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

Previous Article
The impact of migrant workers on the tuberculosis transmission: General models and a case study for China
MBE Home
This Issue
Next Article
Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat

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

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

Jianquan Li ^1, , Yanni Xiao ^2, and Yali Yang ^1,

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

Abstract
Related Papers

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.

Keywords: homoclinic orbit, Parasite-host model, global dynamics, reproduction number..

Mathematics Subject Classification: 92D30, 34C37, 37G3.

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), 459-477. doi: 10.1086/303404.
[2]	T.-W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46 (2003), 17-30. doi: 10.1007/s00285-002-0165-7.
[3]	Kaifa Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems, Comm. Pure Appl. Anal., 10 (2011), 1537-1548. doi: 10.3934/cpaa.2011.10.1537.
[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), 573-590. doi: 10.1007/s00285-009-0278-3.
[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), 283-299.
[6]	F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Math. Biosci. Eng., 2 (2005), 133-152.
[7]	Z. Zhang, T. Ding, et al., "Qualitative Theory of Differential Equations," Translations of Mathematical Monographs, Vol. 101, Amer. Math. Soc., Providence, Rhode Island, 1992.
[8]	Zhien Ma and Jia Li, "Dynamical Modeling and Analysis of Epidemics," Singapore, 2009.

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), 459-477. doi: 10.1086/303404.
[2]	T.-W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46 (2003), 17-30. doi: 10.1007/s00285-002-0165-7.
[3]	Kaifa Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems, Comm. Pure Appl. Anal., 10 (2011), 1537-1548. doi: 10.3934/cpaa.2011.10.1537.
[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), 573-590. doi: 10.1007/s00285-009-0278-3.
[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), 283-299.
[6]	F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Math. Biosci. Eng., 2 (2005), 133-152.
[7]	Z. Zhang, T. Ding, et al., "Qualitative Theory of Differential Equations," Translations of Mathematical Monographs, Vol. 101, Amer. Math. Soc., Providence, Rhode Island, 1992.
[8]	Zhien Ma and Jia Li, "Dynamical Modeling and Analysis of Epidemics," Singapore, 2009.

[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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237
[12]	Harrison Watts, Arti Mishra, Dang H. Nguyen, Tran D. Tuong. Dynamics of a vector-host model under switching environments. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6463-6481. doi: 10.3934/dcdsb.2021029
[13]	Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999
[14]	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
[15]	Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[16]	Rafael Bravo De La Parra, Luis Sanz. A discrete model of competing species sharing a parasite. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2121-2142. doi: 10.3934/dcdsb.2019204
[17]	Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328
[18]	Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6229-6252. doi: 10.3934/dcdsb.2021016
[19]	Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261
[20]	Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259

2018 Impact Factor: 1.313

Tools

Metrics

Other articles
by authors

on AIMS
Jianquan Li Yanni Xiao Yali Yang
on Google Scholar
Jianquan Li Yanni Xiao Yali Yang

[Back to Top]