×

zbMATH — the first resource for mathematics

A competitive model in a chemostat with nutrient recycling and antibiotic treatment. (English) Zbl 1401.92069
Summary: A model for competition between antibiotic-sensitive and antibiotic-resistant bacteria with nutrient recycling and antibiotic treatment in a chemostat is considered. The sufficient and necessary conditions for boundedness of the solution and existence of non-negative equilibria are derived. The extinction and uniform persistence of antibiotic-sensitive and antibiotic-resistant bacteria are also carried out. Numerical simulations are then given to illustrate our main results.

MSC:
92C37 Cell biology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Austin, D.J.; White, N.J.; Anderson, R.M., The dynamics of drug action on the within-host population growth of infectious agents: melding pharmacokinetics with pathogen population dynamics, J. theoret. biol., 194, 3, 313-339, (1998)
[2] Cogan, N.G., Persisters, tolerance and dosing, J. theoret. biol., 238, 3, 694-703, (2006)
[3] Elmojtaba, I.; Mugisha, J.Y.T.; Hashim, M., Modelling of role of cross-immunity between two different strains of leishmania, Nonlinear anal. RWA, 11, 2175-2189, (2010) · Zbl 1196.34059
[4] Corvaisier, S.; Maire, P.; Bouvier D’Yvoire, M.; Barbaut, X.; Bleyzac, N.; Jelliffe, R., Comparisons between antimicrobial pharmacodynamic indices and bacterial Killing as described by using the zhi model, Antimicrob. agents chemother., 42, 7, 1731-1737, (1998)
[5] Nikolaou, M.; Tam, V.H., A new modeling approach to the effect of antimicrobial agents on heterogeneous microbial population, J. math. biol., 52, 154-182, (2006) · Zbl 1091.92053
[6] Regoes, R.R.; Wiuff, C.; Zappala, R.M.; Garner, K.N.; Baquero, F.; Levin, B.R., Pharmacodynamic functions: a multiparameter approach to the design of antibiotic treatment regimens, Antimicrob. agents chemother., 48, 10, 3670-3676, (2004)
[7] Wiuff, C.; Zappala, R.M.; Regoes, R.R.; Garner, K.N.; Baquero, F.; Levin, B.R., Phenotypic tolerance: antibiotic enrichment of noninherited resistance in bacterial populations, Antimicrob. agents chemother., 49, 4, 1483-1494, (2005)
[8] H. Smith, H. Thieme, Persistence of bacteria and phages in a chemostat, J. Math. Biol., in press (http://dx.doi.org/10.1007/s00285-011-0434-4). · Zbl 1303.92123
[9] Zou, L.; Chen, X.; Ruan, S.; Zhang, W., Dynamics of a model of allelopathy and bacteriocin with a single mutation, Nonlinear anal. RWA, 12, 658-670, (2011) · Zbl 1209.34061
[10] Wang, L.; Chen, L.; Nieto, J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal. RWA, 11, 1374-1386, (2010) · Zbl 1188.93038
[11] Smith, H., Bacteria competition in serial tranfer culture, Math. biosci., 229, 149-159, (2011)
[12] Webb, G.F.; D’Agata, E.M.C.; Magal, P.; Ruan, S., A model of antibiotic-resistant bacterial epidemics in hospitals, Proc. natl. acad. sci., 102, 13343-13348, (2005)
[13] D’Agata, E.M.C.; Magal, P.; Olivier, D.; Ruan, S.; Webb, G.F., Modeling antibiotic resisance in hospitals: the impact of minimizing treatment duration, J. theoret. biol., 249, 487-499, (2007)
[14] Webb, G.F.; Horn, M.A.; D’Agata, E.M.C.; Moellerring, R.C.; Ruan, S., Competition of hospital-acquired and community-acquired methicillin-resistant staphylococcus aureus strains in hospitals, J. biol. dyn., 4, 115-129, (2010) · Zbl 1315.92071
[15] Sun, H.; Lu, X.; Ruan, S., Qualitative analysis of model with different treatment protocols ot prevent antibiotic resistance, Math. biosci., 227, 56-67, (2010) · Zbl 1194.92045
[16] Leenheer, P.; Dockery, J., Senescence and antibiotic resistance in an agestructured population model, J. math. biol., 61, 475-499, (2010) · Zbl 1204.92065
[17] Imran, M.; Smith, H., The pharmacodynamics of antibiotic treatment, Comput. math. methods med., 7, 4, 229-263, (2006) · Zbl 1111.92033
[18] Smith, H.; Waltman, P., The theory of the chemostat, (1995), Cambridge University Press Cambridge, UK
[19] Tagashira, O., Permanent coexistence in chemostat models with delayed feedback control, Nonlinear anal. RWA, 10, 1443-1452, (2009) · Zbl 1162.34334
[20] Nie, H.; Wu, J., Coexistence of an unstirred chemostat model with beddington deangelis functional response and inhibitor, Nonlinear anal. RWA, 11, 3639-3652, (2010) · Zbl 1203.35128
[21] Zhang, H.; Georgescu, P.; Nieto, J.; Chen, L., Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield, Appl. math. mech. (English ed.), 30, 933-944, (2009) · Zbl 1178.34053
[22] Meng, X.; Li, Z.; Nieto, J., Dynamic analysis of michaelis – menten chemostat-type competition models with time delay and pulse in a polluted environment, J. math. chem., 47, 123-144, (2010) · Zbl 1194.92075
[23] Yuan, S.; Zhang, W.; Han, M., Global asymptotic behavior in chemostat-type competition models with delay, Nonlinear anal. RWA, 10, 1305-1320, (2009) · Zbl 1162.34335
[24] Nie, H.; Wu, J., Positive solutions of a competition model for two resources in the unstirred chemostat, J. math. anal. appl., 355, 231-242, (2009) · Zbl 1184.35159
[25] Beretta, E.; Bischi, G.I.; Solimano, F., Stability in chemostat equations with delayed nutrient recycling, J. math. biol., 28, 99-111, (1990) · Zbl 0665.45006
[26] Freedman, H.I.; Xu, Y., Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. math. biol., 31, 513-527, (1993) · Zbl 0778.92022
[27] Lu, Z.; Hadeler, K.P., Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. biosci., 148, 147-159, (1998) · Zbl 0930.92029
[28] Yuan, S.; Xiao, D.; Han, M., Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor, Math. biosci., 202, 1-28, (2006) · Zbl 1112.34062
[29] Ruan, S.; He, X.Z., Global stability in chemostat-type competition models with nutrient recycling, SIAM J. appl. math., 58, 1, 170-192, (1998) · Zbl 0912.34062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.