Mathematical Biosciences and Engineering, 2014, 11(3): 547-571. doi: 10.3934/mbe.2014.11.547

Primary: 49J15, 92C50; Secondary: 92C45, 34D15.

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A model of optimal dosing of antibiotic treatment in biofilm

1. Department of Mathematics, Syed Babar Ali School of Science and Engineering, Lahore University of Management Sciences, Lahore
2. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804

Biofilms are heterogeneous matrix enclosed micro-colonies ofbacteria mostly found on moist surfaces. Biofilm formation is theprimary cause of several persistent infections found in humans. Wederive a mathematical model of biofilm and surrounding fluiddynamics to investigate the effect of a periodic dose of antibioticon elimination of microbial population from biofilm. The growth rateof bacteria in biofilm is taken as Monod type for the limitingnutrient. The pharmacodynamics function is taken to be dependentboth on limiting nutrient and antibiotic concentration. Assumingthat flow rate of fluid compartment is large enough, we reduce thesix dimensional model to a three dimensional model. Mathematicallyrigorous results are derived providing sufficient conditions fortreatment success. Persistence theory is used to derive conditionsunder which the periodic solution for treatment failure is obtained. We also discuss the phenomenon of bi-stability where both infection-free state and infection state are locally stable when antibiotic dosing is marginal. In addition, we derive the optimal antibiotic application protocols for different scenarios using control theory and show that such treatments ensure bacteria elimination for a wide variety of cases. The results show that bacteria are successfully eliminated if the discrete treatment is given at an early stage in the infection or if the optimal protocol is adopted. Finally, we examine factors which if changed can result in treatment success of the previously treatment failure cases for the non-optimal technique.
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