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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 number

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 [

Many articles have studied brucellosis through dynamic models [

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 [

In this paper, we study an SI model of animal brucellosis with transport. We show that transport has dual effects. In Section

We study a two-dimensional model: densities of the susceptible (

Flow diagram of disease transmission. Here,

We assume that all parameters are nonnegative in this model.

When

When

(

Based on Ref. [

System (

Evidently, we have

Next, we calculate the positive solution of system (

Assume

If

Consequently, we have the following expression:

If the condition

This is equivalent to

By calculating, (

(

As a conclusion, when

If

It is equivalent to an inequality as follows:

We have that (

According to the discussions above, we can obtain two theorems as follows.

Note that

If

If

If

According to Theorems

Next, we give the relationship between the transport rate

The density of the infected animal populations as a function of basic reproduction number

The relationship between the transport rate

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 (

If

The unique endemic equilibrium

Define the Lyapunov function [

The derivative of

Since

When

In an epidemic model, the basic reproduction number

Figure

Time series of the infected animal populations with different initial values. Parameter values:

Next, we give sensitivity analysis of

Table

The sensitivity index of

Parameter | Sensitivity index |
---|---|

1 | |

1 | |

−1.836 | |

−4.282 |

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 number

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 number

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 [

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This research was supported by the Natural Science Foundation of Taiyuan Institute of Technology (grant no. 2016LZ02), Program for the (Reserved) Discipline Leaders of Taiyuan Institute of Technology (no. 2018008), and Natural Science Foundation of Shanxi Province (no. 201901D111322).