Conceptual Framework

For clarity, we present our analysis in the context of a hypothetical infection, but this framework can also be applied to cancer. We assume a patient can be considered “healthy,” or the infection can be considered “managed,” provided the pathogen density does not exceed a certain maximum acceptable density. We call this density the acceptable burden and denote it by P_max (that is, pathogen maximum). Treatment failure occurs if the total pathogen density surpasses the acceptable burden. We return to the concept of an acceptable burden in the discussion. For now, we simply postulate that it exists.

Now, consider a generic scenario. When a patient first becomes infected, there is a latent period during which the pathogen density is low, and the patient does not experience any noticeable morbidity. Once the pathogen density is large enough, the patient will feel ill and eventually seek treatment. During this critical treatment period, the first clinical necessity is to lower morbidity as quickly as possible, and this is often achieved by treating aggressively to rapidly lower the pathogen density. Once the pathogen density has been lowered to the acceptable burden, the management period begins (Fig 1).

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Fig 1. Generic model of infection under aggressive treatment (A) and containment (B).

Grey shading indicates the drug-sensitive density. Red shading indicates the drug-resistant density. Once a patient is infected, the total pathogen density (black curve) will increase until the patient experiences symptoms and seeks treatment. The infection will be treated to rapidly lower the total pathogen density to the acceptable burden (blue line). At this point (black dot), the management period begins and the time to treatment failure then depends on the subsequent treatment strategy. Under aggressive treatment (A), the total pathogen density continues to decline sharply until the infection consists only of completely drug-resistant pathogens. Under containment (B), the total pathogen density is maintained at the acceptable burden until the infection consists only of completely drug-resistant pathogens. Containment will modify the expansion of the resistant population, increasing or decreasing it (patterned areas), depending on the rate at which sensitive cells become resistant and the strength of competitive suppression.

https://doi.org/10.1371/journal.pbio.2001110.g001

From this point on, decisions need to be made about whether the best strategy is to continue aggressive treatment or whether instead a containment strategy should be adopted, and it is these decisions that are the focus of our analysis. Aggressive treatment involves removing the entire sensitive density as quickly as possible (Fig 1A). The objective of this treatment strategy is to prevent sensitive pathogens from acquiring resistance. This is standard practice in most clinical settings. The alternate strategy, containment, involves maintaining the largest clinically acceptable population of sensitive pathogens. For containment, chemotherapy is adjusted so that just enough sensitive pathogens are removed at each instant to prevent the total pathogen density from exceeding the acceptable burden (Fig 1B). The objective of containment is to maximize the competitive suppression of resistant cells. Under ideal circumstances aggressive treatment will immediately remove the entire sensitive density; if there is no resistance, then aggressive treatment will clear the infection. For this reason, we focus on the situation where there is at least some resistance when the management period begins.

There are numerous scenarios where either (or both) aggressive treatment and containment will completely prevent treatment failure. Although our analysis applies to these scenarios, to simplify the exposition, we focus in the main text on the case where treatment failure is inevitable. In S1 Text, we describe how to extend these results to scenarios where either aggressive treatment or containment completely prevent treatment failure. The analysis and results under these scenarios are essentially unchanged.

Mathematical Framework

The dynamics of a pathogen population are determined by a time-varying combination of pathogen removal and replication. We assume for simplicity that pathogen removal occurs from the combined effects of immunity and baseline pathogen mortality and that while the average “per pathogen” rate of removal μ may increase over time (if for instance immunity becomes more effective), this increase depends only on the time since infection and not the details of past or current pathogen densities. We also assume that immunity is equally effective against drug-resistant and drug-sensitive pathogens. These assumptions and others are considered in detail in the Discussion.

Pathogen replication will be constrained by competition. Most infections exhibit at least some density dependence. The precise form of density dependence will vary. For simplicity, and in accord with many other studies in this area [29, 34, 41, 66–87], we here focus on density dependence that has immediate impact, whereby the rate of population expansion at any instant depends on the density of the pathogens at that instant, and not on any previous pathogen densities in the patient. We also assume that competition scales with the size of the pathogen population. Thus, the larger the total pathogen population, the lower the average replication rate per pathogen.

Under these assumptions, the expansion rate of the resistant density R in a purely resistant infection can be described by a basic logistic growth equation (S2 Text) (1) in which r is the per capita replication rate of the resistant population in the absence of competition (the intrinsic replication rate) and (1−δR(t)) is the reduction in replication due to competition. The competition coefficient δ is a constant that determines how strongly competition can impact replication.

Now consider how a containment strategy changes the resistant expansion rate. The presence of sensitive pathogens has two contrasting effects on the expansion rate of the resistant population. First, sensitive pathogens increase the total pathogen population and hence increase competition, which lowers the resistant replication rate. We refer to this reduction in replication rate as the resistance management benefit of sensitive pathogens due to competitive suppression (or simply, the benefit of sensitive pathogens). Second, sensitive pathogens directly contribute to the resistant density when they acquire resistance by mutation or horizontal gene transfer. If we assume that a fixed proportion ε of sensitive progeny acquire a mutation that confers complete drug-resistance, the amount of mutational input is proportional to the rate of replication of the drug-sensitive population. We refer to this direct contribution as the resistance management cost of sensitive pathogens due to mutational input (or simply, the cost of sensitive pathogens). For simplicity, we focus on mutational input and consider horizontal gene transfer in the Discussion.

During containment, the total pathogen density is maintained at the acceptable burden P_max and so the resistant expansion rate is described by (2) in which P_max−R(t) is the sensitive density at time t (see S3 Text for mathematical details).

Resistance mutations are often associated with fitness costs, which can impact replication rate or competitive ability or both [10, 51, 88, 89]. If drug resistance reduces a pathogen’s intrinsic replication rate (by a factor (1−c_I)) or increases its sensitivity to competition (by a factor (1 + c_c)), then Eq 2 becomes (3)

Although our model includes the possibility that drug-resistance carries a fitness cost, our analysis and results do not require any fitness costs. Just as two identical sensitive pathogens may compete with each other, a sensitive and a resistant pathogen may compete even if they are equally fit. If there are no fitness costs (c_I = c_c = 0), our analysis still holds.

Aggressive Treatment or Containment?

Whenever the resistance management benefit of sensitive pathogens exceeds the cost, sensitive pathogens are advantageous because they will slow the expansion of the resistant population and ultimately delay treatment failure. Conversely, whenever the cost exceeds the benefit, sensitive pathogens increase the resistant expansion rate and are detrimental. A simple comparison of the benefit and cost in Eq 3 indicates that maintaining the largest acceptable sensitive density (P_max−R(t)) will be advantageous whenever the resistant density is sufficiently high. “Sufficiently high” is described by the balance threshold (see S3 Text for mathematical derivation), (4)

When the resistant density is equal to the balance threshold, the benefit and cost of maximizing the sensitive density are exactly balanced, and the sensitive population does not impact the expansion of the resistant population. Whenever the resistant density is above this critical threshold, the benefit of competition exceeds the cost of mutation, and hence maximizing the sensitive density is advantageous. Whenever the resistant density is below this threshold, the cost of mutation exceeds the benefit of competition and hence maximizing the sensitive density is detrimental.

During the management period, the resistant density may sometimes be below the balance threshold and sometimes above it. This means that sensitive pathogens may sometimes be detrimental and sometimes advantageous, making it unclear whether aggressive treatment or containment will manage the infection for longer. The resistant density at the start of the management period R(0), which we call the starting resistant density, plays an important role in determining whether aggressive treatment or containment should be adopted. There are four distinct scenarios (Fig 2).

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Fig 2. Schematic comparing aggressive treatment to containment.

Sensitive cells competitively suppress resistance (resistance management benefits) but can also mutate to resistance (resistance management costs). The shaded blue region is where benefits dominate costs. Red and black lines are densities of resistant and total pathogens, respectively, under aggressive (solid lines) and containment strategies (dashed lines); dots indicate densities at the start of the management period. Panel A: The acceptable burden (blue solid line) is below the balance threshold (blue dashed line), so there is no region where competition is strong enough to offset the mutational dangers of sensitive pathogens. Consequently, containment is never advantageous. Panel B: Resistant density at the start of the management period exceeds the balance threshold (red dot is inside blue shaded area). In this case, competition is strong enough to outweigh the cost of mutation, and so containment delays treatment failure longer than aggressive treatment. Panels C and D: Resistant density at the start of the management period is below the balance threshold (red dot is below blue shaded area). In Panel C, aggressive treatment manages infection longer than containment. In Panel D, the converse is true.

https://doi.org/10.1371/journal.pbio.2001110.g002

If the acceptable burden is too low, sensitive pathogens are never advantageous and aggressive treatment is best. This scenario is the one in which standard practice aimed at eliminating all sensitive pathogens as quickly as possible is indeed the best thing to do. If the patient cannot tolerate any pathogen burden, rapid pathogen clearance is the only acceptable treatment aim and aggressive treatment should be used. Even if the patient can tolerate some burden, maintaining sensitive pathogens will never be advantageous if the acceptable burden is too low. In particular, if it is below the balance threshold, sensitive pathogens cannot create enough competition to offset the cost of mutational input (Fig 2A). In these cases, even if resistance emergence eventually causes aggressive treatment to fail, it will not fail as quickly as containment.

On the other hand, sensitive pathogens will always be advantageous for at least a portion of the infection if the acceptable burden is high enough to generate sufficient competition (Fig 2B–2D; blue shaded regions). This occurs when (5) (see S4 Text for details). When Eq 5 is satisfied, there are three possible scenarios (Fig 2B–2D).

If the starting resistant density exceeds the balance threshold, then containment is best (Fig 2B). In this case, the competitive suppression that comes from maximizing the density of sensitive pathogens outweighs the mutational inputs. Containment will delay treatment failure longer than aggressive treatment (Fig 2B).

The final two possibilities occur when the starting resistant density is below the balance threshold (Fig 2C and 2D). Here, sensitive pathogens are initially detrimental (mutational inputs dominate), but if the resistant pathogen density increases sufficiently, sensitive pathogens become advantageous (competition dominates). In this scenario, containment will increase the resistant expansion rate while the resistant density is low and then decrease the expansion rate when the resistant density is high (i.e., once it enters the blue shaded region). If sufficient time is gained during this latter stage to compensate for the time lost during the initial stage, then the overall effect of containment will be to delay treatment failure. In this case, containment is better than aggressive treatment (Fig 2D). On the other hand, if the time lost during the beginning of the management period exceeds any time gains later in the infection, aggressive treatment is better than containment (Fig 2C). Thus, if the starting resistant density is below the balance threshold, containment may or may not be advantageous. The lower the starting resistant density, the more likely that aggressive treatment is best. The closer the starting resistant density is to the balance threshold, the more likely it is that containment is advantageous (see S5 Text with S1, S2 and S3 Figs and S6 Text with S4 Fig for details).

The above results hold regardless of whether or not there are any fitness costs associated with resistance. Contrary to conventional wisdom, the presence of a fitness cost will not necessarily enhance competitive suppression and increase the likelihood that containment is better than aggressive treatment. For example, if resistant pathogens have a lower intrinsic replication rate (c_I > 0), this will increase the balance threshold (see Eq 4) and hence decrease the number of scenarios in which sensitive pathogens are always advantageous (i.e., the blue region in Fig 2 becomes smaller). On the other hand, if resistant pathogens have a reduced ability to compete (c_C > 0), then they are more sensitive to competition, and this increases the range of scenarios in which sensitive pathogens are always advantageous (i.e., the balance threshold is decreased and the blue region in Fig 2 becomes larger). When assessing whether or not a fitness cost will tip the balance towards preferring containment, the nature of the fitness cost matters.

Clinical Gains

The above analysis defines the situations where containment delays treatment failure longer than aggressive treatment, but in these situations, how much more effective is containment? The gains associated with containment will depend on the specific details of the patient, target cells, and drug (the parameter values in Eqs 3–5), as well as the density of resistant pathogens at the start of the management period. It is possible, however, to get some analytic insights using the above model if we assume that the immune response μ is constant in time. In this case, after aggressive treatment, the resistant density will continue to expand until it reaches the self-limiting density R_lim, at which point the competition between resistant pathogens coupled with the constant immune response μ prevents further expansion of the resistant population (see S2 Text for mathematical details). The relative performance of containment and aggressive treatment (i.e., the ratio of their times to treatment failure) is completely characterized by the way three key pathogen densities compare to the self-limiting density R_lim: the starting resistant density R(0), the balance threshold R_balance, and the acceptable burden P_max (see S6 Text for mathematical details). When the acceptable burden is high and the initial resistant density is low (but not too low), the benefit of containment can be substantial (Fig 3). For example, if the acceptable burden is 80% of R_lim (black curves), containment can delay treatment failure more than 2.5 times longer than aggressive treatment. If the acceptable burden exceeds 60% of R_lim (purple curves), containment can delay treatment failure more than 1.5 times longer than aggressive treatment.

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Fig 3. Ratio of duration of management period under containment to duration of management period under aggressive treatment.

The horizontal axis is the starting resistant density R(0) divided by the self-limiting density R_lim. Each colour corresponds to a different acceptable burden (blue, green, red, purple, and black correspond to acceptable burdens of 10%, 20%, 30%, 60%, and 80% of R_lim). The balance threshold is varied from 0% to 1% of R_lim. For each colour the upper curve corresponds to R_balance = 0, and the lower curve corresponds to a R_balance equal to 1% of R_lim. This range for R_balance will cover the actual expected range unless the mutation rate is quite large (see S6 Text for details). Values are plotted for when the starting resistant density exceeds the balance threshold (i.e., for cases described by Fig 2B). This figure was generated using an analytic expression for the ratio of times to treatment failure (see S6 Text for the mathematical derivation).

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Theoretical Development

There is a long history of mathematical modelling of resistance in both infections and cancer [14, 23, 38, 91–96]. Some of this work has explicitly modelled the competitive suppression of resistance by sensitive cells and studied treatment regimens which kill sensitive cells at slower rates than standard practice [e.g., 29, 30, 34, 36, 40, 43, 44], so called “light-touch therapies” [35, 39, 65]. Almost all of this work, however, has focused on particular treatment regimens (low dose versus high dose, pulsing, cycling, etc.), and much of it has involved objective functions that may actually exacerbate the resistance management problem (e.g., minimising tumour burden at the end of treatment [72, 76]) or has involved acute infections in which immunity rapidly controls resistance [e.g., 43, 44, 61]. Surprisingly, little effort has been directed at defining what the actual aim of treatment should be, given that drugs can only control sensitive populations and sensitive pathogens can both inhibit and contribute to resistant populations. Even those explicitly studying containment strategies have not defined the conditions under which containment would be more effective than aggressive chemotherapy and, as important, when containment makes things worse [33, 34]. Indeed, so far as we are aware, only Martin et al. [29] use an objective function like ours to study the resistance management costs and benefits of sensitive cells. Our analysis builds on theirs by explicitly defining the balance threshold (Eq 4) and hence the size of the tumour or pathogen burden required before containment has the possibility of being better than aggressive treatment (Eq 5). We also extend their analysis to include fitness costs of resistance, which, while not necessary for containment to be more effective than aggressive treatment [cf. 32, 45, 51], may increase the range of situations in which it is. Importantly, our analysis also reveals that certain types of fitness costs can actually decrease the range of scenarios in which containment is better than standard practice. Moreover, an examination of the balance threshold (Eq 4) suggests a range of novel approaches that may delay resistance emergence under either strategy as well as increase the range of scenarios in which containment is better than standard practice (Box 1).

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Fig 4. The impact of alternative therapies.

Therapies that either decrease competitive ability (left box) or reduce the intrinsic replication rate (right box) of resistant (R) and/or sensitive (S) populations may increase (↑), decrease (↓), or leave unchanged (—) the resistance management benefits of sensitive cells. Therapies that reduce competitive ability will decrease the balance threshold, making it more likely that containment is indicated. Decreasing intrinsic replication may increase, decrease or have no effect on the balance threshold depending on whether the alternative therapy targets the sensitive cells, the resistant cells, or both. For mathematical details, see S12 Text.

https://doi.org/10.1371/journal.pbio.2001110.g004

We analysed our conceptual framework (Fig 1) using a specific model of the process (Eq 3). This specific model made a number of key assumptions. The first is that the dynamics of the target cells at any particular time depend only on the time and the densities of the target cells at that time. This assumption allowed us to conclude that maximizing the sensitive density will be advantageous whenever its immediate effect is to decrease the resistant expansion rate (S7 Text with S5 Fig). This assumption will not hold if the pathogen densities experienced by a patient during earlier stages of the infection impact the pathogen dynamics during later stages of the infection. This can occur if, for example, the rate at which protective immunity develops depends on antigen load (rather than simply time, as we assumed). Related issues arise if resource replenishment in the patient depends on past pathogen densities. Although these scenarios require a separate detailed analysis, some insight into the likely outcomes in such situations can be made by applying the general principles outlined here (S8 Text).

A second key assumption is that competition is modelled with a basic logistic formulation, in which the competitive impact of a given cell is unlinked to its resistance phenotype. In S9 Text, we consider more complex situations in which intra- and inter-strain competition differs and depends on resistance phenotype. We also consider the case of Gompertz competition, more frequently considered in models of cancer [34, 38]. Although the mathematical details differ, both of these alternative competitive formulations generate threshold conditions analogous to Eqs 4 and 5, which also depend on the biology of the patient, target cells, and drug.

There is considerable potential for further theoretical analysis. For instance, we have modelled the ecological and mutational processes deterministically, but when resistance is very rare, stochastic processes will become important [e.g., 92]. Similarly, we considered just two extreme treatment options (immediate removal of the sensitive population or containment of the entire cell population at the acceptable burden). There are other possibilities. Complexities are also introduced by relaxing the assumption that resistance is an all-or-nothing trait. Resistance that renders cells impervious to treatment remains the primary clinical concern, but if several mutational steps are required to full resistance, this will introduce history dependence to the mutational processes (S10 Text). Likewise, if resistance can be acquired through horizontal gene transfer, then things can become a great deal more complex depending on whether the resistance comes from the resistant population or from other species in the microbiota (which may nor may not be impacted by drug treatment) (S10 Text). It may also be interesting to explore the impact of more complex assumptions about immunity (S10 Text). We also note that our analysis of the potential clinical gains of containment (Fig 3) is specific to the particular model formulation (Eq 3); although we expect the general trends to be similar for other formulations, the quantitative predictions will be different.

Practicalities

A core premise of our analysis, and one likely to make many clinicians uneasy, is the concept of an “acceptable” tumour or pathogen burden. Clearly, there are situations where there is no acceptable burden (e.g., bacterial meningitis). We note, however, that there is abundant justification for the idea of an acceptable burden in nonsterile site infections (asymptomatic bacteriuria, gastrointestinal bacteria). Even for sites considered “sterile,” there is increasing evidence that a low burden of pathogen may be tolerated (lung, blood) and clear without antibiotics [106, 107]. Exactly what constitutes a maximum acceptable burden is likely to be a very complex problem, which will depend on numerous factors that have to be carefully considered. In the meantime, for this proof-of-principle analysis, we simply postulate that such a burden exists. We note that we are not alone in assuming this. For cancers, the concept of adaptive therapy [32, 33, 45] also rests on the assumption that there is an acceptable burden. In infectious diseases, tolerance or antidisease drugs are actively being investigated, usually as possible solutions to the resistance problem [98–104]. These drugs work not by killing pathogens but by reducing the damage they do and so are aimed at improving health by raising the “acceptable burden” rather than clearing the infection. Moreover, there are contexts in which adding drug-sensitive microbes is actively under consideration (e.g., microbiome or bacteriotherapy [108, 109], faecal transplants [110–113], addition of competitors [49, 114, 115]). The fundamental premise of these approaches is that the resistance management benefit of drug-sensitive microbes may have much to offer clinically. Finally, we note that our approach has been patient-centred. In infections, an additional concern may be the spread of resistance from patient to patient. Our approach may be adapted in this case by redefining the acceptable burden to be that which reduces transmission to an acceptable level.

Even for the simpler cases considered here (Eq 3), several practical hurdles need to be overcome before resistance management gains can be attained from regimens aimed at containment. Most of the key parameters (Eq 4) are defined by biological properties of the system and are thus likely to generalise across classes of patients, but one critical and highly patient-specific parameter is the resistant density at the start of the management period. In practice, resistant cells will frequently be undetectable when initial treatment decisions need to be made. It might still be possible to generalise in the absence of patient-specific data (e.g., certain types of cancer at certain stages of progression might have predictable resistance densities), or technological improvements might make direct measurement possible. Moreover, our analysis also provides no guidance on the specific treatment regimens required to achieve containment. System-specific pharmacokinetic/dynamic models and experimentation might help. One option is adaptive therapy [33], in which drug dosing and inter-dose intervals are progressively adjusted in response to measurements of the tumour or infection burden. Recent studies have shown that this is possible in at least some clinically relevant settings (e.g., [45]). Note that the effectiveness of containment will be maximized by keeping the tumour or infection biomass at the allowable burden, but gains will continue to accrue provided the biomass is maintained within a defined range (see S11 Text). Thus, from a practical perspective, it is not necessary to keep the total cell population at precisely the acceptable burden. This allows for greater flexibility when implementing containment.

It is tempting to think that containment might be worth trying whenever conventional aggressive chemotherapy is virtually certain to fail due to resistance [33, 45], as it is in many cancers. We note, however, that an important conclusion from Eq 5 is that even when the prognosis for aggressive treatment is not good, there will be situations where attempting to contain the tumour or infection will make things even worse. Thus, patients enrolled in clinical trials of containment strategies need to be chosen carefully. Containment strategies should be first attempted where acceptable burdens are relatively large and easily measured. An attractive possibility is to first investigate containment in patients where the side effects of aggressive chemotherapy can be profound (e.g., palliative care [90]), as containment will likely involve lower doses and/or less frequent dosing. It might also be worth trying in situations where aggressive treatment has failed and no alternative drugs are available. Unless all sensitive cells were removed in the initial bout of chemotherapy, such a situation might accord with Fig 2B and prolong life.

Outlook

Decades of experience in agriculture has led to the belief that the often rapid loss of once highly effective insecticides, pesticides, herbicides, and fungicides can be slowed and even halted if chemicals are used to contain rather than eradicate pest species. That paradigm, widely accepted in agriculture [116, 117], has yet to be seriously investigated in medicine [32]. Our analysis makes clear that there are situations where containment may lead to clinical gains. It also reveals that there are situations where current standard practice, even when it fails, will fail more slowly than a containment strategy. One issue that we have not considered is the intriguing possibility that containment may select for cells that are best able to compete in chronically controlled populations. These might more effectively contain resistant competitors and might themselves have rather low replication rates. Should such evolution occur—and there are suggestions it might [33, 45, 118]—this would be a further argument for investigating chemotherapeutic strategies aimed at containing the target population rather than eliminating it.