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Stability of the Endemic Coexistence Equilibrium for One Hostand Two Parasites

Published online by Cambridge University Press:  08 April 2010

T. Dhirasakdanon  and
H. R. Thieme
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Abstract

For an SI type endemic model with one host and two parasite strains, we study thestability of the endemic coexistence equilibrium, where the host and both parasite strainsare present. Our model, which is a system of three ordinary differential equations,assumes complete cross-protection between the parasite strains and reduced fertility andincreased mortality of infected hosts. It also assumes that one parasite strain isexclusively vertically transmitted and cannot persists just by itself. We give severalsufficient conditions for the equilibrium to be locally asymptotically stable. One of themis that the horizontal transmission is of density-dependent (mass-action) type. If thehorizontal transmission is of frequency-dependent (standard) type, we show that, undercertain conditions, the equilibrium can be unstable and undamped oscillations can occur.We support and extend our analytical results by numerical simulations and bytwo-dimensional plots of stability regions for various pairs of parameters.

Keywords

coexistence of parasite strainsdisease incidenceSI endemic modelRouth-Hurwitz conditionsundamped oscillationsvertical transmissionhorizontal transmissiondisease-related fertility reduction
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 6: Ecology (Part 2) , 2010 , pp. 109 - 138
Copyright
© EDP Sciences, 2010

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References

Allen, L.J.S., Langlais, M. Phillips, C.J.. The dynamics of two viral infections in a single host population with applications to hantavirus . Math. Biosci., 186 (2003), 191–217.CrossRefGoogle Scholar
Anderson, R.M., Jackson, H.C., May, R.M. Smith, A.D.M.. Population dynamics of fox rabies in Europe . Nature, 289 (1981), 765–771.CrossRefGoogle Scholar
V. Andreasen. Multiple times scales in the dynamics of infectious diseases. Mathematical Approaches to Problems in Resource Management and Epidemiology (C. Castillo-Chavez, S.A. Levin, C.A. Shoemaker, eds.), 142–151, Springer, Berlin Heidelberg, 1989.
Andreasen, V., Lin, J. Levin, S.A.. The dynamics of cocirculating influenza strains conferring partial cross-immunity . J. Math. Biol., 35 (1997), 825–842.CrossRefGoogle ScholarPubMed
Banerjee, C., Allen, L.J.S. Salazar-Bravo, J.. Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission . Math. Biosci. Engin., 5 (2008), 617–645.Google Scholar
Bang, F.B.. Epidemiological interference . Intern. J. Epidemiology, 4 (1975), 337–342.CrossRefGoogle ScholarPubMed
Briggs, C.J. Godfray, H.C.J.. The dynamics of insect-pathogen interactions in stage-structured populations . The American Naturalist, 145 (1995), 855–887.CrossRefGoogle Scholar
Castillo-Chavez, C., Hethcote, H.W., Andreasen, V., Levin, S.A. Liu, W.M.. Epidemiological models with age structure, proportionate mixing, and cross-immunity . J. Math. Biol., 27 (1989), 233–258.CrossRefGoogle ScholarPubMed
T. Dhirasakdanon, H.R. Thieme. Persistence of vertically transmitted parasite strains which protect against more virulent horizontally transmitted strains. Modeling and Dynamics of Infectious Diseases (Z. Ma, Y. Zhou, J. Wu, eds.), 187–215, World Scientific, Singapore, 2009.
Diekmann, O. Kretzschmar, M.. Patterns in the effects of infectious diseases on population growth . J. Math. Biol., 29 (1991), 539–570.CrossRefGoogle ScholarPubMed
Dietz, K.. Epidemiologic interference of virus populations . J. Math. Biol., 8 (1979), 291–300.CrossRefGoogle ScholarPubMed
K. Dietz. Overall population patterns in the transmission cycle of infectious disease agents. Population Biology of Infectious Diseases (R.M. Anderson, R.M. May, eds.), 87–102, Springer, Dahlem Konferenzen, Berlin, 1982.
Faeth, S.H., Hadeler, K.P. Thieme, H.R.. An apparent paradox of horizontal and vertical disease transmission . J. Biol. Dyn., 1 (2007), 45–62.CrossRefGoogle ScholarPubMed
Feng, Z. Thieme, H.R.. Recurrent outbreaks of childhood diseases revisited: the impact of isolation . Math. Biosci., 128 (1995), 93–130.CrossRefGoogle ScholarPubMed
Feng, Z. Thieme, H.R.. Endemic models with arbitrarily distributed periods of infection. II. Fast disease dynamics and permanent recovery . SIAM J. Appl. Math., 61 (2000), 983–1012.CrossRefGoogle Scholar
L.Q. Gao, J. Mena-Lorca, H.W. Hethcote. Variations on a theme of SEI endemic models. Differential Equations and Applications to Biology and Industry (M. Martelli, C.L. Cooke, E. Cumberbatch, B. Tang, H.R. Thieme, eds.), 191–207, World Scientific, Singapore, 1996.
Getz, W.M. Pickering, J.. Epidemic models: thresholds and population regulation . The American Naturalist, 121 (1983), 892–898.CrossRefGoogle Scholar
Greenhalgh, D.. Some results for an SEIR epidemic model with density dependence in the death rate . IMA J. Math. Appl. Med. Biol., 9 (1992), 67–106.CrossRefGoogle ScholarPubMed
Greenhalgh, D.. Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity . Math. Comput. Modelling, 25 (1997), 85–107.CrossRefGoogle Scholar
Greenman, J.V. Hudson, P.J.. Infected coexistence instability with and without density-dependent regulation . J. Theor. Biol., 185 (1997), 345–356.CrossRefGoogle Scholar
Haine, E.R.. Symbiont-mediated protection . Proc. R. Soc. B, 275 (2008), 353–361.CrossRefGoogle ScholarPubMed
H.W. Hethcote, S.A. Levin. Periodicity in epidemiological models. Applied Mathematical Ecology (S.A. Levin, T.G. Hallam, L.J. Gross, eds.), 193–211, Springer, Berlin Heidelberg, 1989.
Hethcote, H.W., Stech, H.W. van den Driessche, P.. Nonlinear oscillations in epidemic models . SIAM J. Appl. Math., 40 (1981), 1–9.CrossRefGoogle Scholar
Hethcote, H.W., Wang, W. Li, Y.. Species coexistence and periodicity in host-host-pathogen models . J. Math. Biol., 51 (2005), 629–660.CrossRefGoogle ScholarPubMed
Hethcote, H.W. Pickering, J.. Infectious disease and species coexistence: a model of Lotka-Volterra form . Am. Nat., 126 (1985), 196–211.Google Scholar
Iannelli, M., Martcheva, M. Li, X.-Z.. Strain replacement in an epidemic model with super-infection and perfect vaccination . Math. Biosci., 195 (2005), 23–46.CrossRefGoogle Scholar
Li, J., Zhou, Y., Ma, Z. Hyman, J.M.. Epidemiological models for mutating pathogens . SIAM J. Appl. Math., 65 (2004), 1–23.CrossRefGoogle Scholar
Lin, J., Andreasen, V. Levin, S.A.. Dynamics of influenza A drift: the linear three-strain model . Math. Biosci., 162 (1999), 33–51.CrossRefGoogle ScholarPubMed
Lipsitch, M., Siller, S. Nowak, M.A.. The evolution of virulence in pathogens with vertical and horizontal transmission . Evolution, 50 (1996), 1729–1741.CrossRefGoogle ScholarPubMed
Liu, W.-m.. Dose-dependent latent period and periodicity of infectious diseases . J. Math. Biol., 31 (1993), 487–494.CrossRefGoogle ScholarPubMed
Lively, C.M., Clay, K., Wade, M.J. Fuqua, C.. Competitive co-existence of vertically and horizontally transmitted diseases . Evolutionary Ecology Res., 7 (2005), 1183–1190.Google Scholar
M. Martcheva. On the mechanisms with strain replacement in epidemic models with vaccination. Current Developments in Mathematical Biology (R.C. John Boucher, K. Mahdavi, eds.), 149–165, World Scientific, Hackensack, 2007.
Martcheva, M. Pilyugin, S.S.. The role of coinfection in multidisease dynamics . SIAM J. Appl. Math., 66 (2006), 843–872.CrossRefGoogle Scholar
Meijer, G. Leuchtmann, A.. The effects of genetic and environmental factors on disease expression (stroma formation) and plant growth in Brachypodium sylvaticum infected by Epichloë sylvatica . OIKOS, 91 (2000), 446–458.CrossRefGoogle Scholar
Milner, F.A. Pugliese, A.. Periodic solutions: a robust numerical method for an S-I-R model of epidemics . J. Math. Biol., 39 (1999), 471–492.CrossRefGoogle ScholarPubMed
Nuño, M., Feng, Z., Martcheva, M. Castillo-Chavez, C.. Dynamics of two-strain influenza with isolation and partial cross-immunity . SIAM J. Appl. Math., 65 (2005), 964–982.CrossRefGoogle Scholar
A. Pugliese. An S→E→I epidemic model with varying population size. Differential Equations Models in Biology, Epidemiology and Ecology (S. Busenberg, M. Martelli, eds.), 121–138, Springer, Berlin Heidelberg, 1991.
Saikkonen, K., Faeth, S.H., Helander, M. Sullivan, T.J.. Fungal endophytes: a continuum of interactions with host plants . Annu. Rev. Ecol. Syst., 29 (1998), 319–343.CrossRefGoogle Scholar
Swart, J.H.. Hopf bifurcation and stable limit cycle behavior in the spread of infectious disease, with special application to fox rabies . Math. Biosci., 95 (1989), 199–207.CrossRefGoogle ScholarPubMed
H.R. Thieme. Stability change of the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases. Differential Equations Models in Biology, Epidemiology and Ecology (S. Busenberg, M. Martelli, eds.), 139–158, Springer, Berlin Heidelberg, 1991.
H.R. Thieme. Mathematics in Population Biology. Princeton University Press, Princeton, 2003.
Thieme, H.R. Castillo-Chavez, C.. How may infection-age dependent infectivity affect the dynamics of HIV/AIDS? . SIAM J. Appl. Math., 53 (1993), 1447–1479.CrossRefGoogle Scholar
Thieme, H.R., Tridane, A. Kuang, Y.. An epidemic model with post-contact prophylaxis of distributed length. II. Stability and oscillations if treatment is fully effective . Math. Model. Nat. Phenom., 3 (2008), 267–293.CrossRefGoogle Scholar
van den Driessche, P. Zeeman, M.L.. Disease induced oscillations between two competing species . SIAM J. Appl. Dyn. Sys., 3 (2004), 601–619.CrossRefGoogle Scholar
Venturino, E.. The effects of diseases on competing species . Math. Biosci., 174 (2001), 111–131.CrossRefGoogle ScholarPubMed