September  2011, 10(5): 1537-1548. doi: 10.3934/cpaa.2011.10.1537

Fluctuation and extinction dynamics in host-microparasite systems

1. 

Department of Mathematics, College of Medicine, Third Military Medical University, Chongqing, 400038

2. 

Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804

Received  August 2009 Revised  August 2010 Published  April 2011

Although experimental and observational studies have shown that microparasites can induce the deterministic reduction, fluctuation and extinction scenarios for its host population, most existing host-parasite interaction models fail to produce such rich dynamical behaviors simultaneously. We explore the effects of explicit dynamics of parasites under logistic host growth and different infection rate function. Our results show that the explicit dynamics of parasites and standard incidence function can induce the host density fluctuation and extinction scenario in the case of logistic host growth.
Citation: Kaifa Wang, Yang Kuang. Fluctuation and extinction dynamics in host-microparasite systems. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1537-1548. doi: 10.3934/cpaa.2011.10.1537
References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature (London), 280 (1979), 361-367.

[2]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Phil. Tran. R. Soc. Lond. B, 291 (1981), 451-524.

[3]

S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278.

[4]

M. Bralet, S. Branchereau, C. Brechot and N. Ferry, Cell lineage study in the liver using retroviral mediated gene transfer, Am. J. Pathol., 144 (1994), 896-905.

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Y. K. Chun, J. Y. Kim, H. J. Woo, S. M. Oh, I. Kang, J. Ha and S. S. Kim, No significant correlation exists between core promoter mutations, viral replication, and liver damage in chronic hepatitis B infection, Hepatology, 32 (2000), 1154-1162. doi: 10.1053/jhep.2000.19623.

[6]

S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, G. Dusheiko and A. S. Perelson, The role of cells refractory to productive infection in acute hepatitis B viral dynamics, Proc. Natl. Acad. Sci. USA, 104 (2007), 5050-5055. doi: 10.1073/pnas.0603626104.

[7]

G. Deng, Z. Wang, Y. Wang, K. Wang and Y. Fan, Dynamic determination and analysis of serum virus load in patients with chronic HBV infection, World Chin. J. Digestol., 12 (2004), 862-865.

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D. Ebert, "Ecology, Epidemiology, and Evolution of Parasitism in Daphnia," [Internet]. Bethesda (MD): National Library of Medicine (US), National Center for Biotechnology Information. Available from: \indenthttp://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books. 2005.

[9]

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, Am. Nat., 156 (2000), 459-477.

[10]

S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth, Math. Biosci. Eng., 6 (2009), 283-299. doi: 10.3934/mbe.2009.6.283.

[11]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153. doi: 10.1080/17513750701769873.

[12]

S. H. 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.

[13]

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.

[14]

T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Math. Biosci. Eng., 2 (2005), 743-751.

[15]

W. M. Lee, Acute liver failure, New Engl. J. Med., 329 (1993), 1862-1872.

[16]

R. A. MacDonald, "Lifespan" of liver cells. Autoradio-graphic study using tritiated thymidine in normal, cirrhotic, and partially hepatectomized rats, Arch. Intern. Med., 107 (1961), 335-343.

[17]

G. K. Michalopoulos, Liver Regeneration, J. Cell. Physiol., 213 (2007), 286-300. doi: 10.1002/jcp.21172.

[18]

L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky Mount. J. Math., 38 (2008), 1573-1585. doi: 10.1216/RMJ-2008-38-5-1573.

[19]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, New York, 2000.

[20]

P. Pontisso, G. Bellati, M. Brunetto, L. Chemello, G. Colloredo, R. DiStefano, M. Nicoletti, M. G. Rumi, M. G. Ruvoletto, R. Soffedini, L. M. Valenza and G. Colucci, Hepatitis C virus RNA profiles in chronically infected individuals: do they relate to disease activity? Hepatology, 29 (1999), 585-589. doi: 10.1002/hep.510290240.

[21]

J. L. Spouge, R. I. Shrager and D. S. Dimitrov, HIV-1 infection kinetics in tissue cultures, Math. Biosci., 138 (1996), 1-22. doi: 10.1016/S0025-5564(96)00064-8.

[22]

H. C. Tuckwell and F. Y. M. Wan, On the behavior of solutions in viral dynamical models, BioSystems, 73 (2004), 157-161. doi: 10.1016/j.biosystems.2003.11.004.

[23]

K. Wang, Z. Qiu and G. Deng, Study on a population dynamic model of virus infection, J. Sys. Sci. $&$ Math. Scis., 23 (2003), 433-443.

[24]

S. A. Whalley, J. M. Murray, D. Brown, G. J. M. Webster, V. C. Emery, G. M. Dusheiko and A. S. Perelson, Kinetics of acute hepatitis B virus infection in humans, J. Exp. Med., 193 (2001), 847-853. doi: 10.1084/jem.193.7.847.

[25]

D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, TRENDS in Immunology, 23 (2002), 194-200. doi: 10.1016/S1471-4906(02)02189-0.

[26]

H. R. Zhu and H. L. Smith, Stable periodic orbits for a class of three-dimensional competitive systems, J. Diff. Equa., 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.

show all references

References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature (London), 280 (1979), 361-367.

[2]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Phil. Tran. R. Soc. Lond. B, 291 (1981), 451-524.

[3]

S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 3275-3278.

[4]

M. Bralet, S. Branchereau, C. Brechot and N. Ferry, Cell lineage study in the liver using retroviral mediated gene transfer, Am. J. Pathol., 144 (1994), 896-905.

[5]

Y. K. Chun, J. Y. Kim, H. J. Woo, S. M. Oh, I. Kang, J. Ha and S. S. Kim, No significant correlation exists between core promoter mutations, viral replication, and liver damage in chronic hepatitis B infection, Hepatology, 32 (2000), 1154-1162. doi: 10.1053/jhep.2000.19623.

[6]

S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, G. Dusheiko and A. S. Perelson, The role of cells refractory to productive infection in acute hepatitis B viral dynamics, Proc. Natl. Acad. Sci. USA, 104 (2007), 5050-5055. doi: 10.1073/pnas.0603626104.

[7]

G. Deng, Z. Wang, Y. Wang, K. Wang and Y. Fan, Dynamic determination and analysis of serum virus load in patients with chronic HBV infection, World Chin. J. Digestol., 12 (2004), 862-865.

[8]

D. Ebert, "Ecology, Epidemiology, and Evolution of Parasitism in Daphnia," [Internet]. Bethesda (MD): National Library of Medicine (US), National Center for Biotechnology Information. Available from: \indenthttp://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books. 2005.

[9]

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, Am. Nat., 156 (2000), 459-477.

[10]

S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth, Math. Biosci. Eng., 6 (2009), 283-299. doi: 10.3934/mbe.2009.6.283.

[11]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153. doi: 10.1080/17513750701769873.

[12]

S. H. 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.

[13]

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.

[14]

T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Math. Biosci. Eng., 2 (2005), 743-751.

[15]

W. M. Lee, Acute liver failure, New Engl. J. Med., 329 (1993), 1862-1872.

[16]

R. A. MacDonald, "Lifespan" of liver cells. Autoradio-graphic study using tritiated thymidine in normal, cirrhotic, and partially hepatectomized rats, Arch. Intern. Med., 107 (1961), 335-343.

[17]

G. K. Michalopoulos, Liver Regeneration, J. Cell. Physiol., 213 (2007), 286-300. doi: 10.1002/jcp.21172.

[18]

L. Min, Y. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky Mount. J. Math., 38 (2008), 1573-1585. doi: 10.1216/RMJ-2008-38-5-1573.

[19]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, New York, 2000.

[20]

P. Pontisso, G. Bellati, M. Brunetto, L. Chemello, G. Colloredo, R. DiStefano, M. Nicoletti, M. G. Rumi, M. G. Ruvoletto, R. Soffedini, L. M. Valenza and G. Colucci, Hepatitis C virus RNA profiles in chronically infected individuals: do they relate to disease activity? Hepatology, 29 (1999), 585-589. doi: 10.1002/hep.510290240.

[21]

J. L. Spouge, R. I. Shrager and D. S. Dimitrov, HIV-1 infection kinetics in tissue cultures, Math. Biosci., 138 (1996), 1-22. doi: 10.1016/S0025-5564(96)00064-8.

[22]

H. C. Tuckwell and F. Y. M. Wan, On the behavior of solutions in viral dynamical models, BioSystems, 73 (2004), 157-161. doi: 10.1016/j.biosystems.2003.11.004.

[23]

K. Wang, Z. Qiu and G. Deng, Study on a population dynamic model of virus infection, J. Sys. Sci. $&$ Math. Scis., 23 (2003), 433-443.

[24]

S. A. Whalley, J. M. Murray, D. Brown, G. J. M. Webster, V. C. Emery, G. M. Dusheiko and A. S. Perelson, Kinetics of acute hepatitis B virus infection in humans, J. Exp. Med., 193 (2001), 847-853. doi: 10.1084/jem.193.7.847.

[25]

D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, TRENDS in Immunology, 23 (2002), 194-200. doi: 10.1016/S1471-4906(02)02189-0.

[26]

H. R. Zhu and H. L. Smith, Stable periodic orbits for a class of three-dimensional competitive systems, J. Diff. Equa., 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.

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