Fig 1.
Comparison of the lowest population minimum nmin for drug treatment sequences of constant overall duration τ = 60 providing the same total exposure to high stress tr and low stress τ − tr distributed over different numbers of identically-shaped pulses N.
Parameters τ and tr (x-axis), both given in units of growth rate, together with the skewness s, describing how tr is positioned within τ, (y-axis), determine a drug pulse (sketched on the right). The colour code signifies the optimal N-pulse sequence for a certain choice of parameters tr and s, and the shade indicates the value of the nmin. The coloured lines mark the skewnesses that give local minima for fixed tr corresponding to an optimal onset time for the respective number of pulses. In the non-blue regions a sequence of pulses is more effective than a single pulse, either because onset times are closer to the optimum value for higher number of pulses compared to one pulse (top left corner) or because the minimum nmin is reached in the last pulse of every sequence (bottom right corner; region surrounded by the black line).
Fig 2.
Top: Inset: Schematic of the two-species model: wild-type w and resistant r grow logistically at rates λw or λr, decay at rate δ and switch between states at rates μw or μr, respectively. Main figure: Time-dependent antibiotic pulse shape with the three parameters τ, tr, and the skewness s as before. During tr the antibiotic concentration c(t) > MICr of the more resistant species ((high) environment), while during the entire treatment duration τ, c(t) > MICw ((low) and (high) environment). Initially, the system is in the stress-free environment (free). Bottom: Dynamical landscapes in population phase space corresponding to these three different environments in antibiotic concentration:(high) environment, with one attractive fixed point (red dot) at n = 0; (low) environment, with a saddle point at n = 0 and an attractive fixed point on the r axis; (free) environment: with unstable fixed point at n = 0 and stable fixed point close to the w axis, , which we use as the initial configuration.
Fig 3.
Which value of s gives the lowest population minimum for fixed tr?
a) Lowest population minimum nmin for a single pulse with constant τ = 60 for all possible pulse shape parameters tr and s. The optimal skewness so = 2to/(τ − tr) − 1 which gives the smallest nmin for each tr is marked in white, while the gray (dashed) line marks the skewness , where nmin has dropped to its smallest value across tr. The constant contours (dotted lines) serve as guides to the eye. b-d) Explaining to (c, cf. b) and
(d), the timescales for which the (high) environment is not more effective than the (low) environment. In b)
is longer than to: the population (black) starts growing during the (low) environment, even though the wild type (blue dotted) decays, as the more resistant species (red) is not affected by the antibiotic. In c),
and so the total population keeps decaying. d) If
and the pulse ends there, a minimal n is achieved, while for
n grows again.
Fig 4.
a-c) For a sequence of three pulses, the global population minimum can occur during any of the pulses, depending on the pulse parameters (a: tr = 6, s = 0, b: tr = 30, s = 0, c: tr = 45, s = −0.2; τ = 60 for all). The local minimum (within one pulse) is marked with a light green dot, the global minimum is marked with a green cross. In a, the value of n at the minima (green dots) increases successively, such that global minimum occurs during the first pulse. This is because τ − tr is large. For b and c, τ − tr decreases, implying that the global minimum shifts into the second and third pulse, respectively. Panels d-e show that for s = −1, the minimum is attained in the second pulse (d), unless is so long that the population can regrow to
(e).
Fig 5.
Panel a) Phase space (w, r) trajectories for parameters τ = 60, tr = 10, s = 0.9 and different numbers of pulses. The trajectories start from the fixed point of the antibiotic-free environment , with r/w increasing with every pulse. Here, the sequence with three pulses gives the lowest minimum [see also Fig 1]. The cross marks the end of the best pulse within each sequence. Panel b) Ratio r/w at the end of the best pulse of the best sequence from Fig 1. For fixed tr, r/w is best (smallest) in the regions where the first pulse in a high N pulse sequence yields the lowest nmin (top left corner, also in Fig 1). Similarly to Fig 3, the white line marks the skewness corresponding to the lowest minimum for each tr, with the absolute value of the ratio dropping drastically at the red dot, shown in c). Panel c) Dependence of r/w on tr along the white line in b). The ratio is approximately constant for tr < 30 and then suddenly drops to the same value that it would also show at tr = τ.