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2011, 8(4): 1099-1115. doi: 10.3934/mbe.2011.8.1099

Dynamics of a delay Schistosomiasis model in snail infections

Chunhua Shan 1, , Hongjun Gao 2, and  Huaiping Zhu 3,

1. 

Department of Mathematics and Statistics, LAMPS and CDM, York University, Toronto, ON, M3J 1P3, Canada
2. 

Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematics, Nanjing Normal University, Nanjing, Jiangsu, 210046
3. 

Department of Mathematics and Statistics, Laboratory of Mathematical Parallel systems (LAMPS) and CDM, York University, Toronto M3J 1P3

Received  October 2010 Revised  May 2011 Published  August 2011

In this paper we modify and study a system of delay differential equations model proposed by Nåsell and Hirsch (1973) for the transmission dynamics of schistosomiasis. The modified stochastic version of MacDonald’s model takes into account the time delay for the transmission of infection. We carry out bifurcation studies of the model. The saddle-node bifurcation of the model suggests that the transmission and spread of schistosomiasis is initial size dependent. The existence of a Hopf bifurcation due to the delay indicates that the transmission can be periodic.
Keywords: time delay, stability, Hopf bifurcation., saddle-node bifurcation, Schistosomiasis.
Mathematics Subject Classification: 92D30, 37N2.
Citation: Chunhua Shan, Hongjun Gao, Huaiping Zhu. Dynamics of a delay Schistosomiasis model in snail infections. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1099-1115. doi: 10.3934/mbe.2011.8.1099
References:
[1]

J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer-Verlag, New York-Berlin, 1981.

[2]

C. Castillo-Chavez, Z. Feng and D. Xu, A schistosomiasis model with mating structure and time delay, Math. Biosci., 211 (2008), 333-341. doi: 10.1016/j.mbs.2007.11.001.

[3]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982.

[4]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcial. Ekvac., 29 (1986), 77-90.

[5]

C. L. Cosgrove and V. R. Southgate, Mating interactions between Schistosoma mansoni and S. margrebowiei, Parasitology, 125 (2002), 233-243. doi: 10.1017/S0031182002002111.

[6]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

[7]

T. Faria, Normal forms and bifurcations for delay differential equations, in "Delay Differential Equations and Applications," NATO Science Series II: Mathematics, Physics and Chemistry, 205, Springer, Dordrecht, (2006), 227-282.

[8]

Z. Feng, C.-C. Li and F. A. Milner, Schistosomiasis models with density dependence and age of infection in snail dynamics, Math. Biosci., 177/178 (2002), 271-286. doi: 10.1016/S0025-5564(01)00115-8.

[9]

Z. Feng, C.-C. Li and F. A. Milner, Schistosomiasis models with two migrating human groups, Math. Comput. Modelling., 41 (2005), 1213-1230. doi: 10.1016/j.mcm.2004.10.023.

[10]

M. Golubitsky and D. G Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Applied Mathematical Sciences, 51, Springer-Verlag, New York, 1985.

[11]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[12]

B. Gryseels, K. Polman, J. Clerinx and L. Kestens, Human schistosomiasis, The Lancet., 368 (2006), 1106-1118. doi: 10.1016/S0140-6736(06)69440-3.

[13]

J. K. Hale, "Theory of Functional Differential Equations," Second edition, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.

[14]

J. K. Hale and S. M. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[15]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Application of Hopf Bifurcation," London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.

[17]

G. MacDonald, The dynamics of helminth infections with spatial reference to schistosomes, Trans. Roy. Soc. Trop. Med. Hyg., 59 (1965), 489-506. doi: 10.1016/0035-9203(65)90152-5.

[18]

I. Nåsell, A hybrid model of schistosomiasis with snail latency, Theor. Popul. Biol., 10 (1976), 47-69.

[19]

I. Nåsell and W. M. Hirsch, The transmission dynamics of schistosomiasis, Comm. Pure. Appl. Math., 26 (1973), 395-453.

[20]

E. M. T. Salvana and C. H. King, Schistosomiasis in travelers and immigrants, Current Infectious Disease Reports, 10 (2008), 42-49. doi: 10.1007/s11908-008-0009-8.

[21]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[22]

World Health Organization, The control of schistosomiasis, Second report of the WHO Expert Committee, World Health Organ Tech Rep Ser., 830 (1993), 1-86.

[23]

J. Wu, N. Liu and S. Zuo, The qualitative analysis of model of the transmission dynamics of Japanese schistosomiasis, Applied Mathematics-A Journal of Chinese Universities, 2 (1987), 352-362.

[24]

J. Wu and Z. Feng, Mathematical models for schistosomiasis with delays and multiple definitive hosts, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory" (Minneapolis, MN, 1999), IMA Vol. Math. Appl., 126, Springer, New York, (2002), 215-229.

[25]

H. M. Yang, Comparison between schistosomiasis transmission modelings considering acquired immunity and age-structured contact pattern with infested water, Math. Biosci., 184 (2003), 1-26. doi: 10.1016/S0025-5564(03)00045-2.

[26]

P. Zhang, Z. Feng and F. A. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Math. Biosci., 205 (2007), 83-107. doi: 10.1016/j.mbs.2006.06.006.

[27]

X. Zhou, T. Wang and L. Wang, The current status of schistosomiasis epidemics in China (in Chinese), Zhonghua Liu Xing Bing Xue Za Zhi, 25 (2004), 555-558.

show all references

References:
[1]

J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer-Verlag, New York-Berlin, 1981.

[2]

C. Castillo-Chavez, Z. Feng and D. Xu, A schistosomiasis model with mating structure and time delay, Math. Biosci., 211 (2008), 333-341. doi: 10.1016/j.mbs.2007.11.001.

[3]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982.

[4]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcial. Ekvac., 29 (1986), 77-90.

[5]

C. L. Cosgrove and V. R. Southgate, Mating interactions between Schistosoma mansoni and S. margrebowiei, Parasitology, 125 (2002), 233-243. doi: 10.1017/S0031182002002111.

[6]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

[7]

T. Faria, Normal forms and bifurcations for delay differential equations, in "Delay Differential Equations and Applications," NATO Science Series II: Mathematics, Physics and Chemistry, 205, Springer, Dordrecht, (2006), 227-282.

[8]

Z. Feng, C.-C. Li and F. A. Milner, Schistosomiasis models with density dependence and age of infection in snail dynamics, Math. Biosci., 177/178 (2002), 271-286. doi: 10.1016/S0025-5564(01)00115-8.

[9]

Z. Feng, C.-C. Li and F. A. Milner, Schistosomiasis models with two migrating human groups, Math. Comput. Modelling., 41 (2005), 1213-1230. doi: 10.1016/j.mcm.2004.10.023.

[10]

M. Golubitsky and D. G Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Applied Mathematical Sciences, 51, Springer-Verlag, New York, 1985.

[11]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[12]

B. Gryseels, K. Polman, J. Clerinx and L. Kestens, Human schistosomiasis, The Lancet., 368 (2006), 1106-1118. doi: 10.1016/S0140-6736(06)69440-3.

[13]

J. K. Hale, "Theory of Functional Differential Equations," Second edition, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.

[14]

J. K. Hale and S. M. Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[15]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Application of Hopf Bifurcation," London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.

[17]

G. MacDonald, The dynamics of helminth infections with spatial reference to schistosomes, Trans. Roy. Soc. Trop. Med. Hyg., 59 (1965), 489-506. doi: 10.1016/0035-9203(65)90152-5.

[18]

I. Nåsell, A hybrid model of schistosomiasis with snail latency, Theor. Popul. Biol., 10 (1976), 47-69.

[19]

I. Nåsell and W. M. Hirsch, The transmission dynamics of schistosomiasis, Comm. Pure. Appl. Math., 26 (1973), 395-453.

[20]

E. M. T. Salvana and C. H. King, Schistosomiasis in travelers and immigrants, Current Infectious Disease Reports, 10 (2008), 42-49. doi: 10.1007/s11908-008-0009-8.

[21]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[22]

World Health Organization, The control of schistosomiasis, Second report of the WHO Expert Committee, World Health Organ Tech Rep Ser., 830 (1993), 1-86.

[23]

J. Wu, N. Liu and S. Zuo, The qualitative analysis of model of the transmission dynamics of Japanese schistosomiasis, Applied Mathematics-A Journal of Chinese Universities, 2 (1987), 352-362.

[24]

J. Wu and Z. Feng, Mathematical models for schistosomiasis with delays and multiple definitive hosts, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory" (Minneapolis, MN, 1999), IMA Vol. Math. Appl., 126, Springer, New York, (2002), 215-229.

[25]

H. M. Yang, Comparison between schistosomiasis transmission modelings considering acquired immunity and age-structured contact pattern with infested water, Math. Biosci., 184 (2003), 1-26. doi: 10.1016/S0025-5564(03)00045-2.

[26]

P. Zhang, Z. Feng and F. A. Milner, A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Math. Biosci., 205 (2007), 83-107. doi: 10.1016/j.mbs.2006.06.006.

[27]

X. Zhou, T. Wang and L. Wang, The current status of schistosomiasis epidemics in China (in Chinese), Zhonghua Liu Xing Bing Xue Za Zhi, 25 (2004), 555-558.

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