# American Institute of Mathematical Sciences

2012, 9(4): 767-784. doi: 10.3934/mbe.2012.9.767

## Global analysis of a simple parasite-host model with homoclinic orbits

 1 Faculty of Science, Air Force Engineering University, Xi'an 710051, China, China 2 Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China

Received  January 2012 Revised  May 2012 Published  October 2012

In this paper, a simple parasite-host model proposed by Ebert et al.(2000) is reconsidered. The basic epidemiological reproduction number of parasite infection ($R_0$) and the basic demographic reproduction number of infected hosts ($R_1$) are given. The global dynamics of the model is completely investigated, and the existence of heteroclinic and homoclinic orbits is theoretically proved, which implies that the outbreak of parasite infection may happen. The thresholds determining the host extinction in the presence of parasite infection and variation in the equilibrium level of the infected hosts with $R_0$ are found. The effects of $R_0$ and $R_1$ on dynamics of the model are considered and we show that the equilibrium level of the infected host may not be monotone with respect to $R_0$. In particular, it is found that full loss of fecundity of infected hosts may lead to appearance of the singular case.
Citation: Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model with homoclinic orbits. Mathematical Biosciences & Engineering, 2012, 9 (4) : 767-784. doi: 10.3934/mbe.2012.9.767
##### References:
 [1] D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: Experimental epidemiology with Daphnia and six microparasites,, American Naturalist, 156 (2000), 459.  doi: 10.1086/303404.  Google Scholar [2] T.-W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations,, J. Math. Biol., 46 (2003), 17.  doi: 10.1007/s00285-002-0165-7.  Google Scholar [3] Kaifa Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems,, Comm. Pure Appl. Anal., 10 (2011), 1537.  doi: 10.3934/cpaa.2011.10.1537.  Google Scholar [4] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth,, J. Math. Biol., 60 (2010), 573.  doi: 10.1007/s00285-009-0278-3.  Google Scholar [5] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth,, Math. Biosc. Eng., 6 (2009), 283.   Google Scholar [6] F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics,, Math. Biosci. Eng., 2 (2005), 133.   Google Scholar [7] Z. Zhang, T. Ding, et al., "Qualitative Theory of Differential Equations,", Translations of Mathematical Monographs, (1992).   Google Scholar [8] Zhien Ma and Jia Li, "Dynamical Modeling and Analysis of Epidemics,", Singapore, (2009).   Google Scholar

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##### References:
 [1] D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: Experimental epidemiology with Daphnia and six microparasites,, American Naturalist, 156 (2000), 459.  doi: 10.1086/303404.  Google Scholar [2] T.-W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations,, J. Math. Biol., 46 (2003), 17.  doi: 10.1007/s00285-002-0165-7.  Google Scholar [3] Kaifa Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems,, Comm. Pure Appl. Anal., 10 (2011), 1537.  doi: 10.3934/cpaa.2011.10.1537.  Google Scholar [4] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth,, J. Math. Biol., 60 (2010), 573.  doi: 10.1007/s00285-009-0278-3.  Google Scholar [5] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of HBV infection with logistic hepatocyte growth,, Math. Biosc. Eng., 6 (2009), 283.   Google Scholar [6] F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics,, Math. Biosci. Eng., 2 (2005), 133.   Google Scholar [7] Z. Zhang, T. Ding, et al., "Qualitative Theory of Differential Equations,", Translations of Mathematical Monographs, (1992).   Google Scholar [8] Zhien Ma and Jia Li, "Dynamical Modeling and Analysis of Epidemics,", Singapore, (2009).   Google Scholar
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