβI+d+β2IA−BCβI+d=d+uβI+d2<dβA−BC.IfdetJ<0, then E1 is a saddle. Note that I<I0, and we have the following results [25–27].Theorem 4.
(1)
E1 is a saddle if βI0+d2d+μ<dβA−BC.(2)
E2 is stable if(24)βA<βBC+β2I02+3dβI0+2d2+μβI0+μd,d+μβI0+d2+dβBC>maxdd+μ+2βI0d+μ+Bβ,d2+dμ+βdI0+βμI0+Bβ+1−CBdI0.(3)
E2 is unstable if either(25)βA>βBC+β2I02+3dβI0+2d2+μβI0+μd,d+μβI0+d2+dβBC>maxdd+μ+2βI0d+μ+Bβ,d2+dμ+βdI0+βμI0+Bβ+1−CBdI0,or(26)d+μβI0+d2+dβBC>maxdd+μ+2βI0d+μ+Bβ,d2+dμ+βdI0+βμI0+Bβ+1−CBdI0.Theorem 5.
The unique endemic equilibriumE∗S∗,I∗ of system (1) is globally asymptotically stable when ℛ0>1+β/dI0.Proof.
Define the Lyapunov function [28, 29]:(27)VS,I=S−S∗−S∗lnSS0+I−I∗−I∗lnII0.
The derivative ofVS,I along solutions of system (1) is(28)dVdt=S˙1−S∗S+I˙1−I∗I=A−βSI−dS1−S∗S+βSI−d+μI1−I∗I=dS∗2−SS∗−S∗S+βS∗I∗2−SS∗−S∗S.
SinceS/S∗−S∗/S<2, dV/dt=dS∗+βS∗I∗2−S/S∗−S∗/S≤0.
Whenℛ0>1+β/dI0, we have dV/dt<0. dV/dt=0 holds if and only if S=S∗,I=I∗. By Lyapunov–Lasalle theorem [30–32], the endemic equilibrium E∗S∗,I∗ is globally asymptotically stable when ℛ0>1+β/dI0.


## 4. Numerical Results

In an epidemic model, the basic reproduction numberℛ0 is an important parameter which can predict the dynamics of the disease. If ℛ0<1, it indicates that the disease can be controlled and can even disappear. Otherwise, the disease will outbreak when the basic reproduction number ℛ0>1. In this paper, when the endemic equilibrium exists, the basic reproduction number ℛ0>1. Next, we analyze the spread of the disease over time.Figure4 represents time series of the infected animal populations with different initial values. We take B=50,I0=110,d=0.9,μ=0.3,β=0.01,c=0.5,andA=441.99 with different initial values: (a) S0=200,I0=130; (b) S0=100,I0=80; (c) S0=300,I0=180; (d) S0=300,I0=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. Therefore, initial values play an important role in the control of the disease.Figure 4
Time series of the infected animal populations with different initial values. Parameter values:B=50,I0=110,d=0.9,μ=0.3,β=0.01,c=0.5,andA=441.99. Initial values: (a) S0=200,I0=130; (b) S0=100,I0=80; (c) S0=300,I0=180; (d) S0=300,I0=100.
(a)(b)(c)(d)Next, we give sensitivity analysis ofℛ0 in order to find better control strategies. Different parameters are substituted into the expression ℛ0=βA/dd+u. We use the normalized forward sensitivity index and derive an analytical expression [33]:(29)γpℛ0=∂ℛ0∂p×∂p∂ℛ0.Table1 shows the sensitivity index of ℛ0 with respect to different parameters. We can conclude from Table 1 that ℛ0 is more sensitive to the recruitment rate A and the infection coefficient β, which means that the main influencing factors of ℛ0 are A and β. Since A and β are proportional to ℛ0, the most effective way to reduce ℛ0 is to reduce the recruitment rate and the infection coefficient.Table 1
The sensitivity index ofℛ0 with respect to different parameters.
ParameterSensitivity indexβ1A1μ−1.836d−4.282


## 5. 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. This 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. Through mathematical analysis, we obtain the basic reproduction numberℛ0 and the positive equilibria. The 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. This is an interesting result. This indicates that transport can help control the disease for a time. Besides, sensitivity of the basic reproduction numberℛ0 is calculated. The 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.The 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. Therefore, we will integrate the diffusion term into the disease model in the future study [34–36].



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