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2014, 11(5): 1115-1137. doi: 10.3934/mbe.2014.11.1115

## Transmission dynamics and control for a brucellosis model in Hinggan League of Inner Mongolia, China

 1 Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China 2 Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030051, China, China 3 Department of mathematics, North University of China, Taiyuan 030051, PR 4 China Animal Health and Epidemiology Center, Qingdao, Shandong 266032, China, China, China 5 Hinggan League Animal Sanitation Supervision Stations, Ulanhot, Inner Mongolia, 137400, China

Received  July 2013 Revised  January 2014 Published  June 2014

Brucellosis is one of the major infectious and contagious bacterial diseases in Hinggan League of Inner Mongolia, China. The number of newly infected human brucellosis data in this area has increased dramatically in the last 10 years. In this study, in order to explore effective control and prevention measures we propose a deterministic model to investigate the transmission dynamics of brucellosis in Hinggan League. The model describes the spread of brucellosis among sheep and from sheep to humans. The model simulations agree with newly infected human brucellosis data from 2001 to 2011, and the trend of newly infected human brucellosis cases is given. We estimate that the control reproduction number $\mathcal{R}_{c}$ is about $1.9789$ for the brucellosis transmission in Hinggan League and compare the effect of existing mixed cross infection between basic ewes and other sheep or not for newly infected human brucellosis cases. Our study demonstrates that combination of prohibiting mixed feeding between basic ewes and other sheep, vaccination, detection and elimination are useful strategies in controlling human brucellosis in Hinggan League.
Citation: Mingtao Li, Guiquan Sun, Juan Zhang, Zhen Jin, Xiangdong Sun, Youming Wang, Baoxu Huang, Yaohui Zheng. Transmission dynamics and control for a brucellosis model in Hinggan League of Inner Mongolia, China. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1115-1137. doi: 10.3934/mbe.2014.11.1115
##### References:
 [1] B. Alnseba, B. Chahrazed and M. Pierre, A model for ovine brucellosis incorporating direct and indirect transmission, J. Biol. Dyn., 4 (2010), 2-11. doi: 10.1080/17513750903171688. [2] S. M. Blower, A. R. McLean and T. C. Porco, et al., The intrinsic transmission dynamics of tuberculosis epidemics, Nature Med., 1 (1995), 815-821. doi: 10.1038/nm0895-815. [3] S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model as an example, Int. Stat. Rev., 62 (1994), 229-243. doi: 10.2307/1403510. [4] M. J. Corbel, Brucellosis: An overview, Emerg. Infect. Dis., 3 (1997), 213-221. doi: 10.3201/eid0302.970219. [5] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R^{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [6] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873-885. doi: 10.1098/rsif.2009.0386. [7] V. Dobrean, A. Opris and S. Daraban, An epidemiological and surveillance overview of brucellosis in Romania, Vet. Mic., 90 (2002), 157-163. doi: 10.1016/S0378-1135(02)00251-1. [8] 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. [9] , Hinggan League's annual statistical bulletin,, , (). [10] Q. Hou, Z. Jin and S. Ruan, Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China, J. Theoret. Biol., 300 (2012), 39-47. doi: 10.1016/j.jtbi.2012.01.006. [11] Q. Hou, X. D. Sun, J. Zhang, Y. J. Liu, Y. M. Wang and Z. Jin, Modeling the transmission dynamics of brucellosis in Inner Mongolia Autonomous Region, China, Math. Biosci., 242 (2013), 51-58. doi: 10.1016/j.mbs.2012.11.012. [12] Z. Huang, et al., A study on the prevalent type of human Brucellosis caused by $B.suis$ in Guangxi applying fuzzy fathematics and markov forecast, End. Dise. Bul., 5 (1990), 101-105. [13] 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, Appl. Math. Com., 237 (2014), 582-594. doi: 10.1016/j.amc.2014.03.094. [14] M. T. Li, G. Q. Sun, J. Zhang and Z. Jin, Global Dynamic Behavior of a Multigroup Cholera model with Indirect Transmission, Discrete Dynamics in Nature and Society, Volume 2013, Article ID 703826. [15] Z. Li, et al., Application of mathematical models to forecast for inspection district of brucellosis in China (II), Chin. J. Ctrl. Endem. Dis., 15 (2000), 273-275. [16] A. Lu, et al., Establishment of Brucellosis infection rate forcasting models and their accuracy comparison, Chinese Journal of Animal Health Inspection, 17 (2000), 21-22. [17] J. C. Mi, Q. H. Zhang, L. T. Song and Z. Zheng, The epidemiological characteristics of human Brucellosis in Inner Mongolia, Chin. J. Ctrl. Endem. Dis., 25 (2010), 34-36. [18] Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. Glenn Morris, Jr., Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767-8772. [19] , National bureau of statistics of China (2012) China demographic yearbook of 2012,, , (). [20] S. Niu and R. Liu, Hinggan League 2001-2009 human brucellosis epidemiological analysis, Med. Inform., 3 (2010), 473-474. [21] G. Pappas, N. Akritidis, M. Bosilkovski and E. Tsianos, Brucellosis, N. Engl. J. Med., 352 (2005), 2325-2536. doi: 10.1056/NEJMra050570. [22] H. L. Ren, et al., The current research, prevention and control on brucellosis, China Animal Husbandry Veterinary Medicine, 36 (2009), 139-143. [23] E. J. Richey and C. Dix Harrell, Brucella Abortus Disease (Brucellosis) in Beef Cattle, University of Florida, 100 (1997), 1-6. [24] M. A. Sanchez and S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: Tuberculosis as an example, American Journal of Epidemiology, 145 (1997), 1127-1137. doi: 10.1093/oxfordjournals.aje.a009076. [25] D. Q. Shang, D. L. Xiao and J. M. Yin, Epidemiology and control of brucellosis in China, Vet. Microbiol., 69 (2002), 77 pp. [26] K. J. Sharkey, R. G. Bowers, K. L. Morgan, S. E. Robinson and R. M. Christley, Epidemiological consequences of an incursion of highly pathogenic H5N1 avian influenza into the British poultry flock, Proc. R. Soc. B, 275 (2008), 19-28. doi: 10.1098/rspb.2007.1100. [27] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043. [28] , The 2010 census bulletin of the main data of Hinggan League,, , (). [29] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [30] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Biosci., 24 (1993), 407-435. doi: 10.1137/0524026. [31] G. Wang, et al., China livestock brucellosis popular characteristics and cause analysis, Chinese Journal of Animal Health Inspection, 27 (2010), 62-63. [32] W. D. Wang, P. Fergola and C. Tenneriello, Innovation diffusion model in patch environment, Appl. Math. Com., 134 (2003), 51-67. doi: 10.1016/S0096-3003(01)00268-5. [33] W. D. Wang and X. Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [34] J. Zhang, Z. Jin, G. Sun, T. Zhou and S. Ruan, Analysis of rabies in China: Tranmission dynamics and control, PLoS ONE, 6 (2011), e20891. doi: 10.1371/journal.pone.0020891. [35] J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal rabies epidemics in China, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. [36] S. Y. Zhang and D. Zhu, In review of China brucellosis prevention to 50 years, Chin. J. Ctrl. Endem. Dis., 18 (2003), 275-278. [37] X. Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. [38] X. Q. Zhao and Z. J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. [39] Y. B. Zhou and X. L. Liu, The research progress in terms of prevalence, incidence reason and control strategies of brucellosis, J. Liaoning Medical University, 1 (2010), 81-85. [40] 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.

show all references

##### References:
 [1] B. Alnseba, B. Chahrazed and M. Pierre, A model for ovine brucellosis incorporating direct and indirect transmission, J. Biol. Dyn., 4 (2010), 2-11. doi: 10.1080/17513750903171688. [2] S. M. Blower, A. R. McLean and T. C. Porco, et al., The intrinsic transmission dynamics of tuberculosis epidemics, Nature Med., 1 (1995), 815-821. doi: 10.1038/nm0895-815. [3] S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model as an example, Int. Stat. Rev., 62 (1994), 229-243. doi: 10.2307/1403510. [4] M. J. Corbel, Brucellosis: An overview, Emerg. Infect. Dis., 3 (1997), 213-221. doi: 10.3201/eid0302.970219. [5] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R^{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [6] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873-885. doi: 10.1098/rsif.2009.0386. [7] V. Dobrean, A. Opris and S. Daraban, An epidemiological and surveillance overview of brucellosis in Romania, Vet. Mic., 90 (2002), 157-163. doi: 10.1016/S0378-1135(02)00251-1. [8] 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. [9] , Hinggan League's annual statistical bulletin,, , (). [10] Q. Hou, Z. Jin and S. Ruan, Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China, J. Theoret. Biol., 300 (2012), 39-47. doi: 10.1016/j.jtbi.2012.01.006. [11] Q. Hou, X. D. Sun, J. Zhang, Y. J. Liu, Y. M. Wang and Z. Jin, Modeling the transmission dynamics of brucellosis in Inner Mongolia Autonomous Region, China, Math. Biosci., 242 (2013), 51-58. doi: 10.1016/j.mbs.2012.11.012. [12] Z. Huang, et al., A study on the prevalent type of human Brucellosis caused by $B.suis$ in Guangxi applying fuzzy fathematics and markov forecast, End. Dise. Bul., 5 (1990), 101-105. [13] 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, Appl. Math. Com., 237 (2014), 582-594. doi: 10.1016/j.amc.2014.03.094. [14] M. T. Li, G. Q. Sun, J. Zhang and Z. Jin, Global Dynamic Behavior of a Multigroup Cholera model with Indirect Transmission, Discrete Dynamics in Nature and Society, Volume 2013, Article ID 703826. [15] Z. Li, et al., Application of mathematical models to forecast for inspection district of brucellosis in China (II), Chin. J. Ctrl. Endem. Dis., 15 (2000), 273-275. [16] A. Lu, et al., Establishment of Brucellosis infection rate forcasting models and their accuracy comparison, Chinese Journal of Animal Health Inspection, 17 (2000), 21-22. [17] J. C. Mi, Q. H. Zhang, L. T. Song and Z. Zheng, The epidemiological characteristics of human Brucellosis in Inner Mongolia, Chin. J. Ctrl. Endem. Dis., 25 (2010), 34-36. [18] Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D. L. Smith and J. Glenn Morris, Jr., Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA, 108 (2011), 8767-8772. [19] , National bureau of statistics of China (2012) China demographic yearbook of 2012,, , (). [20] S. Niu and R. Liu, Hinggan League 2001-2009 human brucellosis epidemiological analysis, Med. Inform., 3 (2010), 473-474. [21] G. Pappas, N. Akritidis, M. Bosilkovski and E. Tsianos, Brucellosis, N. Engl. J. Med., 352 (2005), 2325-2536. doi: 10.1056/NEJMra050570. [22] H. L. Ren, et al., The current research, prevention and control on brucellosis, China Animal Husbandry Veterinary Medicine, 36 (2009), 139-143. [23] E. J. Richey and C. Dix Harrell, Brucella Abortus Disease (Brucellosis) in Beef Cattle, University of Florida, 100 (1997), 1-6. [24] M. A. Sanchez and S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate: Tuberculosis as an example, American Journal of Epidemiology, 145 (1997), 1127-1137. doi: 10.1093/oxfordjournals.aje.a009076. [25] D. Q. Shang, D. L. Xiao and J. M. Yin, Epidemiology and control of brucellosis in China, Vet. Microbiol., 69 (2002), 77 pp. [26] K. J. Sharkey, R. G. Bowers, K. L. Morgan, S. E. Robinson and R. M. Christley, Epidemiological consequences of an incursion of highly pathogenic H5N1 avian influenza into the British poultry flock, Proc. R. Soc. B, 275 (2008), 19-28. doi: 10.1098/rspb.2007.1100. [27] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043. [28] , The 2010 census bulletin of the main data of Hinggan League,, , (). [29] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [30] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Biosci., 24 (1993), 407-435. doi: 10.1137/0524026. [31] G. Wang, et al., China livestock brucellosis popular characteristics and cause analysis, Chinese Journal of Animal Health Inspection, 27 (2010), 62-63. [32] W. D. Wang, P. Fergola and C. Tenneriello, Innovation diffusion model in patch environment, Appl. Math. Com., 134 (2003), 51-67. doi: 10.1016/S0096-3003(01)00268-5. [33] W. D. Wang and X. Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001. [34] J. Zhang, Z. Jin, G. Sun, T. Zhou and S. Ruan, Analysis of rabies in China: Tranmission dynamics and control, PLoS ONE, 6 (2011), e20891. doi: 10.1371/journal.pone.0020891. [35] J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal rabies epidemics in China, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6. [36] S. Y. Zhang and D. Zhu, In review of China brucellosis prevention to 50 years, Chin. J. Ctrl. Endem. Dis., 18 (2003), 275-278. [37] X. Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), 473-495. [38] X. Q. Zhao and Z. J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. [39] Y. B. Zhou and X. L. Liu, The research progress in terms of prevalence, incidence reason and control strategies of brucellosis, J. Liaoning Medical University, 1 (2010), 81-85. [40] 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.
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