WITHIN-HOST EVOLUTIONARY DYNAMICS OF ANTIMICROBIAL QUANTITATIVE RESISTANCE

. Antimicrobial eﬃcacy is traditionally described by a single value, the minimal inhibitory concentration (MIC), which is the lowest concentration that prevents visible growth of the bacterial population. As a consequence, bacteria are classically qualitatively categorized as resistant if therapeutic concentrations are below MIC and susceptible otherwise. However, there is a continuity in the space of the bacterial resistance levels. Here, we introduce a model of within-host evolution of resistance under treatment that considers resistance as a continuous quantitative trait, describing the level of resistance of the bacterial population. The use of integro-diﬀerential equations allows to simultaneously track the dynamics of the bacterial population density and the evolution of its level of resistance. We analyze this model to characterize the conditions; in terms of (a) the eﬃciency of the drug measured by the antimicrobial activity relatively to the host immune response, and (b) the cost-beneﬁt of resistance; that (i) prevents bacterial growth to make the patient healthy, and (ii) ensures the emergence of a bacterial population with a minimal level of resistance in case of treatment failure. We investigate how chemotherapy ( i.e. , drug treatment) impacts bacterial population structure at equilibrium, focusing on the level of evolved resistance by the bacterial population in presence of antimicrobial pressure. We show that this level is explained by the reproduction number R 0 . We also explore the impact of the initial bacterial population size and their average resistance level on the minimal duration of drug administration in preventing bacterial growth and the emergence of resistant bacterial population.


Introduction
In addition to its impact on ecological dynamics, human activities are major drivers of the evolution of species interacting with us [48].An example of such impact, the evolution of antimicrobial resistance (AMR) among parasites of medical importance, is a growing concern across the world [2,23].An antimicrobial substance is a chemical agent that has the potential to interfere with the physiology of a bacterial cell.Because of their relative size and mechanisms of action (at least for the antimicrobial families currently used to treat infections), a single antimicrobial molecule does not cause any damage to a bacterium, while no bacterial population can survive in a medium fully saturated with antimicrobials.In other words, the negative effect of an antimicrobial substance on a given bacterium's survival, referred to here as the antimicrobial activity and denoted A, is an increasing function of its concentration in the medium (denoted C), with boundaries A (C) = 0 when C = 0 and A (C) → A sat when C → C sat , where A sat and C sat are saturating threshold levels.Here, A is measured as the antimicrobial-related mortality rate.From this intuitive approach, it follows that there exists C in (0, C sat ) such that A (C ) is equal to the intrinsic rate of increase and reverses the growth of a bacterial population, all else being equal.This threshold concentration at which a bacterial population does not grow in vitro is called the Minimum Inhibitory Concentration (MIC).
Resistance is then a continuous trait by nature referred to as antimicrobial quantitative resistance (qAMR).Indeed, because of their short generation times and large population sizes, bacterial populations show a great intraspecific genetic diversity generated through random mutations.These mutations define distinct strains which therefore can differ by their relative susceptibility to a given antimicrobial [30,31].As a consequence, the MIC can be seen as a distributed variable within the same bacterial species, underpinned by a mapping of each strain genome to a unique MIC.These MIC distributions are experimentally assessed on a log 2 -discretised scale (see e.g. the EUCAST database [18], usually with a low skewness that spans over two or three order of magnitudes of antimicrobial concentrations).For instance, a recent statistical model of MIC explained by genomic data has shown, in the case of Neisseria gonorrhoeae, that independent exponential contributions of distinct substitutions provide a good set of regressors for estimating MIC [11].Therefore, we here use the log difference in MIC as a phenotypic distance between bacterial strains, with respect to antimicrobial susceptibility.This is particularly suitable because the log scale allows the additivity of independent mutation effects, which will later support symmetric mutation kernels.
Quantitative resistance is key to better understand the within-host evolutionary dynamics of AMR because intermediate resistance can allow bacterial populations to survive drug concentrations below those considered therapeutic [50], and allows the coexistence of multiple strains within the host.Here, we introduce a continuous phenotypic trait x ∈ R, describing the level of resistance between −∞ and +∞.We also treat this quantitative descriptor x as the label of the bacterial strain with resistance level x.Note that any interval (a, b) with a < b and x ∈ (a, b) is also valid within the context of the model and results developed here.However, it is important to keep in mind that, intuitively there exist two threshold levels x 0 and x 1 (called reference 'sensitive' and 'resistant' strains) such that each strain with resistance level (labelled by x) can be classified as 'sensitive', 'intermediate', or 'resistant' depending on whether x < x 0 , x 0 < x < x 1 , or x > x 1 respectively (Fig. 1).
Many mathematical models have been developed to study antimicrobial resistance evolution within a treated host [4, 10, 14, 22, 25-27, 33, 42, 54, 55, 58].We also think that the literature is so vast that we would not know where to begin since the model used then strongly depends on the question asked.However, most of the modelling approaches devoted to AMR tackling the case of qualitative (or "binary") resistance are generally based on the dynamical interaction between two parasite strains leading to a binary MIC formulation [4].This analysis ignores the evolutionary short-term transient dynamics which lead to the emergence of resistance.
To our knowledge, no study has considered the continuous nature of AMR as for the approach developed here.However, a similar formalism has been developed in the context of anticancer treatments [35].There are also parallels with work on linking drug-target binding kinetics with bacterial replication by modelling the number of target molecules per bacterial cell as a positive continuous variable [7].We use a system of integrodifferential equations modeling the dynamics of bacterial population with density b(•, x) and resistance level x ∈ R. Resistance has a cost and thus growth and death rates depend on the bacterial resistance level x.In addition to those effects on the death and birth rates, bacterial population resistance level also mitigates the antimicrobial efficiency with respect to that population.From a theoretical point of view, some of the properties of this model build on previous analytical quantitative genetics results developed in [6,15].
We first describe our model and its main parameters.Next, we investigate how chemotherapy (i.e., drug treatment) impacts bacterial population structure at equilibrium.This includes the characterization of the resistance level acquired by the bacterial population in the presence of antimicrobial pressure.We show that such a characterization is simply based on the reproduction number R 0 [12], which we prove to play the role of the invasion fitness in evolution [21].Next, we investigate in what conditions of the drug efficiency (measured by the antimicrobial activity relatively to the host immune response) and the cost-benefit of resistance; we can (i) prevent bacterial growth to make the patient healthy, and (ii) ensure the emergence of a bacterial population with a minimal level of resistance in case of treatment failure.This is called thereafter the treatment objective.Finally, we investigate the minimal duration of drug administration to achieve our treatment objective as a function of the initial bacterial population size and their average resistance level.

Scaling considerations and model overview
Of course, anyone can claim to model resistance as a quantitative trait x but this is purely a theoretical thought exercise unless it can be clearly linked with existing nomenclature for sensitive and resistant strains, and with existing quantitative metrics related to drug resistance, especially MIC and growth rate.A bacterial strain is said to be resistant to a given antimicrobial if a treatment, the posology of which does not exceed tolerance limits, is likely to fail [2,18].Therefore, each strain can be classified as "sensitive" or "resistant" (R) respectively, depending on whether or not their MIC (i.e., the threshold concentration at which a bacterial population does not grow) is below or above a therapeutic threshold C 1 defined from clinical and pharmacokinetics investigations.Following the EUCAST 2019 nomenclature [18], sensitive strains can be classified as "normal exposure" (S) or "increased exposure" (previously "intermediate", but still denoted by I) depending on whether their MIC is respectively below or above the pharmacologic threshold C 0 corresponding to the antimicrobial concentration reached by a standard posology.They respectively, correspond to the concentration thresholds at usual (i.e.normal) and maximum tolerable posologies and are known as the two clinical breakpoints.
Based on these definitions, for any strain of a given bacterial species exposed to a given antimicrobial, we can define a scale-free quantitative descriptor of AMR varying in a symmetric manner at each mutation step such that where C x is the MIC of the strain with respect to this antimicrobial.With this definition, the EUCAST 2019 typology [18] implies that S < 0 < I < 1 < R. With the above equation, notice that having a negative value for the resistance level x (i.e.x < 0) just means that the given bacterial strain is more sensitive than the reference 'sensitive' strain (i.e.C x < C 0 ).
The model follows the dynamics of bacterial population and antimicrobial concentrations.The bacterial population is assumed to be phenotypically (and genetically) diverse, with a structuration through the level of antimicrobial resistance, here defined as a continuous trait x and referred to as quantitative antimicrobial resistance.This quantitative antimicrobial resistance level x ranges from −∞ to +∞, and affects different components of the bacterial population life cycle, such as growth and death rate.Bacterial populations with a Antimicrobial activity on the sensitive reference strain x = 0 (0, ∞) p1/p0 Reference resistant and sensitive growth rate ratio (0,1) k1/k0 Reference resistant and sensitive drug efficiency ratio (0,1) For the general analysis µ = µ(x) is a function of x.But, for all our illustrations, and with fixed and variable parameters defined in the table above, we have µ = p 0 R 0 0 (0) .Moreover, the ratio p 1 /p 0 and k 1 /k 0 are assumed to be given parameters, such that p 1 and k 1 are determined through the formulas p 1 = p 0 × (p 1 /p 0 ) and k 1 = k 0 × (k 1 /k 0 ).Finally, we assume a Gaussian distribution resistance level x have a density b(t, x) at time t.The main variables and parameters of the model are listed in Table 1.

Model parameters and general hypothesis
For our model formulation and analysis, the killing rate function of the antimicrobial k(•) will be -quite naturally-a decreasing function with respect to the resistance level x.Our primary goal here is to define the function k(•) with two parameters, namely, k 0 and k 1 representing the antimicrobial activity undergone by strains the MIC of which are exactly C 0 and C 1 and hereafter called reference strains 0 and 1.Therefore, we assume that the killing k(x) of the antimicrobial on the bacterial population with resistance level x takes the form The qualitative shape of the curve k(x) is shown in Figure 2. Likewise, one can define a bacterial intrinsic growth rate that incorporates the cost of resistance (for empirical evidence of such costs (e.g., [19]).This intrinsic growth rate, denoted p, should be upper bounded due to physiological constraints, otherwise, a strain not investing at all in AMR would have an infinite growth rate p (−∞) = ∞, which is biologically unrealistic.Therefore, we set p (−∞) =: p m < ∞.On the other side, a strain that takes an infinite concentration of antimicrobial to inhibit would pay an infinite cost then compromising its growth itself, hence p (∞) = 0. Knowing p 0 and p 1 , the intrinsic growth rate of reference strains 0 and 1 (which can be expressed as function of k 0 , k 1 ), a suitable expression for p is with 0 < p 1 < p 0 < p m .The qualitative shape of the curve p(x) is shown in Figure 2. Importantly, the above functional form for p is not strictly important for our model formulation and analysis.The main important property is that p should be a decreasing function with respect to the resistance level x.

Bacterial population model with quantitative resistance level
We use an integro-differential equation to model the demographic and evolutionary dynamics of the bacterial population.At any time t, the total bacterial population density is B(t) = R b(t, y)dy.Next, bacterial population with resistance level y ∈ R give birth to the bacterial population with resistance level x ∈ R at a per-capita rate α b(t, y), where J(x − y) is the probability for a bacterial population with resistance level y to mutate towards a level x during the reproduction process, p(y) is the bacterial intrinsic growth rate, is the effective growth rate, and α > 0 is a scaling constant.Thus, the number of bacteria produced at time t with resistance level x is 1 (1+B(t)) α R J(x − y)p(y)b(t, y)dy.The clearance of the bacterial population with resistance level x due to the immune system occurs at a rate µ(x).Here, we assume that the immune response µ is constant in time.The presence of antimicrobials generates an additional mortality rate k(x), which depends on the level of bacterial resistance.Therefore, the fraction p(y) (1+B(t)) α accounts for the density dependence of the reproduction rate.Such a formalism is a suitable alternative in regulating the growth of a structured population without reference to the concept of carrying capacity, which we think is not necessarily a measurable factor for this type of population.Thus, the parameter α > 0 is introduced only to impose the population homeostasis and does not impact our downstream results.Taking α = 0 leads to a population with infinite growth if no effect of the immune response nor of the antimicrobial is taken into account.Overall, the bacterial evolutionary dynamics is described by the following differential equation The mutation kernel J is such that J(x − y) is the probability of mutation from resistance level y to x. System (2.1) is considered under the general assumption in Appendix A. Preliminary results on the model (2.1), including the existence of a unique maximal bounded dissipative semiflow, are shown in Appendix E. The formulation of model (2.1) allows to follow evolutionary parameters such as the average level of resistance η(t) expressed by the whole bacterial population and the related variance σ 2 (t) at any time t, as so: Furthermore, the model (2.1) can be used to recover the classical model formulation for the qualitative (or "binary") resistance.Indeed, if we denoted by B S and B R the total densities of highly sensitive (i.e.x = 0) and resistant (i.e.x = 1) bacterial populations, model (2.1) can be rewritten as where ε 0 is the mutation probability.We briefly sketch the interpretation of System (2.2), which will also help in better understanding of Model (2.1).Sensitive bacteria B S growth at effective rate p(0)/(1 + B S + B R ) α .Furthermore, while a proportion ε 0 corresponds to a mutant growth (i.e.mutations away from the subpopulation B S ), the remainder (1 − ε 0 ) corresponds to a faithful growth.Next, the sensitive population B S is cleared at rate (µ(0) + k(0)) accounting for actions of the immune response µ(0) and antimicrobial k(0).The same interpretation holds for the resistant population B R .Finally, we refer to Appendix B for more details on the derivation of System (2.2).

Initial conditions
The initial bacterial population b 0 (x) (at t = 0) is assumed to be composed by a sensitive bacterial population, with average resistance level x = 0.This population is characterized by two parameters: its size (m 0 ) and the variance (σ 2 0 ) of its level of resistance.The higher σ 2 0 , the more frequent resistant bacteria are in the initial population.Formally, we set where N (0, σ 0 , x) stands for the normalized density function of the Gaussian distribution at x with mean 0 and variance σ 2 0 .

Results
We illustrate how to use the model to simultaneously capture the bacterial population dynamics and the evolution of antimicrobial resistance.The spread of a bacterial population in a bacteria-free environment is classically determined by calculating the basic reproduction number of this bacterial population.However, the outcome of the evolutionary dynamics of a rare bacterial population with resistance level y in a resident population with resistance level x is determined by the invasion fitness based on standard adaptive dynamics methodology.Furthermore, we show that the level of the bacterial population at the evolutionary equilibrium of Model (2.1) will coincide with the local maximum of the basic reproduction number.We will also show how the outcome of the treatment (success or unsuccess) and the evolutionary bacterial resistance level strongly relies on two parameters: (i) the resistance's cost-benefit ratio, and (ii) the drug efficiency of the reference sensitive strain, quantified relatively to the host immune response.Finally, notice that for all simulations, we randomly set the parameters (Tab.1), with the only purpose to illustrate our theoretical results.

Basic reproduction number R 0 and invasion fitness
Following classical studies, we define the basic reproduction number R 0 as the expected number of bacteria arising from one bacterium in a bacteria-free environment [3,12].As shown in Appendix C, for a bacterial population with resistance level x, this basic reproduction number is We use R 0 (x) to measure the fitness (or effective growing capacity) of a bacterial population with resistance level x.This R 0 can be seen as a product between (i) the intrinsic growth rate of new bacterial population during their natural life time, p(x), and (ii) the lifespan of a bacterial population with resistance level x, 1/(µ(x) + k(x)).In the following, we denote by R 0 0 , the basic reproduction number as in model (2.1) in absence of antimicrobials (i.e. when k ≡ 0).
As state in the introduction, let us first recall that the quantitative descriptor x for the bacterial resistance level is also treated as the label of the bacterial strain with resistance level x.Then, the spread of a rare bacterial population with resistance level y in a resident population with resistance level x is studied using adaptive dynamics.Quite naturally, we assume R 0 (x) > 1, otherwise, the resident population x is not persistent, which a bit contradicts the concept of 'resident population'.Next, to find the evolutionary attractors, we calculate the invasion fitness f x (y), and the rare population with resistance level y will invade the population x if and only if f x (y) > 0. The sign of this two-dimensional function f x (y) is classically visualized using Pairwise Invasibility Plot (PIP) [13,21,39,44].As shown in Appendix C, the invasion fitness f x (y) is written as The environmental feedback of the resident with resistance level x conditions the ability of a rare population with resistance level y to invade the resident population.It depends on the conditions set out by the resident, and by (3.1), the equality (3.2) is rewritten (see Appendix C for details) It follows that the model (2.1) admits an optimisation principle based on R 0 [13,21,39,44].Indeed, the sign of the invasion fitness f x (y) is given by the sign of the difference between R 0 (y) and R 0 (x) and thus, the evolutionary attractors of the model (2.1) coincide with the local maxima of the R 0

Typical dynamics simulated with the model
One of the parameters highlighted through our model's analysis is the ratio where ∆ = (pm−p1)/p1 (pm−p0)/p0 > 1, and θ = k0−k1 k1 > 0. The ratio c b can be interpreted as the average fitness cost-benefit ratio of the resistance for a given bacterial population.Indeed, the parameter ∆ quantifies the relative cost of resistance of a given bacterial population, whereas θ quantifies the fitness advantage of the reference resistant strain (x = 1) of that bacterial population.Note that ∆ ≈ 1 corresponds to cases where the cost of resistance of the given bacterial population is negligible, and θ ≈ 0 to cases where the fitness advantage of resistance of that bacterial population is negligible.
Before antimicrobial treatment onset, the fitness of a bacterial population (measured by its basic reproduction number in the absence of antimicrobial, R 0 0 (x)) decreases with the level of resistance x, such that wild type sensitive bacteria (x = 0) overgrow resistant ones.This is due to the cost ∆ (which assumes ∆ > 0) of being resistant (Fig. 3A).
The initiation of chemotherapy induces an average benefit (measured by θ) in the resistant bacterial population.Indeed, the drug efficiency (k) decreases as the level of bacterial resistance x increases (Fig. 3D).Therefore, the treatment can modify the fitness landscape (which obviously will have a very rapid effect on the distribution  1. of x values in the population) by shifting the maximum point of the basic reproduction number R 0 from x = 0 to x = x * > 0 (Fig. 3A).
The model captures the evolutionary dynamics of the system following treatment onset by tracking, at the same time, the bacterial population dynamics and the evolution of antimicrobial resistance (Figs.3B, C, E).In the first phase, the treatment causes a decrease in the total bacterial population density.At the end of this phase, the infection is seemingly under control (Fig. 3B).The second phase begins with an increase in both the population density and the level of resistance.This phase occurs when the average drug resistance reaches an optimum evolutionary threshold x * that depends on the amount of drug and on the fitness cost.Finally, the bacterial population is not controlled (Fig. 3B), and even worse, it completely escapes treatment having evolved a high level of resistance (Fig. 3C). Figure 3E illustrates the joint dynamics of bacterial population density and resistance.

Evolutionary equilibrium and global dynamic
As shown above, the evolutionary attractor (x * ) of the model (2.1), in the set of resistance level R, coincides with the local maximum of the basic reproduction number R 0 (Appendix D).Furthermore, the evolutionary attractor (x * ) characterizes the bacterial evolutionary resistance level, which is the level of the bacterial population at the equilibrium.
An explicit expression of x * is difficult to obtain with our parameter setting.However, using the EUCAST 2019 nomenclature [18] and defining the cost-benefit ratio c b by (3.4), we find that low values of cost-benefit ratio ) correspond to a low resistance levels at the evolutionary attractor (i.e.x * ≤ 0).See Figure 4 and we refer to Appendix D for more details.
Next, we simultaneously study the epidemio-evolutionary dynamics of model (2.1) by relaxing the time-scale separation assumption.Indeed, our analysis allows us to jointly perform (i) the asymptotic behavior of the model's state variable b(t, •), and (ii) the long-term behavior of the system in relation to the space of resistance level x ∈ R. We find that the existence of a positive stationary state b * (•) is strongly related to the spectral property of the linear integral operator H : The global dynamics of Model (2.1) are fully described by the spectral radius r (H) of H as follows: (i) If r (H) < 1, all strains asymptotically die out and the bacterial population cannot persist, i.e., 1) exhibits a unique positive stationary state b * (•) and the bacterial population is persistent, meaning that there exists ν > 0 such that, lim inf t→∞ R b(t, x)dx > ν (Appendixs F and I).
The above result describes the asymptotic behaviour of Model (2.1) for any given probability kernel J satisfying Assumption A. However, we can go further steps in our analysis when the kernel J is highly concentrated with the scaling form where ε > 0 represents the mutation variance in the phenotypic space.
Denoting by H ε , the operator H where the kernel J is replace by J ε , by results in [15] (Thm.2.2), the spectral radius r(H ε ) have an asymptotic expansion of the form where ζ(x * ) depends on the successive derivatives R (l) 0 (x * ), l ≥ 2. Therefore, when the mutation variance ε in the phenotypic space is sufficiently small, we have: is concentrated around the evolutionary attractor x * in the space of resistance level x ∈ R. In other words, the average bacterial resistance level at equilibrium is x * and we have b * ε (•) → δ x * (•) when ε → 0. This convergence holds for the narrow topology, that is, for any continuous function u ∈ C (R) one has lim ε→0 R u(x)b * ε (x)dx = u (x * ) .We refer to Theorem 2.3 in [15] for such a concentration phenomenon.

Achieving a successful treatment
Combining the asymptotic results described above (Fig. 3) with the classification of the evolutionary bacterial resistance level x * allows us to identify a path to achieve successful treatment, that prevents bacterial growth.) with respect to the resistance's costbenefit ratio (log(∆)/ log(1 + θ)) and drug efficiency (k 0 /µ) on the reference sensitive strain, quantified relatively to the host immune response (µ).Areas R, I, and S correspond to parameter combinations where the evolutionary level of resistance x * is such that x * ≥ 1, 0 < x * < 1, and x * ≤ 0 respectively.The treatment success holds above the level set {R 0 (x * ) = 1}, that is, for the zone in gray.The treatment is unsuccessful below the level set {R 0 (x * ) = 1}, that is, for zones R, I and S (below the purple curve).The curves labelled 'x * = 0' (in yellow) and 'x * = 1' (in red) indicate 'sensitive' and 'resistant' thresholds.
In fact, for a given cost-benefit ratio to drug resistance (c b ), our analysis allow us to determine the minimum level of drug activity on the reference strain (k 0 /µ), quantified relatively to the host immune response (µ), that is required to achieve a successful treatment.This can be done because we showed that in the plane (c b , k 0 /µ) it is possible to characterize three level sets {(c b , k 0 /µ) : R 0 (x * ) = 1}, {(c b , k 0 /µ) : x * = 0}, {(c b , k 0 /µ) : x * = 1} that determine the potential persistence of a bacterial population with an evolutionary resistance level x * (Fig. 4).
We find that the threshold value of k 0 /µ for which a successful treatment holds increases non-linearly when the cost-benefit ratio c b becomes small (Fig. 4).Interestingly, the treatment is successful if and only if (c b , k 0 /µ) > {R 0 (x * ) = 1}, which means this can happen if the evolutionary resistance level x * is 'sensitive' (c b , k 0 /µ) ≤ {x * = 0}, 'intermediate' {x * = 0} < (c b , k 0 /µ) < {x * = 1} or even 'resistant' (c b , k 0 /µ) ≥ {x * = 1} (Fig. 4, gray area).The corresponding evolutionary dynamics are similar to that shown in Figure 5 where the total bacterial population dies out.Note that the treatment results in the acquisition of an intermediate level of  1.The vertical dashed line in panel (B) shows the time from which the total bacterial population is always ≤ 10 −10 .resistance x * by the bacterial population (Fig. 5C).However, this population is unable to grow because the treatment imposes, at the evolutionary resistance level x * , a fitness smaller than unity R 0 (x * ) < 1 (Fig. 5D).

Failure in achieving a successful treatment leads to the emergence of a resistant bacterial population whatever the cost-benefit ratio
The treatment is unsuccessful when the point (c b , k 0 /µ) is below the level set {R 0 (x * ) = 1} (Fig. 4).Overall, for a given cost-benefit ratio (c b ), therapeutic failure occurs when the drug activity (k 0 /µ), quantified relatively to the host immune response (µ), is below a threshold characterized by the level set {R 0 (x * ) = 1}.Depending on the order of magnitude of c b , such therapeutic failure leads to the emergence of a bacterial population with high (Fig. 4, area R), moderate (Fig. 4, area I), or low (Fig. 4, area S) levels of resistance.Indeed, with high cost-benefit ratio values, c b > (1 − p 0 /p m ) −1 , therapeutic failures is always associated with the persistence of bacteria with low resistance levels (Fig. 6, zone S).A therapeutic failure with intermediate values of cost-benefit ratios, (1 leads to the emergence of bacterial populations with either low resistance level (Fig. 6, area S) or intermediate (Fig. 6, zone I).Finally, when the cost-benefit ratio is relatively low, c b < (1 − p 1 /p m ) −1 , a therapeutic failure regimen can lead to the evolution of bacterial population with low (as in Fig. 6, area S), intermediate (as in Fig. 6, area I), or high (Fig. 6, zone R) resistance level.

Discussion
Optimizing antimicrobial treatment dosage is important in preventing bacterial growth and the emergence of resistant bacteria (the Twofold Treatment Objective -TTO).Antimicrobial efficacy is traditionally described by a single value, the minimal inhibitory concentration (MIC) for a given bacterial population.The distribution of MICs across bacterial strains is often bimodal and this metric is therefore used to create a qualitative (or 'binary') classification in the two discrete categories sensitive 'S' and resistant 'R'.Most modelling studies model drug resistance as a binary trait but, as shown by the MIC, it is a continuous trait with varying degrees of  1.
intermediate resistance.This antimicrobial quantitative resistance (qAMR) is associated with a reduction in the bacterial killing rate of an antimicrobial and fitness cost.
The first achievement of this work is that we introduce a continuous trait x ∈ R that describes the normalized level of resistance -using clinical breakpoints -between −∞ and +∞.By simultaneously addressing the population and evolutionary dynamics, the model with qAMR does not ignore the evolutionary and epidemic short-term transient dynamics which lead to the emergence of resistance.Furthermore, such a continuous level of resistance is shown to be strongly linked to the MIC or growth rate, which means it can be informed from actual data.
Using an integro-differential model, we precisely investigate how chemotherapy impacts bacterial population structure at equilibrium.We first characterize the level of acquired evolutionary resistance by bacterial populations in the presence of antimicrobial pressure.We show that this level is governed by a single metric, the reproduction number R 0 , which we prove to play the role of invasion fitness in evolution.We then build on our analysis to show which levels of both drug activity on the wild-type sensitive bacterial population and the bacterial resistance cost-benefit ratio are required to achieve our TTO objective.Finally, we compare the effect of lethal and sub-lethal treatments on achieving our TTO objective, and investigate the impact of the initial bacterial population characteristics (i.e., size, initial resistance frequency) on the minimal duration of drug administration to achieve our TTO.
Our analysis emphasizes that the potential success of the treatment does not depend on the antimicrobial activity (k 0 ) alone but should we assessed with respect to the level of host immunity (µ) as well.These results suggest that treatments with low antimicrobial activity should be limited to infections which elicit a weak immune response (e.g.respiratory infections).They also echoed earlier studies on the synergy between chemotherapy and immune response, e.g.[22,26].Our model formulation assumes that the immune response µ is constant in time, which allows getting some precise analytical insights into the model's evolutionary dynamics.Furthermore, this assumption of constant immunity is quite plausible in the early moments after the initiation of treatment.However, it is a potential limitation and constitutes one possible extension of the model presented here.
The antimicrobial concentration in the host must not be too low, to clear the bacterial population efficiently, but it cannot be too high without toxic effects in a patient [47].A sub-lethal treatment is defined here as a treatment where the drug activity k 0 /µ is not sufficient to avoid the persistence of bacterial population with the evolutionary resistance level x * .Mathematically, we have R 0 (x * ) > 1.Such a configuration can occurs whatever the value of cost-benefit ratio c b for which the point (c b , k 0 /µ) is below the level set {R 0 (x * ) = 1} (Fig. 4).The corresponding evolutionary dynamics are similar to that shown in Figure 6.
We define a lethal treatment when the drug activity k 0 is enough to ensure that no bacterial population is persistent, i.e. that R 0 (x * ) < 1.The threshold of this feasible range with respect to the initial drug activity k 0 and cost-benefit ratio of resistance c b is such that (c b , k 0 /µ) is above the level set {R 0 (x * ) = 1} (Fig. 4), and our TTO objective always holds in such configurations.In other words, for any value of cost-benefit ratio c b (low, intermediate, or high), there exists a minimum drug activity k 0 /µ that guarantees a lethal treatment (Fig. 4, gray area).The corresponding evolutionary dynamics are similar to that shown in Figure 5 where the total bacterial population dies out.
As pointed by some theoretical studies [10,20,28], a high drug dose ('hitting hard' or 'aggressive chemotherapy') is not necessarily the best strategy to limit the spread of resistant strains.We find that a high antimicrobial dose is necessarily to achieve our TTO objective if and only if antibiotic resistance comes with little cost c b , quantified by the threshold (1 − p 1 /p m ) −1 (Fig. 4, gray zone).However, if the treatment fails for aggressive chemotherapy, it will favor the emergence and spread of a bacterial population with a high resistance level (Fig. 4, zone R).This phenomenon is in accordance with the strong relationship between the resistance level of the emerging bacterial population and the antimicrobial dose [27,33].
The minimal duration of antimicrobial treatment to achieve our TTO objective is a debated question in the literature [9,20,22,45].Longer treatment duration is associated with a higher frequency of resistance at the end of the experiment [17,37,43,46], leading to the suggestion that short antimicrobial courses may limit the evolution of resistance at the population level, and studies to determine whether such short course duration would lead to good infection outcomes [17,37,43,46].We quantify the minimal duration (T op ) of drug administration to achieve our TTO objective when cost-benefit ratio c b and drug activity k 0 /µ (relatively to the host immune response µ) on the initial bacterial population lie in the plane (c b , k 0 /µ) > {R 0 (x * ) = 1} (Fig. 4).We define T op as the time t from which the total bacterial population R b(t, x)dx is always ≤ 10 −10 (for example, the vertical dashed line in Fig. 5B).This threshold can be view as the point at which the immune response µ prevents further expansion of the bacterial population.Overall, for a fixed initial bacterial population density, our analysis shows that the minimal duration of drug administration to achieve our TTO objective is relatively short as soon as (c b , k 0 /µ) lies in regions that guarantee the TT0 (Fig. 4, gray area).This combined effect of the cost-benefit ratio (c b ) and drug activity (k 0 /µ) on the time T op is shown Figure 7.We see that, T op is relatively large around threshold values of k 0 /µ that guarantee our TTO objective.Next, T op decreases exponentially with a slight increase in k 0 /µ compared to the threshold values for our TTO objective.Finally, except around the threshold values of k 0 /µ that guarantee our TTO objective, the time T op very short and barely varies with c b .
The characteristics of the initial bacterial population (size m 0 and the frequency of resistance σ 0 ) are important for treatment success [9,22,33].We assess the combined effect of m 0 and σ 0 on the minimal duration (T op ) of drug administration to achieve our TTO objective (Fig. 7).Overall, the size m 0 of the initial bacterial population has a marginal effect on T op as soon as the cost-benefit ratio c b and the initial drug activity k 0 /µ (relatively to the host immune response µ) is such that the pair (c b , k 0 /µ) lies above the level set {R 0 (x * ) = 1} of Figure 4. Whatever the initial population size, our analysis suggests that our TTO objective always holds in a relatively short time period, once the pair (c b , k 0 /µ) lies above the level set {R 0 (x * ) = 1}.By contrast, the frequency of resistant strains initially present σ 0 has a strong impact on the minimal duration (T op ) of drug administration to achieve our TTO objective (Fig. 7).Even if our TTO objective is still achieved as soon as (c b , k 0 /µ) lies above the level set {R 0 (x * ) = 1}, the time T op increases nearly exponentially with the frequency of resistance (Fig. 7).
The within-host dynamics is often ignored by classifying hosts according to whether they are infected with a given strain or not [58].A such simplification fails to take into account the genetic diversity of the bacterial resistant population [30,31] and the short-term evolutionary transient dynamics which lead to the emergence of resistance at the within-host level.Adopting a nested models approach [29,36,41] is one option to simultaneously track the level of qAMR within the host and the between-host evolutionary epidemiology.Our precise description of the within-host bacterial dynamics, coupled with antimicrobial activity, immune response, and qAMR, can significantly improve the understanding of how bacteria populations adapt to their host at the between-host scale [1].
The concentration property of model (2.1) around the evolutionary attractor x * is subject to the assumption of a small mutation variance ε in the phenotypic space.More generally, this result holds as soon as the mutation kernel distribution J verifies item 3 of Assumption A. However, that assumption does not mean the mutation kernel has a very fast decay at infinity.We emphasize that the decay of the mutation kernel distribution considered here (namely, Asm.A, item 3) allows considering the tails of a wide variety of distributions.Indeed, the shape of the distribution of mutational effects can belong to the domain of distributions with exponential tails, truncated tails, or heavy tails that decay as a power law [51].
Finally, in the model proposed here, mutations are assumed to be sufficiently frequent during replication (i.e., new mutants occur during growth), and randomly displace strains into the phenotype space at each generation according to a mutation kernel.However, this constitutes another potential limitation in the model formulation.Indeed, in exponentially growing cells, mutations usually occur during replication [34], but some studies indicate that mutations can be substantially higher in non-growing than growing cultures [53].Thus, the occurrence of new mutants depends either on the abundance of parental cells or both the abundance and growth rate of the parental cells [52].Therefore, another potential extension of the model would be to allow both processes for the occurrence of new mutants.Appendix C. The basic reproduction number R 0 and maximization principle By formally taking J(x − y) = δ x (y) into (2.1), this system becomes (C.1) Assume that system (C.1)reaches a monomorphic epidemiological equilibrium E z (•) = b z δ z (•), for some level of resistance z, before a new mutation with the level y occurs.Here, δ z (•) is set for the Dirac mass at z.Note that E z is the environmental feedback of the resident z.Moreover, by (C.1), E z is such that µ(z)+k(z) .Next, we introduce a small perturbation in (C.1) with level y, such that b(t, x) = b z δ z (x) + u(t)δ y (x) and such that the perturbation u is governed by the linearized system of (C.1) around E z .This reads as, for all x ∈ R, Evaluating the above equality at x = y, it comes u(t) = 1 (1 + b z + u(t)) α p(y)u(t) − (µ(y) + k(y))u(t), and which gives for the linear part It follows from the classical adaptive dynamics results [13,21,38] that bacterial reproduction number, R(y, E z ), of a rare mutant strategy, y, in the resident z-population are given by .
The invasion fitness f z (y) of a mutant strategy y in the resident z-population is then given by When the environmental feedback E z is reduced to the bacteria-free environment, we have b z = 0.Then, the epidemiological basic reproduction number of the bacterial population with resistance level y is calculated as R 0 (y) = p(y) µ(y) + k(y) .
Once a bacterial strain has spread and reached a monomorphic equilibrium, the endemic (feedback) environment E z becomes which is defined when R 0 (z) > 1 and satisfies Let us give some details on the derivation of (C.4).At the monomorphic equilibrium E z , from (C.1) we have, where b(x) = b z δ z (x).Taking x = z, (C.6) gives and (C.4) follows.
Next, we show that the model (2.1) admits a maximization principle [39,44] based on the R 0 , such that model's evolutionary attractors (or levels of resistance at equilibrium) are characterized by local maximums points of R 0 .This point is important since, usually, the identification of evolutionary attractors tends more to follow a mini-max procedure on an adaptive fitness landscape (see [32] for further discussion).Indeed, by (C.3) and (C.5) we have .Further, we know that x * ≥ 1 if and only if f (1) ≥ g(1), i.e. .
Similarly, we also have .
We now search for conditions such that R 0 (x * ) < 1.Note that .
We then rewrite Next, setting R 0 0 = R 0 | k≡0 , the basic reproduction number of the model without any treatment, we have R 0 0 (0) = p 0 /µ, that is, µ = p0 R 0 0 (0) and so, (D.1) becomes By the comparison principle (eg., Section 3 in [8]), we have b(t, x) ≥ u(t, x) ≥ 0 for all t ≥ 0 and x ∈ R. Therefore, item 3. will follow if we show that u(t, x) > 0, for all t > 0 and x ∈ R. Setting U [u](x) = c 0 R J(x − y)p(y)u(y)dy on L 1 (R), we find that U is continuous and generates an uniformly continuous and positive semigroup {e U t } t on L 1 (R).Then, for each t ≥ 0, where the series converges in the operator norm.Since b 0 = 0, R J(x)dx > 0 and an iteration argument ensures the existence of l 0 such that U l [b 0 ](x) > 0 for x ∈ R and for all l ≥ l 0 .From where, (E.3) gives that e U t [b 0 ](x) > 0 for all x ∈ R. Setting ζ = sup R ζ(x), we then have

Appendix F. Equilibrium
The bacteria-free environment E 0 = 0 is always an equilibrium of Model (2.1).In this section, we discuss the existence of a nontrivial equilibrium b * (•) > 0. From System (2.1) we find, for all x ∈ R ω(x) where ω(x) = R 0 (x), and Therefore, the existence of b * (•) > 0 is strongly related to the spectral property of the linear integral operator H defined on L p (R), for any p ≥ 1, by We then have the following theorem: Theorem F.1.Let Assumption A be satisfied.Let r (H) the spectral radius of operator H and φ > 0 the associated eigenfunction normalized such that φ L 1 = 1.Define the quantity When r (H) ≤ 1, the bacteria-free equilibrium E 0 = 0 is the unique equilibrium of Model (2.1).
When r (H) > 1, in addition to E 0 , Model (2.1) has a unique nutrient-bacteria equilibrium E * > 0 such that Furthermore, an explicit formula for the spectral radius r (H) of H reads r (H) = r 0 , where J(x − y)ω(x)ω(y)v(x)v(y)dxdy. (F.5) Proof of Theorem F.1.Here, we deal with the existence of the principal eigenpair for the linear operator H, and we proceed by several steps.First, we introduce the following lemma: Lemma F.2.The following statements hold under Assumption A.
1.For each p ≥ 1, the operator H is compact and irreducible on L p (R) with positive spectral radius, r(H) > 1.
Further, there exists a function u p ∈ L p (R) such that u p > 0 a.e. and H[u p ] = r(H)u p .Furthermore, if u ∈ L p + (R) \ {0} is such that H[u] = cu with c ∈ R, then u > 0 a.e., u ∈ span(u p ) and c = r(H).2. The common spectral value of the operator H is characterized by r(H) = r 0 for all p ≥ 1; where r 0 is defined by (F.5).
Before giving details on the proof of Lemma F.2, let us quickly end with the proof of Theorem F.1.Obviously, E 0 = 0 is always an equilibrium point of the model.We now check nontrivial solution b * > 0 of system (F.Note that the last inequality comes from Young's inequality and the fact that J L 1 = 1. H is a compact operator in L p (R) for any p ≥ 1. Denote by τ h f , the translation of f : R → R by h, and defined by τ h f (x) = f (x + h) for all x ∈ R. Let p ∈ [1, ∞) be given.Let u ∈ L p (R) and h ∈ R be given.We have [τ h ω(x)J(x − y) − ω(x)J(x − y)]ω(y)u(y)dy p dx.
Then Young's inequality (for integral operators) yields Since τ h ωJ − ωJ L 1 (R) → 0 as h → 0 one gets that lim The function χ(u) = u L 1 is continuous and the compactness assumption to apply Theorem A.34 of [56] is satisfied because the semiflow v(t, b 0 ) induced by the nonnegative solutions of (2.1) has a compact attractor of bounded sets by Theorem E.1.By Theorem E.1, χ(b 0 ) > 0 implies χ (v(t, b 0 )) > 0 and the result follows from [56].

Figure 1 .
Figure 1.Classification of the resistance level x.Here x 0 and x 1 are reference 'sensitive' and 'resistant' strains.

Figure 2 .
Figure 2. (Left) Intrinsic growth rate p(x) of bacterial population with a level of resistance x ∈ R. (Right) Drug activity k(x) on bacterial population with resistance level x ∈ R.

Figure 3 .
Figure 3.Typical dynamics simulated with the model.(A) The basic reproduction numbers R 0 (x) and R 0 0 (x) with and without drug respectively.(D) Drug efficiency k(x) and the initial bacterial population with average level of resistance x = 0. (B) Time evolution of the total bacterial population R b(t, x)dx.(C) Distribution of the bacterial population b(t, x) with respect to time t and resistance level x.A logarithmic time scale is used to better highlight transient dynamics of the bacterial population density (B,E), and the increase of the bacterial population resistance level (C).Here, we have set σ 0 = 0.05, m 0 = 0.05, k 0 = 3, p 1 /p 0 = 0.5, k 1 /k 0 = 0.01 and other parameters are given by Table1.

Figure 4 .
Figure 4. Evolutionary resistance level (x * ) with respect to the resistance's costbenefit ratio (log(∆)/ log(1 + θ)) and drug efficiency (k 0 /µ) on the reference sensitive strain, quantified relatively to the host immune response (µ).Areas R, I, and S correspond to parameter combinations where the evolutionary level of resistance x * is such that x * ≥ 1, 0 < x * < 1, and x * ≤ 0 respectively.The treatment success holds above the level set {R 0 (x * ) = 1}, that is, for the zone in gray.The treatment is unsuccessful below the level set {R 0 (x * ) = 1}, that is, for zones R, I and S (below the purple curve).The curves labelled 'x * = 0' (in yellow) and 'x * = 1' (in red) indicate 'sensitive' and 'resistant' thresholds.

Figure 7 .
Figure 7.The minimal duration (T op ) of drug administration to achieve our TTO objective.(Left) Combined effect of the cost-benefit ratio (c b ) and drug activity (k 0 /µ), quantified relatively to the host immune response µ, on the time T op .(Right) Combined effect of the initial bacterial population size (m 0 ) and the initial frequency of resistance (σ 0 ) on the time T op .
From the definition of p and k, it follows that sgn(R 0 (y)) = sgn [f (y) − g(y)] , where f and g are positive function defined on R by f (x) = k(x) ln d µ + k(x) , and g(x) = ba x ln a 1 + ba x , with d = k 0 /k 1 , b = p m /p 0 − 1 and a = p 0 (p m − p 1 )/(p 1 (p m − p 0 )).Functions f , resp.g, are decreasing, resp.nondecreasing, monotonously on R. Therefore, there exists a unique global maximum of R 0 at x * ∈ R: R 0 (x * ) = max x∈R R 0 (x)

Table 1 .
Model state variables and parameters.