Skip to main content

Advertisement

SpringerLink
Log in

Selecting Against Antibiotic-Resistant Pathogens: Optimal Treatments in the Presence of Commensal Bacteria

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Using optimal control theory as the basic theoretical tool, we investigate the efficacy of different antibiotic treatment protocols in the most exacting of circumstances, described as follows. Viewing a continuous culture device as a proxy for a much more complex host organism, we first inoculate the device with a single bacterial species and deem this the ‘commensal’ bacterium of our host. We then force the commensal to compete for a single carbon source with a rapidly evolving and fitter ‘pathogenic bacterium’, the latter so-named because we wish to use a bacteriostatic antibiotic to drive the pathogen toward low population densities. Constructing a mathematical model to mimic the biology, we do so in such a way that the commensal would be eventually excluded by the pathogen if no antibiotic treatment were given to the host or if the antibiotic were over-deployed. Indeed, in our model, all fixed-dose antibiotic treatment regimens will lead to the eventual loss of the commensal from the host proxy. Despite the obvious gravity of the situation for the commensal bacterium, we show by example that it is possible to design drug deployment protocols that support the commensal and reduce the pathogen load. This may be achieved by appropriately fluctuating the concentration of drug in the environment; a result that is to be anticipated from the theory optimal control where bang-bang solutions may be interpreted as intermittent periods of either maximal and minimal drug deployment. While such ‘antibiotic pulsing’ is near-optimal for a wide range of treatment objectives, we also use this model to evaluate the efficacy of different antibiotic usage strategies to show that dynamically changing antimicrobial therapies may be effective in clearing a bacterial infection even when every ‘static monotherapy’ fails.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alon, U. (2006). An introduction to systems biology. London: Chapman and Hall.

    MATH  Google Scholar 

  • Andersson, D. I., & Hughes, D. (2010). Antibiotic resistance and its cost: is it possible to reverse resistance? Nat. Rev., Microbiol., 8(4), 260–271.

    Google Scholar 

  • Beers, M. H., & Fletcher, A. J. (2004). The Merck manual of medical information. Rahway: Merck. (2nd home ed., online version ed.).

    Google Scholar 

  • Bergeron, M., & Ouellette, M. (1998). Preventing antibiotic resistance through rapid genotypic identification of bacteria and of their antibiotic resistance genes in the clinical microbiology laboratory. Clin. Microbiol., 36(8), 2169–2172.

    Google Scholar 

  • Boucher, H. W., Talbot, G. H., Bradley, J. S., Edwards, J. E., Gilbert, D., Rice, L. B., Scheld, M., Spellberg, B., & Bartlett, J. (2009). Bad bugs, no drugs: no eskape! an update from the infectious diseases society of America. Clin. Infect. Dis., 48(1), 1–12.

    Article  Google Scholar 

  • Chan, C. X., Beiko, R. G., & Ragan, M. A. (2011). Lateral transfer of genes and gene fragments in staphylococcus extends beyond mobile elements. J. Bacteriol., 193(15), 3964–3977.

    Article  Google Scholar 

  • Chastre, J., Wolff, M., Fagon, J.-Y., Chevret, S., Thomas, F., Wermert, D., Clementi, E., Gonzalez, J., Jusserand, D., Asfar, P., Perrin, D., Fieux, F., & Aubas, S. (PneumA Trial Group) (2003). Comparison of 8 vs 15 days of antibiotic therapy for ventilator-associated pneumonia in adults: a randomized trial. JAMA, 290(19), 2588–2598.

    Article  Google Scholar 

  • Dancer, S. J. (2004). How antibiotics can make us sick: the less obvious adverse effects of antimicrobial chemotherapy. Lancet Infect. Dis., 4(10), 611–619.

    Article  Google Scholar 

  • Dethlefsen, L., & Relman, D. A. (2011). Incomplete recovery and individualized responses of the human distal gut microbiota to repeated antibiotic perturbation. Proc. Natl. Acad. Sci. USA, 108(Suppl), 4554–4561. doi:10.1073/pnas.1000087107.

    Article  Google Scholar 

  • Eagle, H., Fleischman, R., & Levy, M. (1953). “Continuous” vs. “discontinuous” therapy with penicillin; the effect of the interval between injections on therapeutic efficacy. N. Engl. J. Med., 248(12), 481–488.

    Article  Google Scholar 

  • Ehrlich, P. (1913). Address in pathology on chemotherapeutics: Scientific principles, methods, and results. Lancet, 182, 445–451. doi:10.1016/S0140-6736(01)38705-6.

    Article  Google Scholar 

  • el Moussaoui, R., de Borgie, C. A. J. M., van den Broek, P., Hustinx, W. N., Bresser, P., van den Berk, G. E. L., Poley, J.-W., van den Berg, B., Krouwels, F. H., Bonten, M. J. M., Weenink, C., Bossuyt, P. M. M., Speelman, P., Opmeer, B. C., & Prins, J. M. (2006). Effectiveness of discontinuing antibiotic treatment after three days versus eight days in mild to moderate-severe community acquired pneumonia: randomised, double blind study. BMJ, 332(7554), 1355.

    Article  Google Scholar 

  • Fleming, A. (1964). Penicillin. In Nobel lectures, physiology or medicine 1942–1962. Amsterdam: Elsevier.

    Google Scholar 

  • Gagneux, S., Long, C. D., Small, P. M., Van, T., Schoolnik, G. K., & Bohannan, B. J. M. (2006). The competitive cost of antibiotic resistance in mycobacterium tuberculosis. Science, 312(5782), 1944–1946.

    Article  Google Scholar 

  • Hegreness, M., Shoresh, N., Damian, D., Hartl, D., & Kishony, R. (2008). Accelerated evolution of resistance in multidrug environments. Proc. Natl. Acad. Sci. USA, 105(37), 13977–13981.

    Article  Google Scholar 

  • Kierzenka, J., & Shampine, L. F. (2001). A BVP solver based on residual control and the Matlab PSE. ACM Trans. Math. Softw., 27(3), 299–316.

    Article  MathSciNet  MATH  Google Scholar 

  • Kirby, W. M., & Craig, W. A. (1981). Theory and applications of pulse dosing: a summary of the symposium. Rev. Infect. Dis., 3(1), 1–3.

    Article  Google Scholar 

  • Kunin, C. M. (1981). Dosage schedules of antimicrobial agents: a historical review. Rev. Infect. Dis., 3(1), 4–11.

    Article  Google Scholar 

  • Lenski, R. E. (1998). Bacterial evolution and the cost of antibiotic resistance. Int. Food Microbiol., 1(4), 265–270.

    MathSciNet  Google Scholar 

  • McFarland, L., Elmer, G., & Surawicz, C. (2002). Breaking the cycle: treatment strategies for 163 cases of recurrent clostridium difficile disease. Am. J. Gastroenterol., 97(7), 1769–1775.

    Article  Google Scholar 

  • Michael, M., Hodson, E. M., Craig, J. C., Martin, S., & Moyer, V. A. (2003). Short versus standard duration oral antibiotic therapy for acute urinary tract infection in children. Cochrane Database Syst. Rev., 1, CD003966.

    Google Scholar 

  • Michel, J.-B., Yeh, P. J., Chait, R., Moellering, R. C. Jr, & Kishony, R. (2008). Drug interactions modulate the potential for evolution of resistance. Proc. Natl. Acad. Sci. USA, 105(39), 14918–14923.

    Article  Google Scholar 

  • Opmeer, B. C., El Moussaoui, R., Bossuyt, P. M. M., Speelman, P., Prins, J. M., & de Borgie, C. A. J. M. (2007). Costs associated with shorter duration of antibiotic therapy in hospitalized patients with mild-to-moderate severe community-acquired pneumonia. J. Antimicrob. Chemother., 60(5), 1131–1136.

    Article  Google Scholar 

  • Pépin, J., Saheb, N., Coulombe, M.-A., Alary, M.-E., Corriveau, M.-P., Authier, S., Leblanc, M., Rivard, G., Bettez, M., Primeau, V., Nguyen, M., Jacob, C.-E., & Lanthier, L. (2005). Emergence of fluoroquinolones as the predominant risk factor for clostridium difficile-associated diarrhea: a cohort study during an epidemic in quebec. Clin. Infect. Dis., 41(9), 1254–1260.

    Article  Google Scholar 

  • Poehlsgaard, J., & Douthwaite, S. (2005). The bacterial ribosome as a target for antibiotics. Nat. Rev., Microbiol., 3(11), 870–881.

    Article  Google Scholar 

  • Rello, J., & Diaz, E. (2001). Optimal use of antibiotics for intubation-associated pneumonia. Intensive Care Med., 27(2), 337–339.

    Article  Google Scholar 

  • Roede, B. M., Bresser, P., El Moussaoui, R., Krouwels, F. H., van den Berg, B. T. J., Hooghiemstra, P. M., de Borgie, C. A. J. M., Speelman, P., Bossuyt, P. M. M., & Prins, J. M. (2007). Three vs 10 days of amoxycillin-clavulanic acid for type 1 acute exacerbations of chronic obstructive pulmonary disease: a randomised, double-blind study. Clin. Microbiol. Infect., 13(3), 284–290.

    Article  Google Scholar 

  • Sandegren, L., & Andersson, D. I. (2009). Bacterial gene amplification: implications for the evolution of antibiotic resistance. Nat. Rev., Microbiol., 7(8), 578–588.

    Article  Google Scholar 

  • Segreti, J., House, H. R., & Siegel, R. E. (2005). Principles of antibiotic treatment of community-acquired pneumonia in the outpatient setting. Am. J. Med., 118(Suppl 7A), 21S–28S.

    Article  Google Scholar 

  • Smith, H., & Waltman, P. (1995). The theory of the chemostat. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Stecher, B., & Hardt, W.-D. (2008). The role of microbiota in infectious disease. Trends Microbiol., 16(3), 107–114.

    Article  Google Scholar 

  • Sullivan, A., Edlund, C., & Nord, C. (2001). Effect of antimicrobial agents on the ecological balance of human microflora. Lancet Infect. Dis., 1(2), 101–114.

    Article  Google Scholar 

  • Surawicz, C. M. (2004). Treatment of recurrent clostridium difficile-associated disease. Nat. Clin. Pract. Gastroenterol. Hepatol., 1(1), 32–38.

    Article  Google Scholar 

  • Sussmann, H. T. (1972). The “bang-bang” problem for certain control systems in GL(n,r). SIAM J. Control, 10, 470. doi:10.1137/0310036.

    Article  MathSciNet  MATH  Google Scholar 

  • Torella, J. P., Chait, R., & Kishony, R. (2010). Optimal drug synergy in antimicrobial treatments. PLoS Comput. Biol., 6(6), e1000796.

    Article  MathSciNet  Google Scholar 

  • Trinh, V., Langelier, M.-F., Archambault, J., & Coulombe, B. (2006). Structural perspective on mutations affecting the function of multisubunit rna polymerases. Microbiol. Mol. Biol. Rev., 70(1), 12–36.

    Article  Google Scholar 

  • Wolkowicz, G. S. K., & Lu, Z. (1992). Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates. SIAM J. Appl. Math., 52(1), 222–233.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, M., Zhou, Y. N., Goldstein, B. P., & Jin, D. J. (2005). Cross-resistance of Escherichia coli RNA polymerases conferring rifampin resistance to different antibiotics. J. Bacteriol., 187(8), 2783–2792.

    Article  Google Scholar 

  • Yeh, P. J., Hegreness, M. J., Aiden, A. P., & Kishony, R. (2009). Drug interactions and the evolution of antibiotic resistance. Nat. Rev., Microbiol., 7(6), 460–466.

    Article  Google Scholar 

  • Zimmermann, G. R., Lehár, J., & Keith, C. T. (2007). Multi-target therapeutics: when the whole is greater than the sum of the parts. Drug Discov. Today, 12(1–2), 34–42.

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Rafael Peña-Miller

    Present address: Biosciences, University of Exeter, EX4 4SB, Exeter, UK

Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, SW7 2A7, UK

    Rafael Peña-Miller & Robert Beardmore

  2. Biosciences, University of Exeter, EX4 4SB, Exeter, UK

    Robert Beardmore

  3. Institute of Evolution and Ecology, University of Tübingen, 72076, Tübingen, Germany

    David Lähnemann

  4. Department of Evolutionary Ecology and Genetics, University of Kiel, 24118, Kiel, Germany

    Hinrich Schulenburg

  5. Department of Environmental Sciences, ETH Zurich, Zurich, Switzerland

    Martin Ackermann

  6. Department of Environmental Microbiology, Eawag, Dubendorf, Switzerland

    Martin Ackermann

Authors
  1. Rafael Peña-Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Lähnemann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hinrich Schulenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Martin Ackermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Robert Beardmore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafael Peña-Miller.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peña-Miller, R., Lähnemann, D., Schulenburg, H. et al. Selecting Against Antibiotic-Resistant Pathogens: Optimal Treatments in the Presence of Commensal Bacteria. Bull Math Biol 74, 908–934 (2012). https://doi.org/10.1007/s11538-011-9698-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-011-9698-5

Keywords

Search

Navigation