American Institute of Mathematical Sciences

Advanced
2013, 10(5&6): 1635-1650. doi: 10.3934/mbe.2013.10.1635

Chemostats and epidemics: Competition for nutrients/hosts

Hal L. Smith 1, and  Horst R. Thieme 2,

1. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804
2. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287

Received  February 2013 Revised  April 2013 Published  August 2013

In a chemostat, several species compete for the same nutrient, while in an epidemic, several strains of the same pathogen may compete for the same susceptible hosts. As winner, chemostat models predict the species with the lowest break-even concentration, while epidemic models predict the strain with the largest basic reproduction number. We show that these predictions amount to the same if the per capita functional responses of consumer species to the nutrient concentration or of infective individuals to the density of susceptibles are proportional to each other but that they are different if the functional responses are nonproportional. In the second case, the correct prediction is given by the break-even concentrations. In the case of nonproportional functional responses, we add a class for which the prediction does not only rely on local stability and instability of one-species (strain) equilibria but on the global outcome of the competition. We also review some results for nonautonomous models.
Keywords: break-even concentrations, Competitive exclusion, cross immunity, Lyapunov functions, basic reproduction number, global stability., cross protection, evolution of virulence, seasonality, coexistence.
Mathematics Subject Classification: Primary: 92D15, 92D25, 92D30; Secondary: 34D23, 37B25, 93D3.
Citation: Hal L. Smith, Horst R. Thieme. Chemostats and epidemics: Competition for nutrients/hosts. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1635-1650. doi: 10.3934/mbe.2013.10.1635
References:
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show all references

References:
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A. S. Ackleh and L. J. S. Allen, Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size,, J. Math. Biol., 47 (2003), 153.  doi: 10.1007/s00285-003-0207-9.  Google Scholar

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[5]

J. Arino, S. S. Pilyugin and G. S. K. Wolkowicz, Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models,, Can. Appl. Math. Q., 11 (2003), 107.   Google Scholar

[6]

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[7]

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[9]

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[10]

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[11]

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[12]

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[13]

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[14]

G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake,, SIAM J. Appl. Math., 45 (1985), 138.  doi: 10.1137/0145006.  Google Scholar

[15]

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[16]

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[17]

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[18]

J. M. Cushing, Two species competition in a periodic environment,, J. Math. Biol., 10 (1980), 385.  doi: 10.1007/BF00276097.  Google Scholar

[19]

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[20]

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[21]

O. Diekmann, A beginners guide to adaptive dynamics,, in, 63 (2004), 47.   Google Scholar

[22]

O. Diekmann, J. A. P. Heesterbeek and T. Britton, "Mathematical Tools for Understanding Infectious Disease Dynamics,", Princeton Series in Theoretical and Computational Biology, (2013).   Google Scholar

[23]

P. W. Ewald and G. De Leo, Alternative transmission modes and the evolution of virulence,, in, (2002), 10.  doi: 10.1017/CBO9780511525728.004.  Google Scholar

[24]

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[25]

H. I. Freedman and Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling,, J. Math. Biol., 31 (1993), 513.  doi: 10.1007/BF00173890.  Google Scholar

[26]

P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (): 337.  doi: 10.1137/060654876.  Google Scholar

[27]

B. S. Goh, Global stability in many species systems,, Amer. Nat., 111 (1977), 135.  doi: 10.1086/283144.  Google Scholar

[28]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Engin., 3 (2006), 513.  doi: 10.3934/mbe.2006.3.513.  Google Scholar

[29]

H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[30]

H. Guo, M. Y. Li and Z. Shuai, Global stability in multigroup epidemic models,, in, 11 (2009), 268.   Google Scholar

[31]

W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733.  doi: 10.1002/cpa.3160380607.  Google Scholar

[32]

S.-B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760.  doi: 10.1137/0134064.  Google Scholar

[33]

S.-B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in a continuous culture of micro-organisms,, SIAM J. App. Math., 32 (1977), 366.  doi: 10.1137/0132030.  Google Scholar

[34]

S.-B. Hsu, A competition model for a seasonally fluctuating nutrient,, J. Math. Biol., 9 (1980), 115.  doi: 10.1007/BF00275917.  Google Scholar

[35]

S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese J. Math., 9 (2005), 151.   Google Scholar

[36]

A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle,, SIAM J. Appl. Math., 67 (2006), 260.  doi: 10.1137/050643271.  Google Scholar

[37]

A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays,, Math. Biosci., 209 (2007), 51.  doi: 10.1016/j.mbs.2007.01.008.  Google Scholar

[38]

V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,", Yale University Press, (1955).   Google Scholar

[39]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75.  doi: 10.1007/s11538-008-9352-z.  Google Scholar

[40]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.  doi: 10.1007/s11538-005-9037-9.  Google Scholar

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A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

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A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate,, Math. Med. Biol., 26 (2009), 225.  doi: 10.1093/imammb/dqp009.  Google Scholar

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