^{*}

^{¤}

AH, RRR, and RA conceived and designed the study. AH performed the analysis and simulations and analyzed the data. RRR contributed analysis tools. AH, RRR, and RA wrote the paper.

¤ Current address: Theoretical Biology, Swiss Federal Institute of Technology, Zurich, Switzerland

The authors have declared that no competing interests exist.

Pathogens that evolve resistance to drugs usually have reduced fitness. However, mutations that largely compensate for this reduction in fitness often arise. We investigate how these compensatory mutations affect population-wide resistance emergence as a function of drug treatment. Using a model of gonorrhea transmission dynamics, we obtain generally applicable, qualitative results that show how compensatory mutations lead to more likely and faster resistance emergence. We further show that resistance emergence depends on the level of drug use in a strongly nonlinear fashion. We also discuss what data need to be obtained to allow future quantitative predictions of resistance emergence.

Pathogens that evolve resistance to drugs pose a serious public health problem. Acquisition of drug resistance usually leads to a pathogen which in the absence of the drug is less fit than the nonresistant pathogen. However, the resistant pathogen can undergo additional mutations that compensate for the fitness cost involved with acquisition of resistance. This can result in a drug-resistant pathogen with fitness comparable to that of the nonresistant pathogen. Handel, Regoes, and Antia analyzed a mathematical model to explain how this process of compensatory mutations influences the probability of and time to resistance emergence on a population level. The authors found that with compensatory mutations, resistance emergence is faster and more likely. Their study also shows how small changes in levels of treatment can lead to large changes in the time to resistance emergence. They further discuss how our ability to accurately predict resistance emergence will depend on improved estimates of the parameters governing the processes of resistance acquisition and compensatory mutations. The study suggests that compensatory mutations can play an important role in the evolution and spread of resistant pathogens through a population.

Drug-resistant pathogens are found at an alarming rate and few replacement drugs are in sight, suggesting a rapidly increasing public health problem [

A number of theoretical between-host studies have provided some insights into prudent drug use. Both population-genetic [

Compensatory mutations have been found to play an important role in the evolution of pathogens [

Despite the importance of compensatory mutations for the process of resistance emergence, we are aware of only one population-genetic study investigating the effect of compensatory mutations on the interplay between drug (pesticide) use and resistance [

Ideally, we would like to study a system for which we can obtain quantitative predictions. Unfortunately, despite the existing studies on compensatory mutations, no host–pathogen system currently exists—as far as we are aware—for which such quantitative predictions could be made. There is simply not enough data available to allow estimation of all the parameters that are needed to quantitatively describe the process of population-wide resistance emergence.

Currently missing are estimates for the between-host fitness levels of resistant mutants. In epidemiological models, the between-host fitness of a pathogen can be defined through the basic reproductive number _{0}, the number of secondary infections caused by one infected host introduced into a fully susceptible population [_{0}.

Also unknown are the rates at which hosts infected with one pathogen strain “convert” to hosts infected with another strain. Each pathogen has a certain probability of undergoing mutations. Provided that a mutation confers a competitive advantage, the mutant can outgrow the other pathogens within an infected host. Once the mutant strain dominates, one can consider the infected host as having “converted.” (As we will discuss at the end of this article, assuming that a host is completely dominated by only one strain at a time is probably an oversimplification.) The rates of conversion depend on both the mutation rates and the dynamics of the different strains within a given host. A recent study provides estimates for the probability that a host commensally infected with

Last, it is also not known what routes of compensatory mutation a pathogen can undergo in a clinically relevant in vivo setting. One study showed that

The current lack of data for these three quantities—between-host fitness levels, rates of conversion, and routes of compensatory mutations—unfortunately limits the predictive power of any model. Nevertheless, one can gain important

In the absence of good estimates for all the necessary parameters, we consider a pathogen for which most parameters are well-defined, and then we investigate how changes in the unknown parameters influence the results. To that end, we study a between-host model for gonorrhea transmission dynamics. Many of the parameters defining the transmission dynamics of gonorrhea infections are rather well-known [

The dynamics of gonorrhea transmission has been analyzed in detail [

We include four different pathogen strains in our model, namely a drug-sensitive strain, a resistant mutant, and two further resistant mutants that have undergone compensatory mutations and thereby regained some of the fitness that was lost in the resistance mutation. The model is given by

Not shown are the flows out of each compartment at rate

Variables and Parameters of the Model

The details of the model are as follows. We consider a homogeneous core group of sexually highly active people 15–39 years of age. New susceptible individuals ^{−1}. Gonorrhea does not cause significant mortality, and the general death rate is low in persons aged 15–39, therefore we ignore deaths.

Susceptible individuals can become infected through sexual interactions with infected persons at a rate determined by the transmission parameters for the different pathogen strains _{s}_{1}, _{2}, _{3}. We assume that there are no differences in transmission between treated and untreated individuals; treatment only reduces the duration of infection. We specify the transmission parameters through the between-host fitness as measured by the basic reproductive number _{0} (_{0}, we use these within-host values as a guideline and choose the between-host fitness for the different resistant strains to be between 60% and 95% that of the sensitive strain.

Some infecteds, _{u},_{t},_{t}_{1}. Since we assume that treatment has no effect on hosts infected with the resistant strains, the duration of infection is the same as those of the untreated hosts. The first and second resistant strains can evolve further to convert to the second and third resistant strains at rates _{1} and _{2}, respectively. As mentioned above, clinically relevant routes of compensatory mutation are not known for any pathogen. Here we choose the simple linear pathway _{1} → _{2} → _{3}, with each mutation leading to a strain with increased between-host fitness. Also not known are the rates at which hosts can convert from being dominated by one strain to being dominated by another strain. While it was found in experiments that compensatory mutations are more frequent than mutations resulting in resistance [_{t}_{1} = _{2}. We additionally choose values for the conversion rates between 10^{−6} and 10^{−1}. We investigate how changes in the conversion rates affect the results.

_{0} for the sensitive and resistant pathogen strains. All strains are assumed to have a fitness of _{0} > 1. The sensitive strain can mutate to a resistant strain with strongly reduced fitness. Due to competition between the strains, a given resistant strain can only invade if its fitness is larger than that of the sensitive strain. Therefore, for all but the highest levels of treatment, once the resistant strain is created, it is outcompeted and goes extinct. However, while stuttering to extinction, the resistant strain can undergo further compensatory mutations and thereby increase in fitness. Once the level of fitness is above that of the sensitive strain, the resistant strain can emerge. The level of drug use determines which of the resistant strains can invade.

The bars indicate between-host fitness levels of the different strains. Solid curved arrows show conversion events that occur frequently due to large or expanding source populations.

Dashed arrows show conversion events that occur infrequently due to small source populations.

(A) Without treatment, all resistant strains are less fit than the sensitive strain. Therefore, resistance emergence is not possible.

(B) Treatment of a small fraction of the population reduces fitness of the sensitive strain enough to allow for emergence of the fittest resistant strain. For that to happen, one frequent and two rare conversions need to occur.

(C) Further increase in treatment level allows both the second and third resistant strains to emerge. For the second resistant strain to emerge, one frequent and one rare conversion need to occur. Subsequently, the third resistant strain is rapidly generated and will outcompete all other strains.

(D) Treatment of a large fraction of the population results in all conversion events being frequent and in rapid emergence of resistance.

Gonorrhea is thought to persist in the population in an endemic state [_{u}_{u}_{t}_{t}_{t}_{t}λ_{u}fλ_{u}fμ_{t}_{t}fλ_{t}β_{s}_{t}c_{u}_{s}c_{t}f_{s}c_{u}f_{t}_{t}f_{u}_{1} = _{1} + _{1} and _{2} = _{2} + μ_{2}. The relative fitness of the resistant strains compared with the sensitive strain can be defined as

The dynamics of the resistant strains can be computed if we assume that during the process of resistance emergence, the endemic steady state does not change. This is a good approximation if the resistant strains are at a low level, which corresponds to the initial phase of resistance emergence that we consider here. If we assume that the system is at steady state and that initially no resistant pathogens are present, we find the explicit expressions for _{1}(_{2}(_{3}(_{i}_{i},_{i}_{i}_{1}. For the times _{2} and _{3}, the equation needs to be solved numerically once the level of resistance

We also study a stochastic version of the model (_{2}

We consider a scenario where the infection is at an endemic equilibrium, which is believed to be the case for gonorrhea [_{u},_{t}_{t} Î_{t}_{t},^{*}, at which treatment drives the pathogen to extinction. The maximum number of introductions is found for some intermediate value of

Parameter choices are described in the text and in

A first guess might suggest that one should try to minimize the number of introductions _{0}) is larger than that of the sensitive strain. If emergence is possible, one can compute approximations for the times at which resistant strain

The black diamonds show the earliest time at which any one of the resistant strains emerges (reaches a level of 5% of total infecteds), obtained from simulations of the full deterministic system (_{0}) of the respective resistant strain is the same as that of the sensitive. For the top panels, fitness levels of the resistant strains are 75%, 85%, and 95% of the sensitive strain in the absence of treatment, resulting in values for the basic reproductive numbers as indicated. For the bottom panels, fitness levels of the resistant strains are 60%, 75%, and 90% of the sensitive strain. The left panels show results for conversion rates _{t}_{1} = _{2} = 10^{−1}, the right panels show results for _{t}_{1} = _{2} = 10^{−3}. Other parameter choices are given in

The results shown in _{1}, the probability of emergence within one year, as a function of the level of treatment

The black diamonds show the probability that any one of the resistant strains emerges (reaches a level of 5% of total infecteds), obtained from stochastic simulations of the full system (stochastic version of

We can use the expression obtained for the probability of emergence to estimate the time until emergence occurs. If the probability that a single introduction leads to emergence is _{i}

Boxplots show distribution of times to emergence (resistance at a level of 5% of total infecteds) for 5,000 simulations of the stochastic model. The red dashed, green dash-dotted, and blue solid lines show the analytic approximations

We end our analysis by briefly studying how changes in the number of resistant strains, levels of fitness, and conversion probabilities influence the results. Some examples are shown in

(A) All conversion probabilities are _{i}^{−3}. The green dashed line shows a situation with two resistant strains with fitness 60% and 90% that of the sensitive strain. The blue dash-dotted line shows three resistant strains with fitness 60%, 75%, and 90%, and the red solid line shows four resistant strains with fitness 60%, 70%, 80%, and 90%.

(B) Same number of strains and fitness levels as (A) but the product of all conversion probabilities is kept the same. We choose _{t}^{−2} for all three cases and _{1} = 10^{−6} for the two-strain scenario (dashed green line), _{1} = _{2} = 10^{−3} for the three-strain scenario (dash-dotted blue line), and _{1} = _{2} = _{3} = 10^{−2} for the four-strain scenario (solid red line).

(C) Three resistant strains with fitness of 60%, 75%, and 90%. Conversion rates are _{i}^{−1} (dashed green line), _{i}^{−2} (dash-dotted blue line), and _{i}^{−3} (solid red line).

(D) Same as (C) but with conversion rates _{t}^{−2}, _{1} = _{2} = 10^{−3} (dashed green line), _{t}^{−4}, _{1} = _{2} = 10^{−2} (dash-dotted blue line), and _{t}_{2} = 10^{−3}, _{1} = 10^{−2} (solid red line).

Before we discuss in detail the results, we want to stress that while we chose a model and parameters to describe gonorrhea transmission dynamics, the qualitative results we obtained are not limited to gonorrhea and should hold for other diseases as well.

While not unexpected, our first and arguably most important finding is that compensatory mutations significantly influence the process of resistance emergence. With the inclusion of compensatory mutations, it is possible that a nonfit resistant strain undergoes further mutations before it goes extinct, until it has regained enough fitness to be able to compete with the sensitive strain (see

A second important finding is the nonlinear dependence of the probability of and time to emergence on the treatment level. In general, small changes in treatment levels at low

An additional interesting result is that for treatment levels close to the extinction threshold, it can be beneficial to increase treatment further. Not only might it lead to eradication of the pathogen, but even if eradication does not happen, the low prevalence of infecteds will lead to very few introductions of resistance (

Our final result concerns the choice of modeling framework. Since the process of compensatory mutations has low numbers and low rates, stochastic effects become important. As

To quantitatively predict resistance emergence, as well as to design optimal drug-use strategies, one needs detailed knowledge of the parameters and dynamics of the underlying system.

As discussed in the introduction, current lack of data unfortunately limits the quantitative and predictive power of our model. Several steps need to be taken to move toward quantitative results. First and foremost, it is important to obtain good estimates for the parameters involved in resistance emergence. The most straightforward way of doing so would be through epidemiological studies. One would need to identify individuals infected with resistant mutants and determine the clinically relevant routes of compensatory mutation a particular resistant pathogen can follow. Further, one would need to determine the rate at which hosts infected with each mutant arise, and measure the number of secondary infections a host infected with a specific mutant causes. These measurements would directly give parameter estimates for _{i},_{0} for pathogens that infect many hosts. Since potentially few hosts are being infected with the different resistant strains, it is pretty much impossible to detect those infections and obtain useful parameter estimates. There might be a better chance to obtain results in controlled transmission experiments in animals. It would likely be possible to investigate the compensatory mutation paths involved in the emergence of resistance. However, measurements for _{0} and the conversion rates would still be difficult to obtain. We think that a more promising approach is to try and obtain the needed parameter estimates by utilizing within-host data and appropriate theoretical models. Measurements of in vivo mutation rates, fitness levels, and pathogen dynamics, combined with appropriate models, might allow estimation of the relative between-host fitness levels for the resistant mutants. Some preliminary work in this direction has been done [

Once better and more complete parameter estimates are available, another step toward quantitative predictions can be made by creating more complex epidemiological models. For gonorrhea transmission dynamics, a more realistic model would accurately describe the population and disease life-history by including factors such as gender, sexual activity, sexual orientation, ethnicity, social status, and others [

Another step toward quantitative predictions will involve more detailed models of the process of resistance emergence. For instance, in our model we assume that treatment does not influence the resistant pathogen. However, in vitro studies of

As always with biological systems, including every detail would lead to a model that is very complicated and difficult to analyze. Often, simplified models are able to capture the dynamics of complex systems, even on a quantitative level. Without additional studies, it is difficult to predict which biological details will need to be included if one wants to obtain quantitative predictions. However, currently the most serious limitation concerning quantitative results is not the lack of model complexity but the almost complete lack of data for the parameters that govern resistance emergence.

In summary, we obtained important qualitative insights into the effects that compensatory mutations have on the population-wide emergence of drug resistance. Our model used parameters applicable for gonorrhea; however, the results we obtained are rather general. Our finding that the inclusion of compensatory mutations significantly influences the dynamics of resistance emergence has important implications for the design of detailed, predictive models. Especially the fact that changes in the level of treatment can lead to disproportionate changes in the probability of and time to resistance emergence has important public health implications. In future work, it will be necessary to obtain estimates for the currently unknown parameters. This knowledge, combined with more complex and detailed epidemiological models, can ultimately lead to quantitative predictions of how the level of treatment influences emergence of resistance. Such knowledge will be extremely valuable for the rational implementation of treatment strategies.

We thank A. Yates for helpful discussions on the topic of branching processes, E. Margolis for feedback on an earlier version of the manuscript, and two anonymous referees for excellent criticism that helped to significantly improve the paper.