Journal of Applied Mathematics

Existence of Periodic Solutions and Stability of Zero Solution of a Mathematical Model of Schistosomiasis

Lin Li and Zhicheng Liu

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A mathematical model on schistosomiasis governed by periodic differential equations with a time delay was studied. By discussing boundedness of the solutions of this model and construction of a monotonic sequence, the existence of positive periodic solution was shown. The conditions under which the model admits a periodic solution and the conditions under which the zero solution is globally stable are given, respectively. Some numerical analyses show the conditional coexistence of locally stable zero solution and periodic solutions and that it is an effective treatment by simply reducing the population of snails and enlarging the death ratio of snails for the control of schistosomiasis.

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J. Appl. Math., Volume 2014 (2014), Article ID 765498, 10 pages.

First available in Project Euclid: 2 March 2015

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Li, Lin; Liu, Zhicheng. Existence of Periodic Solutions and Stability of Zero Solution of a Mathematical Model of Schistosomiasis. J. Appl. Math. 2014 (2014), Article ID 765498, 10 pages. doi:10.1155/2014/765498.

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  • W. Chen, Mathematical models of schistosomiasis epidemic and the simulation of applications in practical [Ph.D. Dissertation], Fudan University, Shanghai, China, 2008 (Chinese).
  • G. Macdonald, “The dynamics of helminth infections, with special reference to schistosomes,” Transactions of the Royal Society of Tropical Medicine and Hygiene, vol. 59, no. 5, pp. 489–506, 1965.
  • N. G. Hairston, “On the mathematical analysis of schistosome populations,” Bulletin of the World Health Organization, vol. 33, pp. 45–62, 1965.
  • G. M. Williams, A. C. Sleigh, Y. Li et al., “Mathematical modelling of schistosomiasis japonica: comparison of control strategies in the People's Republic of China,” Acta Tropica, vol. 82, no. 2, pp. 253–262, 2002.
  • S. Liang, R. C. Spear, E. Seto, A. Hubbard, and D. Qiu, “A multi-group model of Schistosoma japonicum transmission dynamics and control: model calibration and control prediction,” Tropical Medicine and International Health, vol. 10, no. 3, pp. 263–278, 2005.
  • W. F. Lucas, Modules in Applied Mathematics: Life Science Models, vol. 3, Springer, New York, NY, USA, 1983.
  • F. C. Hoppensteadt and C. S. Peskin, Mathematics in Medicine and the Life Science, Springer, New York, NY, USA, 1992.
  • K. Wu, “Mathematical model and transmission dynamics of schistosomiasis and its application,” China Tropical Medicine, vol. 5, no. 4, pp. 837–844, 2005 (Chinese).
  • D. Lu, Q. Jiang, T. Wang et al., “Study on the ecology of snails in Chengcun reservoir irrigation area of Jiangxian county,” Chininese Journal of Parasit Diseases Control, vol. 18, no. 1, pp. 52–55, 2005 (Chinese).
  • B. R. Tang and Y. Kuang, “Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems,” The Tohoku Mathematical Journal, vol. 49, no. 2, pp. 217–239, 1997.
  • X. H. Tang and Y. G. Zhou, “Periodic solutions of a class of nonlinear functional differential equations and global attractivity,” Acta Mathematica Sinica, vol. 49, no. 4, pp. 899–908, 2006 (Chinese).
  • M. Fan and X. Zou, “Global asymptotic stability of a class of nonautonomous integro-differential systems and applications,” Nonlinear Analysis, vol. 57, no. 1, pp. 111–135, 2004.
  • D. Guo, Nonlinear Functional Analysis, Shandong Science and technology Publishing House, Jinan, China, 2001 (Chinese).
  • J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99, Springer, New York, NY, USA, 1993.
  • K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. \endinput