Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China

Linhua Zhou; Meng Fan; Qiang Hou; Zhen Jin; Xiangdong Sun

doi:10.3934/mbe.2018025

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April 2018, 15(2): 543-567. doi: 10.3934/mbe.2018025

Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China

Linhua Zhou ^1, , Meng Fan ^2,, , Qiang Hou ^3, , Zhen Jin ^4, and Xiangdong Sun ^5,

1.	Department of Applied Mathematics, School of Science, Changchun University of Science and Technology, Changchun 130022, China
2.	School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
3.	School of Mechatronic Engineering, North University of China, Taiyuan 030051, China
4.	Complex Systems Research Center, Shanxi University Taiyuan 030006, China
5.	China Animal Health And Epidemiology Center, Qingdao 266032, China

* Corresponding author: Meng Fan

Received November 28, 2016 Accepted March 15, 2017 Published June 2017

A multigroup model is developed to characterize brucellosis transmission, to explore potential effects of key factors, and to prioritize control measures. The global threshold dynamics are completely characterized by theory of asymptotic autonomous systems and Lyapunov direct method. We then formulate a multi-objective optimization problem and, by the weighted sum method, transform it into a scalar optimization problem on minimizing the total cost for control. The existence of optimal control and its characterization are well established by Pontryagin's Maximum Principle. We further parameterize the model and compute optimal control strategy for Inner Mongolia in China. In particular, we expound the effects of sheep recruitment, vaccination of sheep, culling of infected sheep, and health education of human on the dynamics and control of brucellosis. This study indicates that current control measures in Inner Mongolia are not working well and Brucellosis will continue to increase. The main finding here supports opposing unregulated sheep breeding and suggests vaccination and health education as the preferred necessary emergency intervention control. The policymakers must take a new look at the current control strategy, and, in order to control brucellosis better in Inner Mongolia, the governments have to preemptively press ahead with more effective measures.

Keywords: Brucellosis, epidemic model, dynamical analysis, global stability, optimal control.

Mathematics Subject Classification: Primary: 92D30, 34D23; Secondary: 49J15.

Citation: Linhua Zhou, Meng Fan, Qiang Hou, Zhen Jin, Xiangdong Sun. Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China. Mathematical Biosciences & Engineering, 2018, 15 (2) : 543-567. doi: 10.3934/mbe.2018025

References:

[1]	B. Aïnseba, C. Benosman and P. Magal, A model for ovine brucellosis incorporating direct and indirect transmission, J. Biol. Dyn., 4 (2010), 2-11. doi: 10.1080/17513750903171688. Google Scholar
[2]	A. H. Al-Talafhah, S. Q. Lafi and Y. Al-Tarazi, Epidemiology of ovine brucellosis in Awassi sheep in Northern Jordan, Prev. Vet. Med., 60 (2003), 297-306. doi: 10.1016/S0167-5877(03)00127-2. Google Scholar
[3]	H. K. Aulakh, P. K. Patil, S. Sharma, H. Kumar, V. Mahajan and K. S. Sandhu, A study on the Epidemiology of Bovine Brucellosis in Punjab (India) Using Milk-ELISA, Acta Vet. Brun., 77 (2008), 393-399. doi: 10.2754/avb200877030393. Google Scholar
[4]	H. W. Boone, Malta fever in China, China Medical Mission, 19 (1905), 167-173. Google Scholar
[5]	R. S. Cantrell, C. Cosner and W. F. Fagan, Brucellosis, botflies, and brainworms: The impact of edge habitats on pathogen transmission and species extinction, J. Math. Biol., 42 (2001), 95-119. doi: 10.1007/s002850000064. Google Scholar
[6]	W. J. Chen, B. Y. Cui and Q. H. Zhang, Analysis of epidemic characteristics on brucellosis in Inner Mongolia, Chinese Journal of Control of Endemic Disease, 23 (2008), 56-58. Google Scholar
[7]	T. J. Clayton, Optimal Cotnrol of Epidemic Models Involving Rabies and West Nile Viruses Ph. D thesis, University of Tennessee, 2008.Google Scholar
[8]	Committee of China Animal Husbandry Yearbook Editors, China Animal Husbandry Yearbook, China Agriculture Press, Beijing, 2011.Google Scholar
[9]	J. B. Ding, K. R. Mao, J. S. Cheng, Z. H. Dai and Y. W. Jiang, The application and research advances of Brucella vaccines, Acta Microbiol. Sin., 46 (2006), 856. Google Scholar
[10]	A. D'Orazi, M. Mignemi and F. Geraci, Spatial distribution of brucellosis in sheep and goats in Sicily from 2001 to 2005, Vet. Ital., 43 (2007), 541-548. Google Scholar
[11]	J. F. Du and J. Zhang, Analysis of brucellosis monitoring results in Hexigten Banner, Chinese Journal of Epidemiology, 22 (2003), 459-461. Google Scholar
[12]	W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Controls, SpringerVerlag, New York, 1975. Google Scholar
[13]	S. L. Gang, Y. L. Zhao and S. Y. Zhang, Analysis of 1990 to 2001surveillance effect of national major surveilla nce place for brucellosis, Chinese Journal of Control of Endemic Disease, 5 (2002), 285-288. Google Scholar
[14]	J. Gonzíez-Guzmán and R. Naulin, Analysis of a model of bovine brucellosis using singular perturbations, J. Math. Biol., 33 (1994), 211-223. doi: 10.1007/BF00160180. Google Scholar
[15]	W. D. Guo and H. Y. Chi, Epidemiological analysis of human brucellosis in Inner Mongolia Autonomous Region from 2002-2006, China Tropical Medicine, 8 (2008), 604-606. Google Scholar
[16]	Q. Hou, X. D. Sun, J. Zhang, Y. J. Liu, Y. M. Wang and Z. Jin, Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region, China, Math. Biosci., 242 (2013), 51-58. doi: 10.1016/j.mbs.2012.11.012. Google Scholar
[17]	X. W. Hu, What is the best prevention and control of brucella?, Veterinary Orientation, 15 (2015), 16-18. Google Scholar
[18]	A. Konak, D. W. Coitb and A. E. Smithc, Multi-objective optimization using genetic algorithms: A tutorial, Reliab. Eng. Syst. Safe., 91 (2006), 992-1007. doi: 10.1016/j.ress.2005.11.018. Google Scholar
[19]	D. L. Lukes, Differential Equations: Classical to Controlled, vol. 162 of Mathematics in Science and Engineering, Academic Press, New York, 1982. Google Scholar
[20]	J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pa. , 1976. Google Scholar
[21]	S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, 2007. Google Scholar
[22]	T. F. Man, D. L. Wang and B. Y. Cui, Analysis on surveillance data of brucellosis in China, 2009, Disease Surveillance, 25 (2010), 944-946. Google Scholar
[23]	L. Markus, Asymptotically autonomous differential systems, In: S. Lefschetz (ed. ), Contributions to the Theory of Nonlinear Oscillations III, Princeton: Princeton University Press, Annals of Mathematics Studies, 3 (1956), 17–29. Google Scholar
[24]	R. T. Marler and J. S. Arora, Survey of multi-objective optimization methods for engineering, Struct. Multidiscip. Optim., 26 (2004), 369-395. doi: 10.1007/s00158-003-0368-6. Google Scholar
[25]	Ministry of Health of the People's Republic of China, China Health Statistics Yearbook, People's Medical Publishing House Beijing, 2011.Google Scholar
[26]	J. B. Muma, N. Toft, J. Oloya, A. Lund, K. Nielsen, K. Samui and E. Skjerve, Evaluation of three serological tests for brucellosis in naturally infected cattle using latent class analysis, Vet. Microbiol., 125 (2007), 187-192. doi: 10.1016/j.vetmic.2007.05.012. Google Scholar
[27]	National Bureau of Statistics of China, China Population Statistics Yearbook, China Statistical Publishing House, Beijing, 2010, Availiable from: http://www.stats.gov.cn/tjsj/ndsj/2010/indexch.htm.Google Scholar
[28]	R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 75 (2010), 67-81. Google Scholar
[29]	G. Pappas, P. Papadimitriou, N. Akritidis, L. Christou and E. V. Tsianos, The new global map of human brucellosis, Lancet Infect. Dis., 6 (2006), 91-99. doi: 10.1016/S1473-3099(06)70382-6. Google Scholar
[30]	D. R. Piao, Y. L. Li, H. Y. Zhao and B. Y. Cui, Epidemic situation analysis of human brucellosis in Inner Mongolia during 1952 to 2007, Chinese Journal of Epidemiology, 28 (2009), 420-423. Google Scholar
[31]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, The mathematical theory of optimal processes, Trudy Mat.inst.steklov, 16 (1962), 119-158. Google Scholar
[32]	D. Q. Shang, D. L. Xiao and J. M. Yin, Epidemiology and control of brucellosis in China, Vet. Microbiol., 90 (2002), p165. Google Scholar
[33]	X. T. Shi, S. P. Gu, M. X. Zheng and H. L. Ma, The prevalence and control of brucellosis disease, China Animal Husbandry and Veterinary Medicine, 37 (2010), 204-207. Google Scholar
[34]	J. L. Solorio-Rivera, J. C. Segura-Correa and L. G. Sánchez-Gil, Seroprevalence of and risk factors for brucellosis of goats in herds of Michoacan, Mexico, Prev. Vet. Med., 82 (2007), 282-290. doi: 10.1016/j.prevetmed.2007.05.024. Google Scholar
[35]	The Sate Council of China, National animal disease prevention and control for the medium and long term planning (2012-2020), 2012. Availiable from: http://www.gov.cn/zwgk/2012-05/25/content_2145581.htm.Google Scholar
[36]	H. R. Thieme, Convergence results and a Poinca'e-Bendixson trichotomy for asymptotically automous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. Google Scholar
[37]	P. Van Den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar
[38]	D. L. Wang, T. F. Li, S. L. Jiang, F. Q. Liu and J. Q. Wang, Analysis of national brucellosis surveillance in 2007, Chinese Journal of Control of Endemic Disease, 23 (2008), 443-445. Google Scholar
[39]	H. B. Wang, Z. M. Wang and L. Dong, Investigation of brucellosis epidemic situation in northwest of Inner Mongolia chifeng city from 2003 to 2004, Endemic. Dis. Bull., 26 (2006), 67-68. Google Scholar
[40]	L. J. Wu, Prevention and control of livestock brucella in Inner Mongolia, Veterinary Orientation, 9 (2012), 17-19. Google Scholar
[41]	W. W. Yin and H. Sun, Epidemic situation and strategy proposal for human brucellosis in China, Disease Surveillance, 24 (2009), 475-477. Google Scholar
[42]	F. Y. Zhao and T. Z. Li, A study on brucellosis prevention and vaccination, Veterinary Orientation, 151 (2010), 24-27. Google Scholar
[43]	Y. L. Zhao, D. L. Wang and S. L. Giang, Analysis on the surveillance results of the national main monitoring station of the brucellosis in 2001 to 2004, Chinese Journal of Control of Endemic Disease, 42 (2005), 120-134. Google Scholar
[44]	J. Zinsstag, F. Roth, D. Orkhon, G. Chimed-Ochir, M. Nansalmaa, J. Kolar and P. Vounatsou, A model of animal human brucellosis transmission in Mongolia, Prev. Vet. Med., 69 (2005), 77-95. doi: 10.1016/j.prevetmed.2005.01.017. Google Scholar

show all references

References:

[1]	B. Aïnseba, C. Benosman and P. Magal, A model for ovine brucellosis incorporating direct and indirect transmission, J. Biol. Dyn., 4 (2010), 2-11. doi: 10.1080/17513750903171688. Google Scholar
[2]	A. H. Al-Talafhah, S. Q. Lafi and Y. Al-Tarazi, Epidemiology of ovine brucellosis in Awassi sheep in Northern Jordan, Prev. Vet. Med., 60 (2003), 297-306. doi: 10.1016/S0167-5877(03)00127-2. Google Scholar
[3]	H. K. Aulakh, P. K. Patil, S. Sharma, H. Kumar, V. Mahajan and K. S. Sandhu, A study on the Epidemiology of Bovine Brucellosis in Punjab (India) Using Milk-ELISA, Acta Vet. Brun., 77 (2008), 393-399. doi: 10.2754/avb200877030393. Google Scholar
[4]	H. W. Boone, Malta fever in China, China Medical Mission, 19 (1905), 167-173. Google Scholar
[5]	R. S. Cantrell, C. Cosner and W. F. Fagan, Brucellosis, botflies, and brainworms: The impact of edge habitats on pathogen transmission and species extinction, J. Math. Biol., 42 (2001), 95-119. doi: 10.1007/s002850000064. Google Scholar
[6]	W. J. Chen, B. Y. Cui and Q. H. Zhang, Analysis of epidemic characteristics on brucellosis in Inner Mongolia, Chinese Journal of Control of Endemic Disease, 23 (2008), 56-58. Google Scholar
[7]	T. J. Clayton, Optimal Cotnrol of Epidemic Models Involving Rabies and West Nile Viruses Ph. D thesis, University of Tennessee, 2008.Google Scholar
[8]	Committee of China Animal Husbandry Yearbook Editors, China Animal Husbandry Yearbook, China Agriculture Press, Beijing, 2011.Google Scholar
[9]	J. B. Ding, K. R. Mao, J. S. Cheng, Z. H. Dai and Y. W. Jiang, The application and research advances of Brucella vaccines, Acta Microbiol. Sin., 46 (2006), 856. Google Scholar
[10]	A. D'Orazi, M. Mignemi and F. Geraci, Spatial distribution of brucellosis in sheep and goats in Sicily from 2001 to 2005, Vet. Ital., 43 (2007), 541-548. Google Scholar
[11]	J. F. Du and J. Zhang, Analysis of brucellosis monitoring results in Hexigten Banner, Chinese Journal of Epidemiology, 22 (2003), 459-461. Google Scholar
[12]	W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Controls, SpringerVerlag, New York, 1975. Google Scholar
[13]	S. L. Gang, Y. L. Zhao and S. Y. Zhang, Analysis of 1990 to 2001surveillance effect of national major surveilla nce place for brucellosis, Chinese Journal of Control of Endemic Disease, 5 (2002), 285-288. Google Scholar
[14]	J. Gonzíez-Guzmán and R. Naulin, Analysis of a model of bovine brucellosis using singular perturbations, J. Math. Biol., 33 (1994), 211-223. doi: 10.1007/BF00160180. Google Scholar
[15]	W. D. Guo and H. Y. Chi, Epidemiological analysis of human brucellosis in Inner Mongolia Autonomous Region from 2002-2006, China Tropical Medicine, 8 (2008), 604-606. Google Scholar
[16]	Q. Hou, X. D. Sun, J. Zhang, Y. J. Liu, Y. M. Wang and Z. Jin, Modeling the transmission dynamics of sheep brucellosis in Inner Mongolia Autonomous Region, China, Math. Biosci., 242 (2013), 51-58. doi: 10.1016/j.mbs.2012.11.012. Google Scholar
[17]	X. W. Hu, What is the best prevention and control of brucella?, Veterinary Orientation, 15 (2015), 16-18. Google Scholar
[18]	A. Konak, D. W. Coitb and A. E. Smithc, Multi-objective optimization using genetic algorithms: A tutorial, Reliab. Eng. Syst. Safe., 91 (2006), 992-1007. doi: 10.1016/j.ress.2005.11.018. Google Scholar
[19]	D. L. Lukes, Differential Equations: Classical to Controlled, vol. 162 of Mathematics in Science and Engineering, Academic Press, New York, 1982. Google Scholar
[20]	J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pa. , 1976. Google Scholar
[21]	S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, 2007. Google Scholar
[22]	T. F. Man, D. L. Wang and B. Y. Cui, Analysis on surveillance data of brucellosis in China, 2009, Disease Surveillance, 25 (2010), 944-946. Google Scholar
[23]	L. Markus, Asymptotically autonomous differential systems, In: S. Lefschetz (ed. ), Contributions to the Theory of Nonlinear Oscillations III, Princeton: Princeton University Press, Annals of Mathematics Studies, 3 (1956), 17–29. Google Scholar
[24]	R. T. Marler and J. S. Arora, Survey of multi-objective optimization methods for engineering, Struct. Multidiscip. Optim., 26 (2004), 369-395. doi: 10.1007/s00158-003-0368-6. Google Scholar
[25]	Ministry of Health of the People's Republic of China, China Health Statistics Yearbook, People's Medical Publishing House Beijing, 2011.Google Scholar
[26]	J. B. Muma, N. Toft, J. Oloya, A. Lund, K. Nielsen, K. Samui and E. Skjerve, Evaluation of three serological tests for brucellosis in naturally infected cattle using latent class analysis, Vet. Microbiol., 125 (2007), 187-192. doi: 10.1016/j.vetmic.2007.05.012. Google Scholar
[27]	National Bureau of Statistics of China, China Population Statistics Yearbook, China Statistical Publishing House, Beijing, 2010, Availiable from: http://www.stats.gov.cn/tjsj/ndsj/2010/indexch.htm.Google Scholar
[28]	R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 75 (2010), 67-81. Google Scholar
[29]	G. Pappas, P. Papadimitriou, N. Akritidis, L. Christou and E. V. Tsianos, The new global map of human brucellosis, Lancet Infect. Dis., 6 (2006), 91-99. doi: 10.1016/S1473-3099(06)70382-6. Google Scholar
[30]	D. R. Piao, Y. L. Li, H. Y. Zhao and B. Y. Cui, Epidemic situation analysis of human brucellosis in Inner Mongolia during 1952 to 2007, Chinese Journal of Epidemiology, 28 (2009), 420-423. Google Scholar
[31]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, The mathematical theory of optimal processes, Trudy Mat.inst.steklov, 16 (1962), 119-158. Google Scholar
[32]	D. Q. Shang, D. L. Xiao and J. M. Yin, Epidemiology and control of brucellosis in China, Vet. Microbiol., 90 (2002), p165. Google Scholar
[33]	X. T. Shi, S. P. Gu, M. X. Zheng and H. L. Ma, The prevalence and control of brucellosis disease, China Animal Husbandry and Veterinary Medicine, 37 (2010), 204-207. Google Scholar
[34]	J. L. Solorio-Rivera, J. C. Segura-Correa and L. G. Sánchez-Gil, Seroprevalence of and risk factors for brucellosis of goats in herds of Michoacan, Mexico, Prev. Vet. Med., 82 (2007), 282-290. doi: 10.1016/j.prevetmed.2007.05.024. Google Scholar
[35]	The Sate Council of China, National animal disease prevention and control for the medium and long term planning (2012-2020), 2012. Availiable from: http://www.gov.cn/zwgk/2012-05/25/content_2145581.htm.Google Scholar
[36]	H. R. Thieme, Convergence results and a Poinca'e-Bendixson trichotomy for asymptotically automous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. Google Scholar
[37]	P. Van Den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar
[38]	D. L. Wang, T. F. Li, S. L. Jiang, F. Q. Liu and J. Q. Wang, Analysis of national brucellosis surveillance in 2007, Chinese Journal of Control of Endemic Disease, 23 (2008), 443-445. Google Scholar
[39]	H. B. Wang, Z. M. Wang and L. Dong, Investigation of brucellosis epidemic situation in northwest of Inner Mongolia chifeng city from 2003 to 2004, Endemic. Dis. Bull., 26 (2006), 67-68. Google Scholar
[40]	L. J. Wu, Prevention and control of livestock brucella in Inner Mongolia, Veterinary Orientation, 9 (2012), 17-19. Google Scholar
[41]	W. W. Yin and H. Sun, Epidemic situation and strategy proposal for human brucellosis in China, Disease Surveillance, 24 (2009), 475-477. Google Scholar
[42]	F. Y. Zhao and T. Z. Li, A study on brucellosis prevention and vaccination, Veterinary Orientation, 151 (2010), 24-27. Google Scholar
[43]	Y. L. Zhao, D. L. Wang and S. L. Giang, Analysis on the surveillance results of the national main monitoring station of the brucellosis in 2001 to 2004, Chinese Journal of Control of Endemic Disease, 42 (2005), 120-134. Google Scholar
[44]	J. Zinsstag, F. Roth, D. Orkhon, G. Chimed-Ochir, M. Nansalmaa, J. Kolar and P. Vounatsou, A model of animal human brucellosis transmission in Mongolia, Prev. Vet. Med., 69 (2005), 77-95. doi: 10.1016/j.prevetmed.2005.01.017. Google Scholar

Figure 1. Schematic transmission diagram of brucellosis among sheep and two human subpopulations. $S, I, V$ represent susceptible, infectious and vaccinated sheep. $S_1$, $I_1$, $C_1$ and $S_2$, $I_2$, $C_2$ represent susceptible, acute cases and chronic cases of high risk human subpopulation $G_1$ and low risk human subpopulation $G_2$, respectively

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Figure 2. Predicted tendency of brucellosis in Inner Mongolia. The solid line represents the prediction of (1) and the diamonds are the reported data in Inner Mongolia of P. R. China. (a) Number of brucellosis sheep ($I$). (b) Total number of new confirmed human cases ($I_h^{NA}=I_1^{NA}+I_2^{NA}$). (c) Number of new confirmed human cases of high risk subpopulation $G_1$ ($I_1^{NA}$). (d) Number of new confirmed human cases of low risk subpopulation $G_2$ ($I_2^{NA}$). Here, the culling rate $k=0.05$ from 2001 to 2011 [41], $k=0.5$ from 2012 to 2015 [17,35,40], $k=0.15$ after 2015, $\delta=0.67$, $\theta=0.6$, $q_1=q_2=0.4$, other parameter values are listed in Table 1

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Figure 3. Influence of vaccination ($\theta$). The vaccination of sheep can reduce the epidemic situation of brucellosis but can not eliminate it. The values of parameters are the same as those in Fig. 2 except $\theta$

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Figure 4. Influence of the recruitment of sheep ($A$). It shows that reducing the size of sheep population in Inner Mongolia could assert great positive effect on brucellosis control. The values of parameters are the same as those in Fig. 2 except $A$

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Figure 5. Influence of culling of the infected sheep ($k$). Increasing the culling rate of infected sheep could effectively reduce the incidence of brucellosis in both sheep and human and can even eradicate the brucellosis. The values of parameters are the same as those in Fig. 2 except $k$

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Figure 6. Influence of the transmission rate between brucellosis sheep and human($\beta$). Compared to Fig. 4 and Fig. 5, it illustrates that reducing the brucellosis transmission rate between sheep and human has lower efficacy than the controls carried out in sheep stock, since it can not control the source of disease transmission. Here, $\beta_0=4.895\times 10^{-6}$, other values of parameters are the same as those in Fig. 2

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Figure 7. The optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ and the corresponding $I^*(t)$, $I_1^*(t)$, $I_2^*(t)$. The dot-dash curves ($\omega_1=\omega_2=0, \omega_3=\omega_4=0.5$) depict the trends of brucellosis without vaccination programme in sheep stock and health education in human. The dash curves ($\omega_1=\omega_2=0.5, \omega_3=\omega_4=0$) illustrate that the disease can not be eliminated eventually even if the control resources supply is maximal. The solid curves ($\omega_1=\omega_2=\omega_3=0.3,\omega_4=0.1$) show that the situation of brucellosis epidemic will be serious if the control resources supply is limited. Here, $A=3300$, $k=0.15$, $T=34$, $\bar{\theta}=0.85$, $\bar{\varphi_1}=\bar{\varphi_2}=0.9$, and $B_0=0.2$, $B_1=0.3$, $B_2=0.2$, $D_0=1$, $D_1=5$, $D_2=3$, other parameter values are the same as those in Fig. 2

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Figure 8. Simulation of the optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ with varying weighted coefficients. It depicts the significant effect of the weighted coefficients on the optimal control. When $\omega_1$ increase from $0$ to $0.5$, the key minimizing targets switch from vaccination and health education to culling of brucellosis sheep and treatment of human cases, and, accordingly, the optimal control gradually increases from $\theta^*(t)=\varphi_1^*(t)=\varphi_2^*(t)=0$ to $\theta^*(t)=\overline{\theta}$, $\varphi_1^*(t)=\overline{\varphi}_1$ and $\varphi_2^*(t)=\overline{\varphi}_2$. Here $\omega_1=\omega_2, \omega_3=\omega_4$, and $\omega_1+\omega_3=0.5$, other parameter values are the same as those in Fig. 7

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Figure 9. Total economic loss (i.e., the sum of the four integrals in (12)) and prevalence rates vary with the weighted coefficients. The policy-maker should confirm the weight coefficients according to the financial budget for brucellosis ($TELoss^*$) and the prevalence rates in both sheep and human population ($P_s^*,P_{h1}^*,P_{h2}^*$). Here $\omega_1=\omega_2, \omega_3=\omega_4$, and $\omega_1+\omega_3=0.5$, other parameter values are the same as those in Fig. 7

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Figure 10. Simulations of the optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ and the corresponding $I^*(t)$, $I_1^*(t)$, $I_2^*(t)$, with the culling rate $k=0.15$, $k=0.3$ and $k=0.7$, respectively. It reveals the fact again that culling of infected sheep plays an important role on brucellosis control. Here, $\omega_1=\omega_2=\omega_3=0.3, \omega_4=0.1$, other parameter values are the same as those in Fig. 7

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Figure 11. Simulations of the optimal control $\theta^*(t)$, $\varphi_1^*(t)$, $\varphi_2^*(t)$ and the corresponding $I^*(t)$, $I_1^*(t)$, $I_2^*(t)$ with constant recruitment rate of sheep $A=3300$, $A=2200$ and $A=1600$, respectively. Compared to Fig. 7, the brucellosis epidemic situations are effectively controlled in both sheep and human when the constant recruitment rate of sheep $A$ is decreased from $A=3300$ to $A=1600$. It illustrates that reducing size of sheep population in Inner Mongolia is also an effective control measure. Here, $\omega_1=\omega_2=\omega_3=0.3, \omega_4=0.1$, other parameter values are the same as those in Fig. 7

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Table 1. Parameters of model (1) with default values used for numerical studies

Parameter	Value/Range	Unit	Definition	Reference
$A$	3300	$10^4/$year	Constant recruitment of sheep	[8]
$m$	0.6	year$^{-1}$	Natural elimination or death rate of sheep	[8]
$1/\delta$	[1, 3]	year	Mean effective period of vaccination	[9,33,42]
$k$	$0-50\%$	year$^{-1}$	Culling rate of infectious sheep	[17,35,40,41]
$\theta$	[0, 0.85]	year$^{-1}$	Effective vaccination rate of susceptible sheep	[9,42,33]
$\Lambda_1$	12	$10^4$/year	Recruitment of $G_1$ (high risk)	[27]
$\Lambda_2$	11	$10^4$/year	Recruitment of $G_2$ (low risk)	[27]
$d_i$	0.006	year$^{-1}$	Natural death rate of human	[27]
$1/\gamma_i$	0.25	year	Acute onset period of human	[41]
$q_i$	[0.32, 0.74]	year$^{-1}$	Fraction of acute human cases turned into chronic cases	[41]
$\lambda$	$2.55\times 10^{-4}$	year$^{-1}$	Transmission rate of sheep	Fitting
$\beta$	$4.895\times 10^{-6}$	year$^{-1}$	Transmission rate between sheep and $G_1$	Fitting
$\varepsilon$	0.17	year$^{-1}$	Infection risk attenuation coefficient of $G_2$ compared to $G_1$	Fitting
Note: $i=1,2$

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Parameter	Value/Range	Unit	Definition	Reference
$A$	3300	$10^4/$year	Constant recruitment of sheep	[8]
$m$	0.6	year$^{-1}$	Natural elimination or death rate of sheep	[8]
$1/\delta$	[1, 3]	year	Mean effective period of vaccination	[9,33,42]
$k$	$0-50\%$	year$^{-1}$	Culling rate of infectious sheep	[17,35,40,41]
$\theta$	[0, 0.85]	year$^{-1}$	Effective vaccination rate of susceptible sheep	[9,42,33]
$\Lambda_1$	12	$10^4$/year	Recruitment of $G_1$ (high risk)	[27]
$\Lambda_2$	11	$10^4$/year	Recruitment of $G_2$ (low risk)	[27]
$d_i$	0.006	year$^{-1}$	Natural death rate of human	[27]
$1/\gamma_i$	0.25	year	Acute onset period of human	[41]
$q_i$	[0.32, 0.74]	year$^{-1}$	Fraction of acute human cases turned into chronic cases	[41]
$\lambda$	$2.55\times 10^{-4}$	year$^{-1}$	Transmission rate of sheep	Fitting
$\beta$	$4.895\times 10^{-6}$	year$^{-1}$	Transmission rate between sheep and $G_1$	Fitting
$\varepsilon$	0.17	year$^{-1}$	Infection risk attenuation coefficient of $G_2$ compared to $G_1$	Fitting
Note: $i=1,2$

Table 2. Sheep population in Inner Mongolia (unit: 10⁴)

Year	2001	2002	2003	2004	2005
Sheep Breeding[8]	3551.6	3515.9	3951.7	4450.6	5318.48
Sale and Slaughter[8]	2081.2	2146.5	2156	2867.74	3782.99
Brucellosis sheep¹	33.74	35.15	59.27	89.1	132.95
Year	2006	2007	2008	2009	2010
Sheep Breeding[8]	5419.99	5594.44	5063.29	5125.3	5197.2
Sale and Slaughter[8]	4539.6	5011.05	4874.94	5183.7	5339.2
Brucellosis sheep¹	162.57	195.79	202.52	230.63	259.85
¹ Calculated from data of sheep breeding and the annual seroprevalence of sheep brucella in key monitoring regions of Inner Mongolia[11,13,22,30,38,39,43].

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Year	2001	2002	2003	2004	2005
Sheep Breeding[8]	3551.6	3515.9	3951.7	4450.6	5318.48
Sale and Slaughter[8]	2081.2	2146.5	2156	2867.74	3782.99
Brucellosis sheep¹	33.74	35.15	59.27	89.1	132.95
Year	2006	2007	2008	2009	2010
Sheep Breeding[8]	5419.99	5594.44	5063.29	5125.3	5197.2
Sale and Slaughter[8]	4539.6	5011.05	4874.94	5183.7	5339.2
Brucellosis sheep¹	162.57	195.79	202.52	230.63	259.85
¹ Calculated from data of sheep breeding and the annual seroprevalence of sheep brucella in key monitoring regions of Inner Mongolia[11,13,22,30,38,39,43].

Table 3. Annual new confirmed cases of human brucellosis in Inner Mongolia[25,41]

Year	2001	2002	2003	2004	2005	2006	2007
Reported Human Cases	420	610	1280	4140	8740	8050	8117
Year	2008	2009	2010	2011	2012	2013	2014
Reported Human Cases	11105	16551	16935	20845	12817	9310	10538

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Year	2001	2002	2003	2004	2005	2006	2007
Reported Human Cases	420	610	1280	4140	8740	8050	8117
Year	2008	2009	2010	2011	2012	2013	2014
Reported Human Cases	11105	16551	16935	20845	12817	9310	10538

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American Institute of Mathematical Sciences

Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China

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