American Institute of Mathematical Sciences

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

Mathematical modelling and control of echinococcus in Qinghai province, China

Liumei Wu 1, , Baojun Song 2, , Wen Du 1, and  Jie Lou 1,

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.
Keywords: backward bifurcation, global stability., Echinococcosis, mathematical model.
Mathematics Subject Classification: 62J12, 93A30, 97M1.
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. 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), 242-246. 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), 547-555. Google Scholar

[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. 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, New York, 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), 201-236. 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), 333-341. 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), 1105-1112, 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), 1098-1114. 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), 119-125. 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), 35-51. 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), 35-42. Google Scholar

[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).  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), 183-201. 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), Number 2.  Google Scholar

[18]

J. P. LaSalle, "The Stability of Dynamical Systems," Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 227-248. 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), 361-404. 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), 178-196. 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), 167-177. Google Scholar

show all references

References:
[1]

CDC., Parasites and health: Echinococcosis, DPDx, July (2009), Web. February 2010, http://www.dpd.cdc.gov/DPDx/html/Echinococcosis.htm. 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), 242-246. 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), 547-555. Google Scholar

[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. 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, New York, 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), 201-236. 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), 333-341. 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), 1105-1112, 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), 1098-1114. 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), 119-125. 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), 35-51. 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), 35-42. Google Scholar

[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).  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), 183-201. 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), Number 2.  Google Scholar

[18]

J. P. LaSalle, "The Stability of Dynamical Systems," Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 227-248. 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), 361-404. 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), 178-196. 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), 167-177. Google Scholar

[1]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999
[2]

Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749
[3]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[4]

Kai Wang, Zhidong Teng, Xueliang Zhang. Dynamical behaviors of an Echinococcosis epidemic model with distributed delays. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1425-1445. doi: 10.3934/mbe.2017074
[5]

Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863
[6]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5197-5216. doi: 10.3934/dcdsb.2020339
[7]

Shu Liao, Jin Wang. Stability analysis and application of a mathematical cholera model. Mathematical Biosciences & Engineering, 2011, 8 (3) : 733-752. doi: 10.3934/mbe.2011.8.733
[8]

Abdul M. Kamareddine, Roger L. Hughes. Towards a mathematical model for stability in pedestrian flows. Networks & Heterogeneous Media, 2011, 6 (3) : 465-483. doi: 10.3934/nhm.2011.6.465
[9]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445
[10]

Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences & Engineering, 2004, 1 (1) : 81-93. doi: 10.3934/mbe.2004.1.81
[11]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[12]

Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
[13]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[14]

Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3073-3081. doi: 10.3934/dcdss.2020135
[15]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[16]

Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849
[17]

Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056
[18]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
[19]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[20]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy