Simulation Code

Mathematica code for stochastic simulation

Example where conventional strategy of high-dose chemotherapy best prevents the emergence of resistance.

<p>(a) The dose-response curves for the wild type in blue (<i>r</i>(<i>c</i>) = 0.6(1−tanh(15(<i>c</i>−0.3)))) and the resistant strain in red (<i>r</i>_<i>m</i>(<i>c</i>) = 0.59(1−tanh(15(<i>c</i>−0.45)))) as well as the therapeutic window in green. Red dots indicate the probability of resistance emergence. Probability of resistance emergence is defined as the fraction of 5000 simulations for which resistance reached a density of at least 100 (and thus caused disease).(b) and (c) wild type density (blue), resistant density (red), and immune molecule density (black) during infection for 1000 representative realizations of a stochastic implementation of the model. (b) treatment at the smallest effective dose <i>c</i>_<i>L</i>, (c) treatment at the maximum tolerable dose <i>c</i>_<i>U</i>. Parameter values are <i>P</i>(0) = 10, <i>P</i>_<i>m</i>(0) = 0, <i>I</i>(0) = 2, <i>α</i> = 0.05, <i>δ</i> = 0.05, <i>κ</i> = 0.075, <i>μ</i> = 10⁻², and <i>γ</i> = 0.01.</p

Does High-Dose Antimicrobial Chemotherapy Prevent the Evolution of Resistance?

<div><p>High-dose chemotherapy has long been advocated as a means of controlling drug resistance in infectious diseases but recent empirical studies have begun to challenge this view. We develop a very general framework for modeling and understanding resistance emergence based on principles from evolutionary biology. We use this framework to show how high-dose chemotherapy engenders opposing evolutionary processes involving the mutational input of resistant strains and their release from ecological competition. Whether such therapy provides the best approach for controlling resistance therefore depends on the relative strengths of these processes. These opposing processes typically lead to a unimodal relationship between drug pressure and resistance emergence. As a result, the optimal drug dose lies at either end of the therapeutic window of clinically acceptable concentrations. We illustrate our findings with a simple model that shows how a seemingly minor change in parameter values can alter the outcome from one where high-dose chemotherapy is optimal to one where using the smallest clinically effective dose is best. A review of the available empirical evidence provides broad support for these general conclusions. Our analysis opens up treatment options not currently considered as resistance management strategies, and it also simplifies the experiments required to determine the drug doses which best retard resistance emergence in patients.</p></div

Example where low-dose strategy best prevents the emergence of resistance.

<p>(a) The dose-response curves for the wild type in blue (<i>r</i>(<i>c</i>) = 0.6(1−tanh(15(<i>c</i>−0.3)))) and the resistant strain in red (<i>r</i>_<i>m</i>(<i>c</i>) = 0.59(1−tanh(15(<i>c</i>−0.6)))) as well as the therapeutic window in green. Red dots indicate the probability of resistance emergence. Probability of resistance emergence is defined as the fraction of 5000 simulations for which resistance reached a density of at least 100 (and thus caused disease).(b) and (c) wild type density (blue), resistant density (red), and immune molecule density (black) during infection for 1000 representative realizations of a stochastic implementation of the model. (b) treatment at the smallest effective dose <i>c</i>_<i>L</i>, (c) treatment at the maximum tolerable dose <i>c</i>_<i>U</i>. (d) The probability that a resistant strain appears by mutation is indicated by grey bars for low and high dose. The probability of resistance emergence is indicated by the height of the red bars for these cases. The probability of resistance emergence, given a resistant strain appeared by mutation, can be interpreted as the ratio of the red to grey bars. Parameter values are <i>P</i>(0) = 10, <i>P</i>_<i>m</i>(0) = 0, <i>I</i>(0) = 2, <i>α</i> = 0.05, <i>δ</i> = 0.05, <i>κ</i> = 0.075, <i>μ</i> = 10⁻², and <i>γ</i> = 0.01.</p

Frequency distribution of resistant strain outbreak sizes for the simulation underlying Fig 3.

<p>Each distribution is based on 5000 realizations of a stochastic implementation of the model. (a) Low drug dose. (b) High drug dose. Insets show the same distribution on a different vertical scale.</p

Alternative frameworks and definitions.

(DOCX)</p

Derivation of the model and framework.

(DOCX)</p

Pathogen adaptation as the fraction of primed individuals increases.

Plots of the growth rate of all viable variants in a fully naïve and a fully primed population (dots). Large blue dot denotes the phenotype of the current wild type and black arrow indicates direction of selection (i.e., the variants that are most advantageous). Variants in the gray region are disadvantageous. Note that the location of all variants along the ri,P axis is specific to immune response and may be different for natural immunity and different vaccines. Colored regions indicate the 4 different kinds of variants. (a) Early in a novel host–pathogen association when a small fraction of hosts are primed. Many potential new variants will be better adapted to both host types (i.e., they will be generalists). (b) Later in the association, when the pathogen is better adapted to its novel host (and a larger fraction of hosts are primed). The evolutionary trajectory of successive fixation events leading to the new wild type variant is indicated by the succession of blue dots. Note how the change in the location of the blue dot can affect the typology of some variants (i.e., a variant that was identified as a generalist in the early stage of adaption could later become a specialist relative to the more recent form of the pathogen). Once the level of adaptation is high (panel (b)), most advantageous variants that appear will tend to be specialists. Even though generalists are still more strongly favored by selection there are fewer of them that can arise.</p

The fate of a variant (<i>i</i>) is determined by 3 key components of fitness, each of which can be affected by multiple within-host mechanisms of adaptation.

All else being equal, variants with increased infectivity, increased transmissibility, or a long and early infectious period (i.e., long infections and a short generation interval) will have an increased fitness (rate of spread in a population). As indicated in Eq (1), fitness depends on both the degree of adaptation to naïve and primed hosts. Within-host processes affect the 3 components of fitness in each of the host types. Some within-host mechanisms of adaptation can be measured directly using in vitro assays. Some components of pathogen fitness can be inferred from evolutionary epidemiological studies.</p

Four types of immunity-adapted variants.

Solid lines depict the growth rate of the population of infected individuals for the wild type (blue) and for a variant (red) as a function of the fraction of the population that has been primed against infection by vaccination, previous infection, or both. Priming decreases the growth rate of the wild type (rN>rP). Quantities ΔrN and ΔrP are the differences in growth rate between the variant and the wild type in naïve and primed hosts, respectively. Colored shading indicates which type prevails evolutionarily: the wild type (light blue shading) or the variant (light red shading). Panels (a) and (b) show generalists; the variant is also better adapted to naive hosts (ΔrN>0). Generalist variants will outcompete the wild type even in the absence of priming. Panels (c) and (d) show specialists; the variant is maladapted to naïve hosts (ΔrNa) and (c) show immunity-inhibited variants; the growth rate of the variant decreases with increasing fractions of primed hosts. As a result, the growth rate of infections after adaptation (i.e., after fixation of the fittest type) in a fully primed population (black dot) is always lower than that in a fully naïve population (white dot and dashed line). Panels (b) and (d) are immunity-facilitated variants; the growth rate of the variant increases with increasing fractions of primed hosts. As a result, the growth rate of infections after adaptation in a fully primed population (black dot) is always higher than that in a fully naïve population (white dot) for generalist variants (panel (c)) but it can go either way for specialists (panel (d); only the case where it is lower is shown). Panel (e) show a plot of the growth rate of variants in a fully naïve (ri,N) and a fully primed (ri,P) population. Blue dot indicates location of the wild type. Uncolored region corresponds to variants whose growth rate in primed hosts is less than that of the wild type and so are immunity-maladapted (and so ignored in our discussion). Different colored regions correspond to the 4 types of variants from panels (a–d). Finer distinctions within these types are presented in S1 Fig. See S2 Appendix for a discussion of alternative ways to visualize variants.</p