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:
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CDC., Parasites and health: Echinococcosis,, DPDx, (2009). Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. doi: 10.1007/s00285-011-0457-x. Google Scholar

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C. Castillo-Chavez, Z. Feng and D. Xu, A schistosomiasis model with mating structure and time delay,, Mathematical Biosciences, 211 (2008), 333. doi: 10.1016/j.mbs.2007.11.001. Google Scholar

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Z. Feng, A. Eppert, F. Milner and D. Minchella, Estimation of parameters governing the transmission dynamics of schistosomes,, Applied Mathematics Letters, 17 (2004), 1105. doi: 10.1016/j.aml.2004.02.002. Google Scholar

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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. doi: 10.1007/s11538-007-9292-z. Google Scholar

[12]

Z. M. Chen and L. Zou et.al, Mathematical modelling and control of schistosomiasis in Hubei province, China,, Acta Tropica, 115 (2010), 119. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases,, Model building analyis and interpretation, (2000). Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[17]

H. B. Guo and M. Y. Li., Global stability in a mathematical model of tuberculosis,, Canadian applied mathematics quarterly, 14 (2006). Google Scholar

[18]

J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[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. doi: 10.1007/s002850050099. Google Scholar

[20]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Eng., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar

[21]

, China Yearbook., \url{http://tongji.cnki.net/kns55/brief/result.aspx?stab=shuzhi}., (). Google Scholar

[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. Google Scholar

[23]

H. Wang and D. L. A, Analysis of pulmonary echinococcosis cyst excision in 136 cases,, Chinese Journal of Misdiagnostics, 11 (2011). Google Scholar

[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. Google Scholar

show all references

References:
[1]

CDC., Parasites and health: Echinococcosis,, DPDx, (2009). Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

H. Wang, L. Li and B. Zhang etc, Status of human hydatid disease report,, in, (2008), 73. Google Scholar

[5]

CDC., Web. April (2010)., \url{http://www.chinacdc.cn/jkzt/tfggwssj/zzfb/crbjcykz/201004/t20100420_24967.htm}., (). Google Scholar

[6]

X. Zhe, Medlive, Web. April (2011)., \url{http://disease.medlive.cn/wiki/entry/10001076_301_0}., (). Google Scholar

[7]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford Univ. Press, (1991). Google Scholar

[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. doi: 10.1007/s00285-011-0457-x. Google Scholar

[9]

C. Castillo-Chavez, Z. Feng and D. Xu, A schistosomiasis model with mating structure and time delay,, Mathematical Biosciences, 211 (2008), 333. doi: 10.1016/j.mbs.2007.11.001. Google Scholar

[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. doi: 10.1016/j.aml.2004.02.002. Google Scholar

[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. doi: 10.1007/s11538-007-9292-z. Google Scholar

[12]

Z. M. Chen and L. Zou et.al, Mathematical modelling and control of schistosomiasis in Hubei province, China,, Acta Tropica, 115 (2010), 119. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases,, Model building analyis and interpretation, (2000). Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[17]

H. B. Guo and M. Y. Li., Global stability in a mathematical model of tuberculosis,, Canadian applied mathematics quarterly, 14 (2006). Google Scholar

[18]

J. P. LaSalle, "The Stability of Dynamical Systems,", Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[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. doi: 10.1007/s002850050099. Google Scholar

[20]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Eng., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar

[21]

, China Yearbook., \url{http://tongji.cnki.net/kns55/brief/result.aspx?stab=shuzhi}., (). Google Scholar

[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. Google Scholar

[23]

H. Wang and D. L. A, Analysis of pulmonary echinococcosis cyst excision in 136 cases,, Chinese Journal of Misdiagnostics, 11 (2011). Google Scholar

[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. Google Scholar

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