Fig 1.
Population dynamics of the resistant (orange) and sensitive (black) strains.
The sensitive subpopulation declines almost deterministically during treatment, which is administered for seven days (gray shaded region). At the same time the resistant subpopulation, if it survives, increases in size. There is strong variation between the resistant growth curves because of stochastic birth and death events. Non-surviving trajectories are omitted. After treatment, both subpopulations grow until the overall pathogen population reaches the carrying capacity, which happens approximately two days after the end of treatment when the resistant subpopulation reaches its maximum. The overall size of the resistant subpopulation at that point shows variation among the ten sample trajectories, again because of demographic stochasticity. This variation carries over to the time at which the resistant subpopulation is outcompeted by the sensitive subpopulation in the absence of antibiotic treatment.
Table 1.
Birth and death rates in the four studied scenarios.
The overall birth and death rates, denoted by λj and μj, are composed of the birth, death and competition processes that occur at rates βj, δj and γj, respectively. Additionally, the effect of antibiotics, denoted by αj(c), affects either the birth or the death rate, depending on the type of drug administered. The variable c denotes the concentration of the antibiotic and the index j indicates the strain-specificity, j = S or j = R for antibiotic-sensitive or -resistant cells.
Fig 2.
Bacterial growth rate without density-dependent effects for different antibiotic concentrations.
The curves show the effect of a biostatic (bs) and biocidal (bc) drug. For a biostatic drug, we plot max(βj − αj(c), 0) − δj (instead of the expression written on the y-axis label, βj − δj − αj(c)). For clarity, we only show the biostatic drug effect for the sensitive strain (black dot-dashed line), and not for the resistant strains. The effect of a biostatic drug on the resistant strains would yield a minimal growth rate at −δR. Parameters for the sensitive antibiotic response curve are motivated by estimates for ciprofloxacin from [35]: βS = 2.5 per day (d−1), βR = 2.25 (d−1), δS = δR = 0.5 (d−1), κ = 1.1, ψj,min = −156 × log(10) (d−1), ψj,max = βj − δj (d−1).
Fig 3.
Survival probability of the resistant subpopulation for varying drug concentrations, MIC values, drug types and density regulations.
At the beginning of treatment the population consists of the sensitive strain at its carrying capacity and a single resistant cell. Then treatment, either with a biostatic (blue) or biocidal (orange) antibiotic, is applied for seven days (solid lines) or infinitely long (colored dashed lines). The vertical dashed line indicates the MIC of the sensitive strain, micS = 0.017. The survival probability of the resistant subpopulation obtained from 106 stochastic simulations (symbols) agrees perfectly with our theoretical predictions (Section B in S1 Appendix). The different rows show different models (Table 1): top row = birth competition; bottom row = death competition. The columns show survival probabilities for different values of the resistant MIC as a multiple of the sensitive MIC: left = low, middle = intermediate, right = high resistant MIC. Parameter values are: βS = 2.5 per day (d−1), βR = 2.25 (d−1), δS = δR = 0.5 (d−1), κ = 1.1, ψj,min = −156 × log(10) (d−1), ψj,max = βj − δj (d−1), γS = γR = 1 (d−1), K = 1, 000, XR(0) = 1, XS(0) = (βS − δS)K/γ.
Fig 4.
Size of the resistant subpopulation at the end of treatment if the resistant subpopulation survives.
At the onset of treatment there is exactly one resistant cell in the population and treatment lasts for seven days. The top row shows the results for the model of birth competition, the lower row corresponds to death competition. Dotted lines show the deterministic prediction of the resistant subpopulation size, solid lines correspond to the stochastic prediction that incorporates a ‘correction’ due to conditioning on survival. Symbols are the mean resistant subpopulation sizes of 106 stochastic simulations that were conditioned on survival of the resistant subpopulation. Blue and orange colors correspond to biostatic and biocidal treatment, respectively. Parameters are as in Fig 3.
Fig 5.
Mean carriage time of the resistant subpopulation in a host (including the treatment time τ = 7).
The carriage time is set to zero if the resistant subpopulation did not survive antibiotic treatment. Discontinuities are a result of discontinuities in the estimate of the resistant subpopulation size at the end of treatment (discussion in Section D in S1 Appendix). The theoretical predictions (lines) are derived in Section E. Symbols show the average carriage time of 106 stochastic simulations. The color coding and figure structure is the same as in the previous figures. Parameters are the same as in Fig 3.