# American Institute of Mathematical Sciences

2013, 10(2): 425-444. doi: 10.3934/mbe.2013.10.425

## Mathematical modelling and control of echinococcus in Qinghai province, China

 1 Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China, China, China 2 Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043

Received  June 2012 Revised  December 2012 Published  January 2013

In this paper, two mathematical models, the baseline model and the intervention model, are proposed to study the transmission dynamics of echinococcus. A global forward bifurcation completely characterizes the dynamical behavior of the baseline model. That is, when the basic reproductive number is less than one, the disease-free equilibrium is asymptotically globally stable; when the number is greater than one, the endemic equilibrium is asymptotically globally stable. For the intervention model, however, the basic reproduction number alone is not enough to describe the dynamics, particularly for the case where the basic reproductive number is less then one. The emergence of a backward bifurcation enriches the dynamical behavior of the model. Applying these mathematical models to Qinghai Province, China, we found that the infection of echinococcus is in an endemic state. Furthermore, the model appears to be supportive of human interventions in order to change the landscape of echinococcus infection in this region.
Citation: Liumei Wu, Baojun Song, Wen Du, Jie Lou. Mathematical modelling and control of echinococcus in Qinghai province, China. Mathematical Biosciences & Engineering, 2013, 10 (2) : 425-444. doi: 10.3934/mbe.2013.10.425
##### References:
 [1] CDC., Parasites and health: Echinococcosis, DPDx, July (2009), Web. February 2010, http://www.dpd.cdc.gov/DPDx/html/Echinococcosis.htm. [2] S. H. Yu, H. Wang, X. H. Wu, X. Ma, Y. F. Liu, Y. M. Zhao, Y. Morishima and M. Kawanaka, Cystic and alveolar echinococcosis: An epidemiological survey in a Tibetan population in Southeast Qinghai, China, Jpn.J.Infect.Dis., 61 (2008), 242-246. [3] Y. R. Yang, M. C. Rosenzvit, L. H. Zhang, J. Z. Zhang and D. P. Mcmanus, Molecular study of echinococcus in west-central China, Parasitology, 131 (2005), 547-555. [4] H. Wang, L. Li and B. Zhang etc, Status of human hydatid disease report, in "National Survey of Important Human Parasitic Diseases" People's Health Publishing House, Beijing, (2008), 73-79. [5] CDC., Web. April (2010). http://www.chinacdc.cn/jkzt/tfggwssj/zzfb/crbjcykz/201004/t20100420_24967.htm. [6] X. Zhe, Medlive, Web. April (2011). http://disease.medlive.cn/wiki/entry/10001076_301_0. [7] R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford Univ. Press, New York, 1991. [8] Y. Yang, Z. Feng, D. Xu, G. Sandland and D. J. Minchella, Evolution of host resistance to parasite infection in the snail-schistosome-human system, Journal of Mathematical Biology, 65 (2012), 201-236. doi: 10.1007/s00285-011-0457-x. [9] C. Castillo-Chavez, Z. Feng and D. Xu, A schistosomiasis model with mating structure and time delay, Mathematical Biosciences, 211 (2008), 333-341. doi: 10.1016/j.mbs.2007.11.001. [10] Z. Feng, A. Eppert, F. Milner and D. Minchella, Estimation of parameters governing the transmission dynamics of schistosomes, Applied Mathematics Letters, 17 (2004), 1105-1112, doi: 10.1016/j.aml.2004.02.002. [11] S. G. Ruan, D. M. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bulletin of Mathematical Biology, 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [12] Z. M. Chen and L. Zou et.al, Mathematical modelling and control of schistosomiasis in Hubei province, China, Acta Tropica, 115 (2010), 119-125. [13] P. R. Torgersona, D. H. Williamsb and M. N. Abo-Shehada, Modelling the prevalence of Echinococcus and Taenia species in small ruminants of different ages in northern Jordan, Veterinary Parasitology, 79 (1998), 35-51. [14] R. M. Mukbel, P. R. Torgerson and M. N. Abo-Shehada, Prevalence of hydatidosis among donkeys in northern Jordan, Veterinary Parasitology, 88 (2000), 35-42. [15] O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases, Model building analyis and interpretation, Wiley Series in Mathematical and Computational Biology, (2000). [16] P. van den Driessche and J. Watmough, Reproductive numbers and sub-threshold endemic equilibria for compartmentmodels of disease transmission, Math. Biosci., 180 (2002), 183-201. doi: 10.1016/S0025-5564(02)00108-6. [17] H. B. Guo and M. Y. Li., Global stability in a mathematical model of tuberculosis, Canadian applied mathematics quarterly, 14 (2006), Number 2. [18] J. P. LaSalle, "The Stability of Dynamical Systems," Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [19] J. Dushoff, W. Huang and C. Castillo-Chavez, Backward bifurcations and batas- trophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248. doi: 10.1007/s002850050099. [20] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [21] , China Yearbook. http://tongji.cnki.net/kns55/brief/result.aspx?stab=shuzhi. [22] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196. [23] H. Wang and D. L. A, Analysis of pulmonary echinococcosis cyst excision in 136 cases, Chinese Journal of Misdiagnostics, 11 (2011). [24] P. S. Craig, P. Giraudoux, D. Shi and B. Bartholomot, An epidemiological and ecological study of human alveolar echinococcosis transmission in south Gansu, China, Acta Tropica, 77 (2000), 167-177.

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##### References:
 [1] CDC., Parasites and health: Echinococcosis, DPDx, July (2009), Web. February 2010, http://www.dpd.cdc.gov/DPDx/html/Echinococcosis.htm. [2] S. H. Yu, H. Wang, X. H. Wu, X. Ma, Y. F. Liu, Y. M. Zhao, Y. Morishima and M. Kawanaka, Cystic and alveolar echinococcosis: An epidemiological survey in a Tibetan population in Southeast Qinghai, China, Jpn.J.Infect.Dis., 61 (2008), 242-246. [3] Y. R. Yang, M. C. Rosenzvit, L. H. Zhang, J. Z. Zhang and D. P. Mcmanus, Molecular study of echinococcus in west-central China, Parasitology, 131 (2005), 547-555. [4] H. Wang, L. Li and B. Zhang etc, Status of human hydatid disease report, in "National Survey of Important Human Parasitic Diseases" People's Health Publishing House, Beijing, (2008), 73-79. [5] CDC., Web. April (2010). http://www.chinacdc.cn/jkzt/tfggwssj/zzfb/crbjcykz/201004/t20100420_24967.htm. [6] X. Zhe, Medlive, Web. April (2011). http://disease.medlive.cn/wiki/entry/10001076_301_0. [7] R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford Univ. Press, New York, 1991. [8] Y. Yang, Z. Feng, D. Xu, G. Sandland and D. J. Minchella, Evolution of host resistance to parasite infection in the snail-schistosome-human system, Journal of Mathematical Biology, 65 (2012), 201-236. doi: 10.1007/s00285-011-0457-x. [9] C. Castillo-Chavez, Z. Feng and D. Xu, A schistosomiasis model with mating structure and time delay, Mathematical Biosciences, 211 (2008), 333-341. doi: 10.1016/j.mbs.2007.11.001. [10] Z. Feng, A. Eppert, F. Milner and D. Minchella, Estimation of parameters governing the transmission dynamics of schistosomes, Applied Mathematics Letters, 17 (2004), 1105-1112, doi: 10.1016/j.aml.2004.02.002. [11] S. G. Ruan, D. M. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bulletin of Mathematical Biology, 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [12] Z. M. Chen and L. Zou et.al, Mathematical modelling and control of schistosomiasis in Hubei province, China, Acta Tropica, 115 (2010), 119-125. [13] P. R. Torgersona, D. H. Williamsb and M. N. Abo-Shehada, Modelling the prevalence of Echinococcus and Taenia species in small ruminants of different ages in northern Jordan, Veterinary Parasitology, 79 (1998), 35-51. [14] R. M. Mukbel, P. R. Torgerson and M. N. Abo-Shehada, Prevalence of hydatidosis among donkeys in northern Jordan, Veterinary Parasitology, 88 (2000), 35-42. [15] O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases, Model building analyis and interpretation, Wiley Series in Mathematical and Computational Biology, (2000). [16] P. van den Driessche and J. Watmough, Reproductive numbers and sub-threshold endemic equilibria for compartmentmodels of disease transmission, Math. Biosci., 180 (2002), 183-201. doi: 10.1016/S0025-5564(02)00108-6. [17] H. B. Guo and M. Y. Li., Global stability in a mathematical model of tuberculosis, Canadian applied mathematics quarterly, 14 (2006), Number 2. [18] J. P. LaSalle, "The Stability of Dynamical Systems," Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [19] J. Dushoff, W. Huang and C. Castillo-Chavez, Backward bifurcations and batas- trophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248. doi: 10.1007/s002850050099. [20] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [21] , China Yearbook. http://tongji.cnki.net/kns55/brief/result.aspx?stab=shuzhi. [22] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196. [23] H. Wang and D. L. A, Analysis of pulmonary echinococcosis cyst excision in 136 cases, Chinese Journal of Misdiagnostics, 11 (2011). [24] P. S. Craig, P. Giraudoux, D. Shi and B. Bartholomot, An epidemiological and ecological study of human alveolar echinococcosis transmission in south Gansu, China, Acta Tropica, 77 (2000), 167-177.
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