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Dynamical behaviors of an Echinococcosis epidemic model with distributed delays

a. Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, Xinjiang 830054, China
b. College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China

In this paper, a novel spreading dynamical model for Echinococcosis with distributed time delays is proposed. For the model, we firstly give the basic reproduction number $\mathcal{R}_0$ and the existence of a unique endemic equilibrium when $\mathcal{R}_0>1$. Furthermore, we analyze the dynamical behaviors of the model. The results show that the dynamical properties of the model is completely determined by $\mathcal{R}_0$. That is, if $\mathcal{R}_0<1 the="" disease-free="" equilibrium="" is="" globally="" asymptotically="" stable="" and="" if="" mathcal="" r="" _0="">1$, the model is permanent and the endemic equilibrium is globally asymptotically stable. According to human Echinococcosis cases from January 2004 to December 2011 in Xinjiang, China, we estimate the parameters of the model and study the transmission trend of the disease in Xinjiang, China. The model provides an approximate estimate of the basic reproduction number $\mathcal{R}_0=1.23$ in Xinjiang, China. From theoretic results, we further find that Echinococcosis is endemic in Xinjiang, China. Finally, we perform some sensitivity analysis of several model parameters and give some useful measures on controlling the transmission of Echinococcosis.

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References

[1] Baidu, How Long is the Life Expectancy of Dogs? 2011, Available from: http://wenku.baidu.com/view/e7ccf4fec8d376eeaeaa31f9.html.

[2] Baidu, The Sixth Census Data Announced by Xinjiang 2011, Available from: http://wenku.baidu.com/view/83e6d26cb84ae45c3b358cf1.html.

[3] S. A. Berger and J. S. Marr, Human Parasitic Diseases Sourcebook, 1$^{st}$ edition, Jones and Bartlett Publishers: Sudbury, Massachusetts, 2006.

[4] P. A. Cabrera,G. Haran,U. benavidez,S. Valledor,G. Perera,S. Lloyd,M. A. Gemmell,M. Baraibar,A. Morana,J. Maissonave,M. Carballo, Transmission dynamics of Echinococcus granulosus, Taenia hydatigena and Taenia ovis in sheep in Uruguay, Int. J. Parasitol., 25 (1995): 807-813.

[5] Y. Cao,J. Wen,Q. Zheng, Analysis of the epidemic status of Echinococcosis in Xinjiang in 2010, J. Ningxia Med. Univ., 33 (2011): 784-788.

[6] J. Eckert, M. A. Gemmell, F. X. Meslin and Z. S. Pawlowski, WHO/OIE manual on Echinococcosis in humans and animals: A public health problem of global concern, Paris: World Health Organization/World Organization for Animal Health, 2001.

[7] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993.

[8] S. Lahmar,H. Debbek,L. H. Zhang,D. P. McManus,A. Souissi,S. Chelly,P. R. Torgerson, Tansmission dynamics of the Echinococcus granulosus sheep-dog strain(G1 genotype) in camels in Tunisia, Vet. Parasitol., 121 (2004): 151-156.

[9] Malike,Nusilaiti,Zulihumaer,Lv,Wali,Xi,Abudureyimu,Qiu,Abuduaini,Hanati, Investigation of Echinococcus infection in domestic in Xinjiang, Chin. J. Anim.Infect. Dis., 19 (2011): 57-60(in Chinese).

[10] National Bureau of Statistics of China, China Statistical Yearbook, 2011 Available from: http://www.stats.gov.cn/tjsj/ndsj/2011/indexch.htm.

[11] Nusilaiti,Zulihumaer, The infection situation of Echinococcosis and countermeasures of some counties and cities in xinjiang, Xinjiang Livestock Industry, 6 (2011): 45-46.

[12] H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[13] K. Takumi,A. Vires,M. Chu,J. Mulder,P. Teunis,J. Giessen, Evidence for an increasing presence of Echinococcus multilocularis in foxes in The Netherlands, Inter. J. Parasitol., 38 (2008): 571-578.

[14] The Government of Xinjiang Uygur Autonomous Region of China, The improvement of people's living standard, 2010, Available from: http://www.xinjiang.gov.cn/2011/11/15/64.html.

[15] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically automous differential equations, J. Math. Biol., 30 (1992): 755-763.

[16] P. R. Torgerson,K. K. Burtisurnov,B. S. Shaikenov,A. T. Rysmukhambetova,A. M. Abdybekova,A. E. Ussenbayev, Modelling the transmission dynamics of Echinococcus granulosus in sheep and cattle in Kazakhstan, Vet. Parasitol., 114 (2003): 143-153.

[17] S. Wang and S. Ye, Textbook of Medical Microbiology and Parasitology, 1$^{st}$ edition, Science Press, Beijing, 2006.

[18] Z. Wang,X.-Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012): 89-113.

[19] K. Wang,X. Zhang,Z. Jin,H. Ma,Z. Teng,L. Wang, Modeling and analysis of the transmission of Echinococcosis with application to Xinjiang Uygur Autonomous Region of China, J. Theor. Biol., 333 (2013): 78-90.

[20] W. Wu, The Chinese National Plan for the Control Echincoccosis, Urumq: ISUOG's World Congress of Hydatidology Final Programme & Abstracts Book, 2011(in Chinese).

[21] L. Wu,B. Song,W. Du,J. Lou, Mathematical modelling and control of Echinococcus in Qinghai province, China, Math. Biosci. Eng., 10 (2013): 425-444.

[22] Xinjiang CDC, Information Center of Xinjiang Autonomous Region, 2012, Available from: http://www.xjcdc.com/.

[23] Xi,Song,Xue,Nusilaiti,Malike,Zulihumaer, Survey on Echinococcus granulosus infection in shepherd dogs in tianshan mountainous area of Hejing city, Xinjiang province, Chin. J. Anim. Infect. Dis., 18 (2010): 59-62.

[24] Z. Xu,X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012): 2015-2034.

[25] S. Yan,Y. Zhang, The control and prevention of livestock Echinococcosis in Xinjiang, Grass-Feeding Livestock, null (1994): 45-47.

[26] X.-Q. Zhao,Z. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Q., 4 (1996): 421-444.

Copyright Info: © 2017, Kai Wang, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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