Abstract
Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.
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References
Albrich WC, Monnet DL, Harbarth S (2004) Antibiotic selection pressure and resistance in Streptococcus pneumoniae and Streptococcus pyogenes. Emerg Infect Dis 10(3): 514–517
Ayati BP, Klapper I (2007) A multiscale model of biofilm as a senescence-structured fluid. SIAM Multi Model Sim 6: 347–365
Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S (2005) Bacterial persistence as a phenotypic switch. Science 305: 1622–1625
Bigger J (1944) Treatment of staphylococcal infections with penicillin by intermittent sterilisation. The Lancet 244: 497–500
Cogan NG (2006) Effects of persister formation on bacterial response dosing. J Theor Biol 238: 694–703
Cogan NG (2007) Incorporating toxin hypothesis into a mathematical model of persister formation and dynamics. J Theor Biol 248: 340–349
Consortium REX (2007) Structure of the scientific community modelling the evolution of resistance. PLoS ONE 2(12): e1275. doi:10.1371/journal.pone.0001275
Davies J (1994) Inactivation of antibiotics and the dissemination of resistance genes. Science 264: 375–382
D’Agata EMC, Magal P, Olivier D, Ruan S, Webb GF (2007) Modeling antibiotic resistance in hospitals: the impact of minimizing treatment duration. J Theor Biol 249: 487–499
D’Agata EMC, Dupont-Rouzeyrol M, Magal P, Olivier D, Ruan S (2008) The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria. PLoS ONE 3(12): e4036. doi:10.1371/journal.pone.0004036
D’Agata EMC, Webb GF, Horn MA, Moellering RC Jr, Ruan S (2009) Modeling the invasion of community-acquired methicillin-resistant Staphylococcus aureusi into the hospital setting. Clin Infect Dis 48: 274–284
De Leenheer P, Cogan NG (2009) Failure of antibiotic treatment in microbial populations. J Math Biol 59(4): 563–579
Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J Math Biol 28: 365–382
Grundmann H, Hellriegel B (2006) Mathematical modeling: a tool for hospital infection control. Lancet Infect Dis 6: 39–45
Hardin G (1960) Competitive exclusion principle. Science 131: 1292–1297
Hirsch MW, Hanisch H, Gabriel J-P (1985) Differential equation models for some parasitic infections: methods for the study of asymptotic behavior. Comm Pure Appl Math 38: 733–753
Imran M, Smith HL (2006) The pharmacodynamics of antibiotic treatment. J Comp Math Meth Med 7: 229–263
Keren I, Kaldalu N, Spoering A, Wang Y, Lewis K (2004) Persister cells and tolerance to antimicrobials. FEMS Microbiol Lett 230: 13–18
Klapper I, Gilbert P, Ayati BP, Dockery J, Stewart PS (2007) Senescence can explain microbial persistence. Microbiology 153: 3623–3630
Korona R, Nakatsu CH, Forney LJ, Lenski RE (1994) Evidence for multiple adaptive peaks from populations of bacteria evolving in a structured habitat. Proc Natl Acad Sci USA 91: 9037–9041
Kussell E, Leibler S (2005) Phenotypic diversity, population growth, and information in fluctuating environments. Science 309: 275–278
Kussell E, Kishony R, Balaban NQ, Leibler S (2005) Bacterial persistence: a model of survival in changing environments. Genetics 169: 1807–1814
Layden T, Layden J, Ribeiro R, Perelson A (2003) Mathematical modeling of viral kinetics: a tool to understand and optimize therapy. Clin Liver Dis 7: 163–178
Lenas P, Pavlou S (1995) Coexistence of three microbial populations in a chemostat with periodically varying dilution rate. Math Biosci 129: 111–142
Levin BR (2001) Minimizing potential resistance: a population dynamics view. Clin Infect Dis 33: S161–S169
Levy S, Marshall B (2004) Antibacterial resistance worldwide: causes, challenges and responses. Nat Med 10: S122–S129
Lewis K (2001) Riddle of biofilm resistance. Antimicrob Agents Chemoth 45: 999–1007
Lewis K (2007) Persister cells, dormancy and infectious disease. Nat Rev Micro 5: 48–56
Lipsitch M, Levin B (1997) The population dynamics of antimicrobial chemotherapy. Antimicrob Agents Chemother 41: 363–373
Lindner AB, Madden R, Demarez A, Stewart EJ, Taddei F (2008) Asymmetric segregation of protein aggregates is associated with cellular aging and rejuvenation. Proc Natl Acad Sci USA 105: 3076–3081
Martinez JL (2008) Antibiotics and antibiotic resistance genes in natural environments. Science 321: 365–367
McGowan JE (1983) Antimicrobial resistance in hospital organisms and its relation to antibiotic use. Rev Inf Dis 5(6): 1033–1048
Ochman H, Lawrence JG, Groisman EA (2000) Lateral gene transfer and the nature of bacterial innovation. Nature 405: 299–304
Perelson A, Nelson P (1999) Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev 41: 3–44
Siegel JD, Rhinehart E, Jackson M, Chiarello L (2007) Management of multidrug-resistant organisms in health care settings, 2006. Am J Infect Control 35: S165–S193
Spellberg B, Guidos R, Gilbert D, Bradley J, Boucher HW, Scheld WM, Bartlett JG, Edwards J Jr (2008) The epidemic of antibiotic-resistant infections: a call to action for the medical community from the Infectious Diseases Society of America. Clin Infect Dis 46: 155–164
Smith HL, Waltman P (1995) The theory of the chemostat. Cambridge University Press, Cambridge
Smith HL, Waltman P (1999) Perturbation of a globally stable steady state. Proc AMS 127: 447–453
Stewart EJ, Madden R, Paul G, Taddei F (2005) Aging and death in an organism that reproduces by morphologically symmetric division. PLoS Comp Biol 3: 295–300
Thieme HR (1993) Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM J Math Anal 24: 407–435
Thieme HR (2003) Mathematics in population biology. Princeton University Press, Princeton
Temime L, Hejblum G, Setbon M, Valleron AJ (2008) The rising impact of mathematical modeling in epidemiology: antibiotic resistance research as a case study. Epidemiol Infect 136: 289–298
Turner PE, Souza V, Lenski RE (1996) Tests of ecological mechanisms promoting the stable coexistence of two bacterial genotypes. Ecology 77: 2119–2129
Vance RR (1985) The stable coexistence of two competitors for one resource. Am Nat 126: 72–86
van den Driessche P, Watmough J (2002) Reproduction numbers and sub-treshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180: 29–48
Wiuff C, Anderson D (2007) Antibiotic treatment in vitro of phenotypically tolerant bacterial populations. J Antimicrob Chemother 59: 254–263
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P. De Leenheer was supported in part by NSF grant DMS-0614651. T. Gedeon was supported in part by NSF grant DMS-0818785. S. S. Pilyugin was supported in part by NSF grant DMS-0818050.
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De Leenheer, P., Dockery, J., Gedeon, T. et al. Senescence and antibiotic resistance in an age-structured population model. J. Math. Biol. 61, 475–499 (2010). https://doi.org/10.1007/s00285-009-0302-7
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DOI: https://doi.org/10.1007/s00285-009-0302-7