`a`
Mathematical Biosciences and Engineering (MBE)
 

Chemostats and epidemics: Competition for nutrients/hosts
Pages: 1635 - 1650, Issue 5/6, October/December 2013

doi:10.3934/mbe.2013.10.1635      Abstract        References        Full text (414.6K)                  Related Articles

Hal L. Smith - School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, United States (email)
Horst R. Thieme - School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States (email)

1 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), 153-168.       
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), 175-188.       
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), 1-17.       
4 R. M. Anderson and R. M. May, Coevolution of host and parasites, Parasitology, 85 (1982), 411-426.
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), 107-142.       
6 R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur., 115 (1980), 151-170.       
7 F. B. Bader, Kinetics of double-substrate limited growth, in "Microbial Population Dynamics" (ed. M. J. Bazin), CRC Series in Mathematical Models in Microbiology, CRC Press, Boca Raton, FL, (1982), 1-32.
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), 458-490.       
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.       
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.       
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.
13 H.-J. Bremermann and H. R. Thieme, A competition exclusion principle for pathogen virulence, J. Math. Biol., 27 (1989), 179-190.       
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.       
15 V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics, 97, Springer-Verlag, Berlin, 1993.       
16 V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model, Math. Biosci., 42 (1978), 43-61.       
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.       
19 P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol., 11 (1981), 319-335.       
20 O. Diekmann, The many facets of evolutionary dynamics, J. Biol. Systems, 5 (1997), 325-339.
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.
24 A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73.       
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.       
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 (2006/07), 337-353.       
27 B. S. Goh, Global stability in many species systems, Amer. Nat., 111 (1977), 135-142.
28 H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Engin., 3 (2006), 513-525.       
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.       
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.       
32 S.-B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.       
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.       
34 S.-B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115-132.       
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.       
37 A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays, Math. Biosci., 209 (2007), 51-75.       
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.       
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.       
41 A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.       
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), 225-239.       
43 A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, MMB IMA, 22 (2005), 113-128.
44 A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Letters, 15 (2002), 955-960.       
45 B. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates, SIAM J. Appl. Math., 59 (1999), 411-422.       
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), 2434-2448.       
47 M. Y. Li and Z. Shuai, Global stability problem for coupled systems of differential equations on networks, J. Differential Eqns., 248 (2010), 1-20.       
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), 38-47.       
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), 282-295.       
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), 675-685.       
51 P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140.       
52 M. Martcheva, A non-autonomous multi-strain SIS epidemic model, J. Biol. Dyn., 3 (2009), 235-251.       
53 M. Martcheva, S. S. Pilyugin and R. D. Holt, Subthreshold and superthreshold coexistence of pathogen variants: The impact of host structure, Math. Biosci., 207 (2007), 58-77.       
54 C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603-614.       
55 C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, J. Math. Anal. Appl., 338 (2008), 518-535.       
56 C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Engin., 6 (2009), 603-610.       
57 C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59.       
58 C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109.       
59 C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Engin., 7 (2010), 837-850.       
60 J. Mena-Lorca, J. X. Velasco-Hernandez and C. Castillo-Chavez, Density-dependent dynamics and superinfection in an epidemic model, IMA J. Math. Appl. Med. Biol., 16 (1999), 307-317.
61 J. A. J. Metz, S. D. Mylius and O. Diekmann, When does evolution optimise? Evolutionary Ecology Research, 10 (2008), 629-654.
62 J. Prüss, L. Pujo-Menjouet and G. F. Webb, Analysis of a model for the dynamics of prions, Discr. Contin. Dyn. Syst. B, 6 (2006), 225-235.       
63 J. Roughgarden, "Theory of Population Genetics and Evolutionary Ecology: An Introduction," Macmillan, New York, 1979.
64 S. Ruan and X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math., 58 (1998), 170-192.       
65 T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Math. Biosci. Eng., 8 (2011), 827-840.       
66 H. L. Smith, Competitive coexistence in an oscillating chemostat, SIAM J. Appl. Math., 40 (1981), 498-522.       
67 H. L. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995.       
68 S. Tennenbaum, T. G. Kassem, S. Roudenko and C. Castillo-Chavez, The role of transactional sex in spreading HIV in Nigeria, in "Mathematical Studies on Human Disease Dynamics" (eds. Abba B. Gumel, Carlos Castillo-Chavez, Ronald E. Mickens and Dominic P. Clemence), Contemporary Mathematics, 410, Amer. Math. Soc., Providence, RI, (2006), 367-389.       
69 H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.       
70 H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Eqns., 250 (2011), 3772-3801.       
71 H. R. Thieme, Pathogen competition and coexistence and the evolution of virulence, in "Mathematics for Life Sciences and Medicine" (eds. Y. Takeuchi, Y. Iwasa and K. Sato), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 123-153.       
72 V. Volterra, "Leçons sur la Théorie Mathématique de la Lutte pour la Vie," Gauthier-Villars, Paris, 1931.
73 E. B. Wilson and J. Worcester, The law of mass action in epidemiology, Part I, Proc. Nat. Acad. Sci., 31 (1945), 24-34; Part II, Proc. Nat. Acad. Sci., 31 (1945), 109-116.       
74 G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.       
75 G. S. K. Wolkowicz, M. M. Ballyk and S. P. Daoussis, Interaction in a chemostat: Introduction of a competitor can promote greater diversity, Rocky Mountain Journal of Mathematics, 25 (1995), 515-543.       
76 G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.       
77 G. S. K. Wolkowicz and H. Xia, Global asymptotic behavior of a chemostat model with discrete delay, SIAM J. Appl. Math., 57 (1997), 1019-1043.       
78 G. S. K. Wolkowicz and X.-Q. Zhao, $N$-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491.       

Go to top