Bulletin of Mathematical Biology

, Volume 74, Issue 4, pp 908–934 | Cite as

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

  • Rafael Peña-Miller
  • David Lähnemann
  • Hinrich Schulenburg
  • Martin Ackermann
  • Robert Beardmore
Original Article


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.


Evolution of antimicrobial resistance Control theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alon, U. (2006). An introduction to systems biology. London: Chapman and Hall. zbMATHGoogle Scholar
  2. 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
  3. Beers, M. H., & Fletcher, A. J. (2004). The Merck manual of medical information. Rahway: Merck. (2nd home ed., online version ed.). Google Scholar
  4. 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
  5. 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. CrossRefGoogle Scholar
  6. 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. CrossRefGoogle Scholar
  7. 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. CrossRefGoogle Scholar
  8. 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. CrossRefGoogle Scholar
  9. 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. CrossRefGoogle Scholar
  10. 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. CrossRefGoogle Scholar
  11. 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. CrossRefGoogle Scholar
  12. 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. CrossRefGoogle Scholar
  13. Fleming, A. (1964). Penicillin. In Nobel lectures, physiology or medicine 1942–1962. Amsterdam: Elsevier. Google Scholar
  14. 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. CrossRefGoogle Scholar
  15. 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. CrossRefGoogle Scholar
  16. 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. MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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. CrossRefGoogle Scholar
  18. Kunin, C. M. (1981). Dosage schedules of antimicrobial agents: a historical review. Rev. Infect. Dis., 3(1), 4–11. CrossRefGoogle Scholar
  19. Lenski, R. E. (1998). Bacterial evolution and the cost of antibiotic resistance. Int. Food Microbiol., 1(4), 265–270. MathSciNetGoogle Scholar
  20. 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. CrossRefGoogle Scholar
  21. 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
  22. 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. CrossRefGoogle Scholar
  23. 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. CrossRefGoogle Scholar
  24. 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. CrossRefGoogle Scholar
  25. Poehlsgaard, J., & Douthwaite, S. (2005). The bacterial ribosome as a target for antibiotics. Nat. Rev., Microbiol., 3(11), 870–881. CrossRefGoogle Scholar
  26. Rello, J., & Diaz, E. (2001). Optimal use of antibiotics for intubation-associated pneumonia. Intensive Care Med., 27(2), 337–339. CrossRefGoogle Scholar
  27. 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. CrossRefGoogle Scholar
  28. Sandegren, L., & Andersson, D. I. (2009). Bacterial gene amplification: implications for the evolution of antibiotic resistance. Nat. Rev., Microbiol., 7(8), 578–588. CrossRefGoogle Scholar
  29. 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. CrossRefGoogle Scholar
  30. Smith, H., & Waltman, P. (1995). The theory of the chemostat. Cambridge: Cambridge University Press. zbMATHCrossRefGoogle Scholar
  31. Stecher, B., & Hardt, W.-D. (2008). The role of microbiota in infectious disease. Trends Microbiol., 16(3), 107–114. CrossRefGoogle Scholar
  32. 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. CrossRefGoogle Scholar
  33. Surawicz, C. M. (2004). Treatment of recurrent clostridium difficile-associated disease. Nat. Clin. Pract. Gastroenterol. Hepatol., 1(1), 32–38. CrossRefGoogle Scholar
  34. 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. MathSciNetzbMATHCrossRefGoogle Scholar
  35. Torella, J. P., Chait, R., & Kishony, R. (2010). Optimal drug synergy in antimicrobial treatments. PLoS Comput. Biol., 6(6), e1000796. MathSciNetCrossRefGoogle Scholar
  36. 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. CrossRefGoogle Scholar
  37. 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. MathSciNetzbMATHCrossRefGoogle Scholar
  38. 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. CrossRefGoogle Scholar
  39. 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. CrossRefGoogle Scholar
  40. 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. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  • Rafael Peña-Miller
    • 1
    • 2
  • David Lähnemann
    • 3
  • Hinrich Schulenburg
    • 4
  • Martin Ackermann
    • 5
    • 6
  • Robert Beardmore
    • 1
    • 2
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.BiosciencesUniversity of ExeterExeterUK
  3. 3.Institute of Evolution and EcologyUniversity of TübingenTübingenGermany
  4. 4.Department of Evolutionary Ecology and GeneticsUniversity of KielKielGermany
  5. 5.Department of Environmental SciencesETH ZurichZurichSwitzerland
  6. 6.Department of Environmental MicrobiologyEawagDubendorfSwitzerland

Personalised recommendations