Chemostats and epidemics: Competition for nutrients/hosts

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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

Abstract
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Under a Creative Commons license

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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References:

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[6]	R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170. doi: 10.1086/283553.
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[9]	M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources, Math. Biosci., 118 (1993), 127-180. doi: 10.1016/0025-5564(93)90050-K.
[10]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis, 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.
[11]	F. F. Blackman, Optima and limiting factors, Ann. Bot. London, 19 (1905), 281-295.
[12]	C. J. Briggs and H. C. J. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations, Amer. Nat., 145 (1995), 855-887. doi: 10.1086/285774.
[13]	H.-J. Bremermann and H. R. Thieme, A competition exclusion principle for pathogen virulence, J. Math. Biol., 27 (1989), 179-190. doi: 10.1007/BF00276102.
[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), 138-151. doi: 10.1137/0145006.
[15]	V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics, 97, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-70514-7.
[16]	V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.
[17]	C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in "Mathematical Population Dynamics. Analysis of Heterogeneity. Vol. One. Theory of Epidemics" (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Wuerz, Winnipeg, (1995), 33-50.
[18]	J. M. Cushing, Two species competition in a periodic environment, J. Math. Biol., 10 (1980), 385-400. doi: 10.1007/BF00276097.
[19]	P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335. doi: 10.1007/BF00276900.
[20]	O. Diekmann, The many facets of evolutionary dynamics, J. Biol. Systems, 5 (1997), 325-339. doi: 10.1142/S0218339097000205.
[21]	O. Diekmann, A beginners guide to adaptive dynamics, in "Mathematical Modelling of Population Dynamics," Banach Center Publications, 63, Polish Acad. Sci., (2004), 47-86.
[22]	O. Diekmann, J. A. P. Heesterbeek and T. Britton, "Mathematical Tools for Understanding Infectious Disease Dynamics," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2013.
[23]	P. W. Ewald and G. De Leo, Alternative transmission modes and the evolution of virulence, in "Adaptive Dynamics of Infectious Diseases: In Pursuit of Virulence Management" (eds. U. Dieckmann, J. A. J. Metz, M. W. Sabelis and K. Sigmund), International Institute for Applied Systems Analysis, Cambridge University Press, Cambridge, (2002), 10-25. doi: 10.1017/CBO9780511525728.004.
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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), 513-527. doi: 10.1007/BF00173890.
[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.
[27]	B. S. Goh, Global stability in many species systems, Amer. Nat., 111 (1977), 135-142. doi: 10.1086/283144.
[28]	H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Engin., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513.
[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), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.
[30]	H. Guo, M. Y. Li and Z. Shuai, Global stability in multigroup epidemic models, in "Modeling and Dynamics of Infectious Diseases" (eds. Z. Ma, Y. Zhou and J. Wu), Ser. Contemp. Appl. Math. CAM, 11, Higher Ed. Press, Beijing, (2009), 268-288.
[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), 733-753. doi: 10.1002/cpa.3160380607.
[32]	S.-B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.
[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), 366-383. doi: 10.1137/0132030.
[34]	S.-B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132. doi: 10.1007/BF00275917.
[35]	S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwanese J. Math., 9 (2005), 151-173.
[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), 260-278. doi: 10.1137/050643271.
[37]	A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays, Math. Biosci., 209 (2007), 51-75. doi: 10.1016/j.mbs.2007.01.008.
[38]	V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes," Yale University Press, New Haven, 1955.
[39]	A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1007/s11538-008-9352-z.
[40]	A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.
[41]	A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.
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