具有饱和发生率的血吸虫病动力学分析
|
摘要
血吸虫病是我国一种严重的寄生虫病,并且在湖北、安徽、湖南、江苏、四川和云南成地方病.结合中国血吸虫病的现状及特点,考虑人群、牛群以及水环境中的钉螺、尾蚴和毛蚴之间的相互传染,建立了具有饱和发生率的血吸虫病动力学模型,给出了模型的基本再生数.通过构造Lyapunov函数证明了当基本再生数小于1时,模型的无病平衡点全局渐近稳定;当基本再生数大于1时,模型的地方病平衡点也是全局渐近稳定的.最后,利用数值模拟验证了理论结果.
Schistosomiasis is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan,and Yunnan. Combining the current situation and characteristics of schistosomiasis in China,and considering the transmission with human, beef and snail, cercariae and miracidium in the aquatic environment, the schistosomiasis dynamical model with saturating incidence is formulated. We obtain the basic reproduction number of this model. By using Lyapunov functions, we conclude that if the basic reproduction number is less than 1, then the diseasefree equilibrium is globally asymptotically stable, if the basic reproduction number is more than 1, then the unique endemic equilibrium is also globally asymptotically stable. Finally,numerical simulation is used to verify the theoretical results.
Schistosomiasis is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan,and Yunnan. Combining the current situation and characteristics of schistosomiasis in China,and considering the transmission with human, beef and snail, cercariae and miracidium in the aquatic environment, the schistosomiasis dynamical model with saturating incidence is formulated. We obtain the basic reproduction number of this model. By using Lyapunov functions, we conclude that if the basic reproduction number is less than 1, then the diseasefree equilibrium is globally asymptotically stable, if the basic reproduction number is more than 1, then the unique endemic equilibrium is also globally asymptotically stable. Finally,numerical simulation is used to verify the theoretical results.
引文
[1] World Health Organization, et al. Schistosomiasis:progress report 2001-2011, strategic plan 2012-2020[M]. 2013.
[2] Gurarie D, Seto E Y W. Connectivity sustains disease transmission in environments with low potential for endemicity:modelling schistosomiasis with hydrologic and social connectivities[J]. Journal of the Royal Society Interface, 2009, 6(35):495-508.
[3] Hotez P J, Molyneux D H, Fenwick A, Kumaresan J, Sachs S E, Sachs J D, Savioli L. Control of neglected tropical diseases[J]. New England Journal of Medicine, 2007, 357(10):1018-1027.
[4] Macdonald G. The dynamics of helminth infections, with special reference to schistosomes[J].Transactions of the Royal Society of Tropical Medicine and Hygiene, 1965, 59(5):489-506.
[5] Chen Z M, Zou L, Shen D W, Zhang W N, Ruan S G. Mathematical modelling and control of schistosomiasis in Hubei Province China[J]. Acta Tropica, 2010, 115(1):119-125.
[6] Zou L, Ruan S G. Schistosomiasis transmission and control in China[J]. Acta Tropica,2015, 143:51-57.
[7] Gao S J, Liu Y J, Luo Y Q, Xie D H. Control problems of a mathematical model for schistosomiasis transmission dynamics[J]. Nonlinear Dynamics, 2011, 63(3):503-512.
[8] Ngarakana-Gwasira E T, Bhunu C P, Masocha M, Mashonjowa E. Transmission dynamics of schistosomiasis in Zimbabwe:A mathematical and GIS approach[J]. Communications in Nonlinear Science and Numerical Simulation, 2016, 35:137-147.
[9] McManus D P, Gray D J, Li Y S, et al. Schistosomiasis in the People's Republic of China:The Era of the Three Gorges Dam[J]. Clinical Microbiology Reviews, 2010, 23(2):442-466.
[10] Diekmann O, Heesterbeek J A P, Metz J A J, On the definition and the computation of the basic reproduction ratio R_0 in models for infectious diseases in heterogeneous populations[J]. J Math Biol, 1990, 28:365-382.
[11] van den P, Driessche J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J]. Math Biosci, 2002, 180:29-48.
[12] Smith H L, Waltman P. The theory of the chemostat:Dynamics of microbial competition[M].Cambridge university press, 1995.
[13] LaSalle J P. The stability of dynamical systems[C]//Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
[14] Freedman H I, Ruan S G, Tang M X. Uniform persistence and flows near a closed positively invariant set[J]. Journal of Dynamics and Differential Equations, 1994, 6(4):583-600.
[2] Gurarie D, Seto E Y W. Connectivity sustains disease transmission in environments with low potential for endemicity:modelling schistosomiasis with hydrologic and social connectivities[J]. Journal of the Royal Society Interface, 2009, 6(35):495-508.
[3] Hotez P J, Molyneux D H, Fenwick A, Kumaresan J, Sachs S E, Sachs J D, Savioli L. Control of neglected tropical diseases[J]. New England Journal of Medicine, 2007, 357(10):1018-1027.
[4] Macdonald G. The dynamics of helminth infections, with special reference to schistosomes[J].Transactions of the Royal Society of Tropical Medicine and Hygiene, 1965, 59(5):489-506.
[5] Chen Z M, Zou L, Shen D W, Zhang W N, Ruan S G. Mathematical modelling and control of schistosomiasis in Hubei Province China[J]. Acta Tropica, 2010, 115(1):119-125.
[6] Zou L, Ruan S G. Schistosomiasis transmission and control in China[J]. Acta Tropica,2015, 143:51-57.
[7] Gao S J, Liu Y J, Luo Y Q, Xie D H. Control problems of a mathematical model for schistosomiasis transmission dynamics[J]. Nonlinear Dynamics, 2011, 63(3):503-512.
[8] Ngarakana-Gwasira E T, Bhunu C P, Masocha M, Mashonjowa E. Transmission dynamics of schistosomiasis in Zimbabwe:A mathematical and GIS approach[J]. Communications in Nonlinear Science and Numerical Simulation, 2016, 35:137-147.
[9] McManus D P, Gray D J, Li Y S, et al. Schistosomiasis in the People's Republic of China:The Era of the Three Gorges Dam[J]. Clinical Microbiology Reviews, 2010, 23(2):442-466.
[10] Diekmann O, Heesterbeek J A P, Metz J A J, On the definition and the computation of the basic reproduction ratio R_0 in models for infectious diseases in heterogeneous populations[J]. J Math Biol, 1990, 28:365-382.
[11] van den P, Driessche J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J]. Math Biosci, 2002, 180:29-48.
[12] Smith H L, Waltman P. The theory of the chemostat:Dynamics of microbial competition[M].Cambridge university press, 1995.
[13] LaSalle J P. The stability of dynamical systems[C]//Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
[14] Freedman H I, Ruan S G, Tang M X. Uniform persistence and flows near a closed positively invariant set[J]. Journal of Dynamics and Differential Equations, 1994, 6(4):583-600.