JAMP  Vol.7 No.4 , April 2019
Dynamical Analysis of a Schistosomiasis japonicum Model with Time Delay
In this paper, a Schistosomiasis japonicum model incorporating time delay is proposed which represents the developmental time from cercaria penetration through skins of human hosts to egg laying. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equations, the local stability of the equilibria is investigated. And it proves that Hopf bifurcations occur when the time delay passes through a sequence of critical value. Furthermore, the explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using techniques from the normal form theory and Center Manifold Theorem. Some numerical simulations which support our theoretical analysis are also conducted.
Cite this paper: Zhang, F. , Gao, S. , Cao, H. and Luo, Y. (2019) Dynamical Analysis of a Schistosomiasis japonicum Model with Time Delay. Journal of Applied Mathematics and Physics, 7, 948-967. doi: 10.4236/jamp.2019.74064.

[1]   Feng, Z., Eppert, A. and Milner, F. (2004) Estimation of Parameters Governing the Transmission Dynamics of Schistosomes. Applied Mathematics Letters, 17, 1105-1112.

[2]   Malallah, H., Al-Onaizi, T. and Shuaib, A. (2017) Schistosomiasis as a Cause of Acute Appendicitis in Non-Endemic Areas. International Journal of Surgery, 47, S15-S108.

[3]   Angeles, J., Goto, Y. and Kirinoki, M. (2011) Human Antibody Response to Thioredoxin Peroxidase-1 and Tandem Repeat Proteins as Immunodiagnostic Antigen Candidates for Schistosoma Japonicum Infection. The American Journal of Tropical Medicine and Hygiene, 85, 674-679.

[4]   Chistulo, L., Loverde, P. and Engels, D. (2004) Focus: Schistosomiasis. Nature Reviews Microbiology, 2, 2-12.

[5]   Williams, G., Sleigh, A. and Li, Y. (2002) Mathematical Modelling of Schistosomiasis Japonica: Comparison of Control Strategies in the People’s Republic of China. Acta Tropica, 82, 253-262.

[6]   Wu, Y., Li, M. and Sun, G. (2016) Asymptotic Analysis of Schistosomiasis Persistence in Models with General Functions. Journal of the Franklin Institute, 353, 4772-4784.

[7]   Anderson, R. and May, R. (1985) Helminth Infections of Humans: Mathematical Model, Population Dynamics, and Control. Advances in Parasitology, 24, 1-101.

[8]   Barbour, A. (1996) Modeling the Transmission of Schistosomiasis: An Introductory View. The American Journal of Tropical Medicine and Hygiene, 55, 135-143.

[9]   Feng, Z., Li, C. and Milner, F. (2002) Schistosomiasis Models with Density Dependence and Age of Infection in Snail Dynamics. Mathematical Biosciences, 177-178, 271-286.

[10]   Qi, L. and Cui, J. (2013) A Schistosomiasis Model with Mating Structure. Abstract and Applied Analysis, 2, 205-215.

[11]   Gao, S., He, Y. and Liu, Y. (2013) Field Transmission Intensity of Schistosoma Japonicum Measured by Basic Reproduction Ratio from Modified Barbour’s Model. Parasites & Vectors, 6, 1-10.

[12]   Zhang, P. and Milner, F. (2007) A Schistosomiasis Model with An Age-Structure in Human Hosts and Its Application to Treatment Strategies. Mathematical Biosciences, 205, 83-107.

[13]   Chiyaka, E. and Garira, W. (2009) Mathematical Analysis of the Transmission Dynamics of Schistosomiasis in the Human-Snail Host. Journal of Biological Systems, 17, 397-423.

[14]   Zhang, X., Gao, S. and Cao, H. (2014) Threshold Dynamics for a Nonautonomous Schistosomiasis Model in a Periodic Environment. Journal of Applied Mathematics and Computing, 46, 305-319.

[15]   Bhunu, C. and Tchuenche, J. (2010) Modeling the Effects of Schistosomiasis on the Transmission Dynamics of HIV/AIDS. Journal of Biological Systems, 18, 277-297.

[16]   Chen, Z., Zou, L. and Shen, D. (2010) Mathematical Modelling and Control of Schistosomiasis in Hubei Province, China. Acta Tropica, 115, 119-125.

[17]   Qi, L. and Cui, J. (2012) Modeling the Schistosomiasis on the Islets in Nanjing. International Journal of Biomathematics, 5, 189-205.

[18]   Mouhamadou, D., Abderrahman, I. and Mamadou, S. (2014) Global Analysis of a Shistosomiasis Infection with Biological Control. Applied Mathematics and Computation, 246, 731-742.

[19]   Feng, Z. and Velasco-Hernández, X. (1997) Competitive Exclusion in a Vector-Host Model for the Dengue Fever. Journal of Mathematical Biology, 35, 523-544.

[20]   Chiyaka, E., Magombedze, G. and Mutimbu, L. (2010) Modelling within Host Parasite Dynamics of Schistosomiasis. Computational and Mathematical Methods in Medicine, 11, 255-280.

[21]   Liang, S., Maszle, D. and Spear, R. (2002) A Quantitative Framework for a Multi-Group Model of Schistosomiasis Japonicum Transmission Dynamics and Control in Sichuan, China. Acta Tropica, 82, 263-277.

[22]   Ruan, S., Xiao, D. and Beier, J. (2008) On the Delayed Ross-Macdonald Model for Malaria Transmission. Bulletin of Mathematical Biology, 70, 1098-1114.

[23]   Castillo-Chavez, C., Feng, Z. and Xu, D. (2008) A Schistosomiasis Model with Mating Structure and Time Delay. Mathematical Biosciences, 211, 333-341.

[24]   Qi, L. and Cui, J. (2012) The Delayed Barbour’s Model for Schistosomiasis. International Journal of Biomathematics, 5, 117-137.

[25]   Hassard, B., Kazarino, D. and Wan, Y. (1981) Theory and Application of Hopf Bifurcation. Canbrige Universtiy Press, London, 5.,3844179