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

show all references

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

[1]

Tzy-Wei Hwang, Yang Kuang. Host Extinction Dynamics in a Simple Parasite-Host Interaction Model. Mathematical Biosciences & Engineering, 2005, 2 (4) : 743-751. doi: 10.3934/mbe.2005.2.743

[2]

Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989

[3]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[4]

Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060

[5]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883

[6]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

[7]

Yuanshi Wang, Donald L. DeAngelis. A mutualism-parasitism system modeling host and parasite with mutualism at low density. Mathematical Biosciences & Engineering, 2012, 9 (2) : 431-444. doi: 10.3934/mbe.2012.9.431

[8]

Liman Dai, Xingfu Zou. Effects of superinfection and cost of immunity on host-parasite co-evolution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 809-829. doi: 10.3934/dcdsb.2017040

[9]

Adam Sullivan, Folashade Agusto, Sharon Bewick, Chunlei Su, Suzanne Lenhart, Xiaopeng Zhao. A mathematical model for within-host Toxoplasma gondii invasion dynamics. Mathematical Biosciences & Engineering, 2012, 9 (3) : 647-662. doi: 10.3934/mbe.2012.9.647

[10]

Chang Gong, Jennifer J. Linderman, Denise Kirschner. A population model capturing dynamics of tuberculosis granulomas predicts host infection outcomes. Mathematical Biosciences & Engineering, 2015, 12 (3) : 625-642. doi: 10.3934/mbe.2015.12.625

[11]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237

[12]

Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999

[13]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239

[14]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[15]

Rafael Bravo De La Parra, Luis Sanz. A discrete model of competing species sharing a parasite. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019204

[16]

Denise E. Kirschner, Alexei Tsygvintsev. On the global dynamics of a model for tumor immunotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 573-583. doi: 10.3934/mbe.2009.6.573

[17]

Hongying Shu, Xiang-Sheng Wang. Global dynamics of a coupled epidemic model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076

[18]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

[19]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[20]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]