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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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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[2]	A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality,, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175. doi: 10.3934/dcdsb.2005.5.175. Google Scholar
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[7]	F. B. Bader, Kinetics of double-substrate limited growth,, in, (1982), 1. Google Scholar
[8]	M. M. Ballyk, C. C. McCluskey and G. S. K. Wolkowicz, Global analysis of competition for perfectly substituable resources with linear response,, J. Math. Biol., 51 (2005), 458. doi: 10.1007/s00285-005-0333-7. Google Scholar
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[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. doi: 10.1016/S0362-546X(01)00528-4. Google Scholar
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References:

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[2]	A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality,, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175. doi: 10.3934/dcdsb.2005.5.175. Google Scholar
[3]	P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis,, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1. doi: 10.3934/dcdsb.2007.8.1. Google Scholar
[4]	R. M. Anderson and R. M. May, Coevolution of host and parasites,, Parasitology, 85 (1982), 411. doi: 10.1111/j.1095-8312.2009.01256.x. Google Scholar
[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]	R. A. Armstrong and R. McGehee, Competitive exclusion,, Amer. Natur., 115 (1980), 151. doi: 10.1086/283553. Google Scholar
[7]	F. B. Bader, Kinetics of double-substrate limited growth,, in, (1982), 1. Google Scholar
[8]	M. M. Ballyk, C. C. McCluskey and G. S. K. Wolkowicz, Global analysis of competition for perfectly substituable resources with linear response,, J. Math. Biol., 51 (2005), 458. doi: 10.1007/s00285-005-0333-7. Google Scholar
[9]	M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources,, Math. Biosci., 118 (1993), 127. doi: 10.1016/0025-5564(93)90050-K. Google Scholar
[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. doi: 10.1016/S0362-546X(01)00528-4. Google Scholar
[11]	F. F. Blackman, Optima and limiting factors,, Ann. Bot. London, 19 (1905), 281. Google Scholar
[12]	C. J. Briggs and H. C. J. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations,, Amer. Nat., 145 (1995), 855. doi: 10.1086/285774. Google Scholar
[13]	H.-J. Bremermann and H. R. Thieme, A competition exclusion principle for pathogen virulence,, J. Math. Biol., 27 (1989), 179. doi: 10.1007/BF00276102. Google Scholar
[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]	V. Capasso, "Mathematical Structures of Epidemic Systems,", Lecture Notes in Biomathematics, 97 (1993). doi: 10.1007/978-3-540-70514-7. Google Scholar
[16]	V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model,, Math. Biosci., 42 (1978), 43. doi: 10.1016/0025-5564(78)90006-8. Google Scholar
[17]	C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models,, in, (1995), 33. Google Scholar
[18]	J. M. Cushing, Two species competition in a periodic environment,, J. Math. Biol., 10 (1980), 385. doi: 10.1007/BF00276097. Google Scholar
[19]	P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Math. Biol., 11 (1981), 319. doi: 10.1007/BF00276900. Google Scholar
[20]	O. Diekmann, The many facets of evolutionary dynamics,, J. Biol. Systems, 5 (1997), 325. doi: 10.1142/S0218339097000205. Google Scholar
[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]	A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions,, Math. Model. Nat. Phenom., 2 (2007), 55. doi: 10.1051/mmnp:2008011. Google Scholar
[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
[41]	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
[42]	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
[43]	A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models,, MMB IMA, 22 (2005), 113. Google Scholar
[44]	A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,, Appl. Math. Letters, 15 (2002), 955. doi: 10.1016/S0893-9659(02)00069-1. Google Scholar
[45]	B. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates,, SIAM J. Appl. Math., 59 (1999), 411. doi: 10.1137/S003613999631100X. Google Scholar
[46]	M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434. doi: 10.1137/090779322. Google Scholar
[47]	M. Y. Li and Z. Shuai, Global stability problem for coupled systems of differential equations on networks,, J. Differential Eqns., 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar
[48]	M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38. doi: 10.1016/j.jmaa.2009.09.017. Google Scholar
[49]	X. Lin and J. W.-H. So, Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations,, J. Austral. Math. Soc. Ser. B, 34 (1993), 282. doi: 10.1017/S0334270000008900. Google Scholar
[50]	S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. Google Scholar
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