Complexity

Complexity / 2020 / Article
Special Issue

Modelling and Simulation of Complex Biological Systems

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 5050393 | https://doi.org/10.1155/2020/5050393

Juan Liang, Zhirong Zhao, Can Li, "Rich Dynamics of a Brucellosis Model with Transport", Complexity, vol. 2020, Article ID 5050393, 7 pages, 2020. https://doi.org/10.1155/2020/5050393

Rich Dynamics of a Brucellosis Model with Transport

Academic Editor: Xinzhu Meng
Received11 Jun 2020
Accepted14 Jul 2020
Published06 Aug 2020

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 number 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 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.

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 [18].

Many articles have studied brucellosis through dynamic models [912]. 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 ( 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 [1618]. 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 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 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.

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 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, equals 0. The detailed relation between susceptible and infected is given in Figure 1. Then, the model is expressed in the following form:where

We assume that all parameters are nonnegative in this model.

is the recruitment rate of the animal populations, is the infection coefficient, is transport proportion, is the rate of susceptible in this transport, is the natural death rate of the animal populations, and is the disease-related death rate.

When , the steady state of (1) can be written as follows:

When , 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 :

System (4) has a positive equilibrium .

Evidently, we have

Next, we calculate the positive solution of system (3). Through the first equation of system (3), we obtain . Then, we substitute it into the other equation of (3), so we have

Assume . Note that .

If , it is easy to know that there is no positive equilibrium; then, we consider , where

Consequently, we have the following expression:

is equivalent to

is equivalent to

and if and only if (10) holds. If (10) holds, there are two solutions of (3):

If the condition holds, then we have

This is equivalent toor

By calculating, (14) is equivalent to

(15) is equivalent to

As a conclusion, when or , holds.

If holds, we have

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.

Theorem 1. is the unique endemic equilibrium of (1) if and only if .
Note that is equivalent to . Besides, if exist, they must satisfy which is equivalent to . Therefore, we can obtain that is impossible. Note that is equivalent to . So, we have the following.

Theorem 2. (1)If , endemic equilibria and do not exist.(2)If and , then and exist.(3)If and , then exists.

According to Theorems 1 and 2, the changes in the number of equilibrium with the basic reproduction number are shown in Figure 2. When , there are three endemic equilibria; when , two endemic equilibria exist. We can see that the number of infected is proportional to if there is no transport. On the contrary, the number of infected does not always increase with 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 and in Figure 3. As seen from the Figure 3, model (1) has two endemic equilibria when there is transport. When or and , the number of the infected always increases with the increase of ; when and , transport is good for disease control. So, the transport has dual effects on the number of ; it can not only promote the growth of brucellosis but also inhibit the growth of the disease. Consequently, the transport rate plays a very important role.

3. Stability Analysis

is the Jacobian matrix of (1) with respect to :with , . Then, , and we have the following result [2224].

Theorem 3. is locally asymptotically when .
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 iswith

If , then is a saddle. Note that , and we have the following results [2527].

Theorem 4. (1) is a saddle if .(2) is stable if(3) is unstable if eitheror

Theorem 5. The unique endemic equilibrium of system (1) is globally asymptotically stable when .

Proof. Define the Lyapunov function [28, 29]:The derivative of along solutions of system (1) isSince , .
When , we have . holds if and only if . By Lyapunov–Lasalle theorem [3032], the endemic equilibrium is globally asymptotically stable when .

4. Numerical Results

In an epidemic model, the basic reproduction number is an important parameter which can predict the dynamics of the disease. If , it indicates that the disease can be controlled and can even disappear. Otherwise, the disease will outbreak when the basic reproduction number . In this paper, when the endemic equilibrium exists, the basic reproduction number . 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 with different initial values: (a) ; (b) ; (c) ; (d) . 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.

Next, we give sensitivity analysis of in order to find better control strategies. Different parameters are substituted into the expression . We use the normalized forward sensitivity index and derive an analytical expression [33]:

Table 1 shows the sensitivity index of with respect to different parameters. We can conclude from Table 1 that is more sensitive to the recruitment rate and the infection coefficient , which means that the main influencing factors of are and . Since and are proportional to , the most effective way to reduce is to reduce the recruitment rate and the infection coefficient.


ParameterSensitivity index

1
1
−1.836
−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 number 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 number 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 [3436].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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).

References

  1. J. Zinsstag, F. Roth, D. Orkhon et al., “A model of animal-human brucellosis transmission in Mongolia,” Preventive Veterinary Medicine, vol. 69, no. 1-2, pp. 77–95, 2005. View at: Publisher Site | Google Scholar
  2. J. Gonzalez-Guzman and R. Naulin, “Analysis of a model of bovine brucellosis using singular perturbations,” Journal of Mathematical Biology, vol. 33, no. 2, pp. 211–223, 1994. View at: Publisher Site | Google Scholar
  3. G. Pappas, N. Akritidis, M. Bosilkovski, and E. Tsianos, “Brucellosis and the respiratory system,” Clinical Infectious Diseases, vol. 37, no. 7, pp. 95–99, 2003. View at: Publisher Site | Google Scholar
  4. M. Doganay and B. Aygen, “Human brucellosis: an overview,” International Journal of Infectious Diseases, vol. 7, no. 3, pp. 173–182, 2003. View at: Publisher Site | Google Scholar
  5. R. S. Cantrell, C. Cosner, and W. F. Faganantrell, “Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction,” Journal of Mathematical Biology, vol. 42, no. 2, pp. 95–119, 2001. View at: Publisher Site | Google Scholar
  6. M.-T. Li, G.-Q. Sun, Y.-F. Wu, J. Zhang, and Z. Jin, “Transmission dynamics of a multi-group brucellosis model with mixed cross infection in public farm,” Applied Mathematics and Computation, vol. 237, pp. 582–594, 2014. View at: Publisher Site | Google Scholar
  7. B. Ruben, J. D. Band, P. Wong, and J. Colville, “Person-to-person transmission of Brucella melitensis,” Lancet, vol. 337, no. 8732, pp. 14-15, 1982. View at: Publisher Site | Google Scholar
  8. G.-Q. Sun, C.-H. Wang, and Z.-Y. Wu, “Pattern dynamics of a Gierer-Meinhardt model with spatial effects,” Nonlinear Dynamics, vol. 88, no. 2, pp. 1385–1396, 2017. View at: Publisher Site | Google Scholar
  9. Q. Hou, X. Sun, Y. Wang, B. Huang, and Z. Jin, “Global properties of a general dynamic model for animal diseases: a case study of brucellosis and tuberculosis transmission,” Journal of Mathematical Analysis and Applications, vol. 414, no. 1, pp. 424–433, 2014. View at: Publisher Site | Google Scholar
  10. Q. Hou, X. Sun, J. Zhang, Y. Liu, Y. Wang, and Z. Jin, “Modeling the transmission dynamics of sheep brucellosis in inner Mongolia autonomous region, China,” Mathematical Biosciences, vol. 242, no. 1, pp. 51–58, 2013. View at: Publisher Site | Google Scholar
  11. G.-Q. Sun and Z.-K. Zhang, “Global stability for a sheep brucellosis model with immigration,” Applied Mathematics and Computation, vol. 246, pp. 336–345, 2014. View at: Publisher Site | Google Scholar
  12. N. Jing and G. Q. Sun, “Modeling the transmission dynamics of dairy cattle brucellosis in Jilin province, China,” Journal of Biological System, vol. 22, no. 04, pp. 533–554, 2014. View at: Google Scholar
  13. J. Yang, R. Xu, and J. Li, “Threshold dynamics of an age-space structured brucellosis disease model with Neumann boundary condition,” Nonlinear Analysis: Real World Applications, vol. 50, pp. 192–217, 2019. View at: Publisher Site | Google Scholar
  14. C. Y. Yang, P. O. Lolika, S. Mushayabasa, and J. Wang, “Modeling the spatiotemporal variations in brucellosis transmission,” Nonlinear Analysis: Real World Applications, vol. 38, pp. 49–67, 2017. View at: Publisher Site | Google Scholar
  15. B. Aïnseba, C. Benosman, and P. Magal, “A model for ovine brucellosis incorporating direct and indirect transmission,” Journal of Biological Dynamics, vol. 4, no. 1, pp. 2–11, 2010. View at: Publisher Site | Google Scholar
  16. J. Liu, J. Wu, and Y. Zhou, “Modeling disease spread via transport-related infection by a delay differential equation,” Rocky Mountain Journal of Mathematics, vol. 38, no. 5, pp. 1525–1540, 2008. View at: Publisher Site | Google Scholar
  17. X. Liu and Y. Takeuchi, “Spread of disease with transport-related infection and entry screening,” Journal of Theoretical Biology, vol. 242, no. 2, pp. 517–528, 2006. View at: Publisher Site | Google Scholar
  18. Y. Takeuchi, X. Liu, and J. Cui, “Global dynamics of SIS models with transport-related infection,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1460–1471, 2007. View at: Publisher Site | Google Scholar
  19. J. Liu, Z. Bai, and T. Zhang, “A periodic two-patch SIS model with time delay and transport-related infection,” Journal of Theoretical Biology, vol. 437, pp. 36–44, 2018. View at: Publisher Site | Google Scholar
  20. X. Liu and P. Stechlinski, “Transmission dynamics of a switched multi-city model with transport-related infections,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 264–279, 2013. View at: Publisher Site | Google Scholar
  21. O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990. View at: Publisher Site | Google Scholar
  22. A. Q. Khan, “Stability and Neimark-Sacker bifurcation of a ratio-dependence predator-prey model,” Mathematical Methods in the Applied Sciences, vol. 40, pp. 404109–404117, 2017. View at: Publisher Site | Google Scholar
  23. G.-Q. Sun, C.-H. Wang, L.-L. Chang, Y.-P. Wu, L. Li, and Z. Jin, “Effects of feedback regulation on vegetation patterns in semi-arid environments,” Applied Mathematical Modelling, vol. 61, pp. 200–215, 2018. View at: Publisher Site | Google Scholar
  24. L. Li, J. Zhen, and S. Gui-Quan, “Spatial pattern of an epidemic model with cross-diffusion,” Chinese Physics Letters, vol. 25, no. 9, pp. 3500–3503, 2008. View at: Publisher Site | Google Scholar
  25. G.-Q. Sun, J.-H. Xie, S.-H. Huang, Z. Jin, M.-T. Li, and L. Liu, “Transmission dynamics of cholera: mathematical modeling and control strategies,” Communications in Nonlinear Science and Numerical Simulation, vol. 45, pp. 235–244, 2017. View at: Publisher Site | Google Scholar
  26. L. Li, G.-Q. Sun, and Z. Jin, “Bifurcation and chaos in an epidemic model with nonlinear incidence rates,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1226–1234, 2010. View at: Publisher Site | Google Scholar
  27. G.-Q. Sun, “Pattern formation of an epidemic model with diffusion,” Nonlinear Dynamics, vol. 69, no. 3, pp. 1097–1104, 2012. View at: Publisher Site | Google Scholar
  28. X. Zhang and X. Liu, “Backward bifurcation of an epidemic model with saturated treatment function,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433–443, 2008. View at: Publisher Site | Google Scholar
  29. W. D. Wang, “Backward bifurcation of an epidemic model with treatment,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 58–71, 2006. View at: Publisher Site | Google Scholar
  30. J. Hespanha, “Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle,” IEEE Transactions on Automatic Control, vol. 49, no. 4, pp. 470–482, 2004. View at: Publisher Site | Google Scholar
  31. J. P. LaSalle, “Stability theory for ordinary differential equations,” Journal of Differential Equations, vol. 4, no. 1, pp. 57–65, 1968. View at: Publisher Site | Google Scholar
  32. J. P. LaSalle, “Stability theory and invariance principles,” Dynamical Systems, vol. 1, pp. 211–222, 1976. View at: Publisher Site | Google Scholar
  33. N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” Bulletin of Mathematical Biology, vol. 70, no. 5, pp. 1272–1296, 2008. View at: Publisher Site | Google Scholar
  34. G.-Q. Sun, M. Jusup, Z. Jin, Y. Wang, and Z. Wang, “Pattern transitions in spatial epidemics: mechanisms and emergent properties,” Physics of Life Reviews, vol. 19, pp. 43–73, 2016. View at: Publisher Site | Google Scholar
  35. Z. G. Guo, G. Q. Sun, Z. Wang et al., “Spatial dynamics of an epidemic model with nonlocal infection,” Applied Mathematics and Computation, vol. 377, Article ID 125158, 2020. View at: Publisher Site | Google Scholar
  36. Z. G. Guo, L. P. Song, G. Q. Sun, C. Li, and Z. Jin, “Pattern dynamics of an SIS epidemic model with nonlocal delay,” International Journal of Bifurcation and Chaos, vol. 29, no. 2, Article ID 1950027, 2019. View at: Publisher Site | Google Scholar

Copyright © 2020 Juan Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views123
Downloads169
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.