Fig 1.
Periodic presence of a perfect biostatic antimicrobial.
A: Microbial fitness versus genotype with and without antimicrobial. Genotypes are the following: S: sensitive; R: resistant; C: resistant-compensated. δ represents the fitness cost of resistance. B: Periodic presence of antimicrobial (gray: presence, white: absence), and impact on the fitness of S microorganisms. C: Probability p0 that the microbial population goes extinct before resistance gets established versus alternation period T, for various carrying capacities K. Markers: simulation results, with probabilities estimated over 102 − 103 realizations. Horizontal solid lines: analytical predictions from Eq 1. Dashed lines: T/2 = τS. D and E: Numbers of S, R and C microorganisms versus time in example simulation runs for K = 1000, with T = 20 and T = 1000 respectively. In D, resistance takes over, while in E, extinction occurs shortly after antimicrobial is first added. Phases without (resp. with) antimicrobial are shaded in white (resp. gray). Parameter values: fS = 1 without antimicrobial, with antimicrobial, fR = 0.9, fC = 1, gS = gR = gC = 0.1, μ1 = 10−5 and μ2 = 10−3. All simulations start with 10 S microorganisms.
Fig 2.
Periodic presence of a biocidal antimicrobial above the MIC.
A: Probability p0 that the microbial population goes extinct before resistance gets established versus alternation period T, for various carrying capacities K. Markers: simulation results, with probabilities estimated over 102 − 103 realizations. Horizontal solid lines: analytical predictions from Eq 4. Dashed lines: T/2 = τS. B and C: Numbers of sensitive (S), resistant (R) and compensated (C) microorganisms versus time in example simulation runs for K = 1000, with T = 8 and T = 1000 respectively. In B, resistance takes over, while in C, extinction occurs shortly after antimicrobial is first added. Phases without (resp. with) antimicrobial are shaded in white (resp. gray). Parameter values in A, B and C: fS = 1, fR = 0.9, fC = 1, gS = 0.1 without antimicrobial, with antimicrobial, gR = gC = 0.1, μ1 = 10−5 and μ2 = 10−3. All simulations start with 10 S microorganisms. D, E and F: same as A, B and C, but with
. All other parameters are the same.
Fig 3.
Resistance emergence in the presence of a biocidal antimicrobial above the MIC.
A: Numbers of sensitive (S), resistant (R) and compensated (C) microorganisms versus time in an example simulation run for K = 104, with T = 1000. Here resistance takes over. Phases without (resp. with) antimicrobial are shaded in white (resp. gray). B: Zoom showing the emergence of resistance in this realization: an R mutant appears after antimicrobial is added (gray). At this time, the S population is decreasing due to the antimicrobial-induced high death rate, but the surviving S microorganisms are still able to divide. Parameter values and initial conditions are the same as in Fig 2A, 2B and 2C.
Fig 4.
Dependence of the extinction probability p0 on population size and antimicrobial mode of action.
The extinction probability p0 is plotted versus carrying capacity K for the perfect biostatic drug (corresponding to Fig 1), two different concentrations of biocidal drugs yielding two different death rates (corresponding to Fig 2) and a drug with both biostatic and biocidal effects. Markers correspond to simulation results, computed over 103 realizations. Solid lines correspond to our analytical predictions from Eqs 1 and 4, respectively, which hold for K ≪ 1/μ1. Parameter values and initial conditions are the same as in Figs 1 and 2, respectively, and the period of alternations is T = 103, which is in the large-period regime.
Fig 5.
Heatmaps of the extinction probability.
Extinction probability p0 versus alternation period T and with biostatic (A) or biocidal (B) antimicrobial. Heatmap: simulation data, each point computed over 103 realizations of simulation results, and linearly interpolated. Dashed white line: value of
such that
(see main text). Solid white line: T/2 = τS. Parameter values: K = 103, μ1 = 10−5, μ2 = 10−3, fS = 1, fR = 0.9, fC = 1, gS = gR = gC = 0.1, and (A)
and variable
or (B)
and variable
. Dotted line in B:
. All simulations start with 10 S microorganisms.
Fig 6.
Dependence of the extinction transition on population size and antimicrobial mode of action.
Extinction probability p0 versus the ratio with biostatic or biocidal antimicrobial, for different carrying capacities K, either in the small-period regime, with T = 102.5 (A and B) or in the large-period regime, with T = 105 (C). Markers: simulation results, calculated over 103 realizations. Vertical dashed lines: predicted extinction thresholds, i.e. values of
such that T/2 = τS (A and B) or
(C). Solid lines (C): Analytical estimates of p0 from Eq 1 (biostatic) or Eq 4 (biocidal). For K = 102 and 103, the analytical predictions in the biostatic and biocidal case are confounded, while for K = 104 we used two shades of green to show the slight difference (light: biostatic, dark: biocidal). Parameter values: μ1 = 10−5, μ2 = 10−3, fS = 1, fR = 0.9, fC = 1, gS = gR = gC = 0.1, and
(biostatic) or
(biocidal). All simulations start with 10 S microorganisms.