How native antibiotic use, carriage period, resistance prices and worldwide journey form resistance frequency in E. coli in France

The worldwide rise within the prevalence of extended-spectrum beta-lactamase (ESBL) producing micro organism is a serious well being concern. These micro organism are proof against many prescribed antibiotics, akin to penicillins or third-generation cephalosporins and are regularly co-resistant to different antibiotic households like fluoroquinolones or trimethoprime/sulfamethoxazole, which leaves only a few remedy choices upon an infection. Third-generation cephalosporin-resistant Escherichia coli brought on an estimated 33,100 extra-deaths worldwide in 2021, making it one of many prime ten deadliest pathogen-drug resistance mixtures (Naghavi et al., 2024).

Epidemiological analysis has offered a complete understanding of the danger components and mechanisms underlying the unfold of ESBL-resistant E. coli. While ESBL strains have been initially largely present in hospitals, from 2000 onward these strains have largely unfold in the neighborhood as an entire (Pitout et al., 2005; Woerther et al., 2013). Frequencies of ESBL-producing E. coli are extremely heterogeneous globally. In Europe, the ESBL carriage frequency in the neighborhood gently raised (Woerther et al., 2013; Bezabih et al., 2021) to stabilize from the 2010s at about 6-8 % (Blanquart, 2019; Bezabih et al., 2021; Castanheira et al., 2021; Pennings, 2025). In distinction, ESBL carriage elevated to greater ranges in different international locations, particularly in South-East Asia, the place research generally report a carriage of round 50%; carriage can attain values of 90% to 100% in some research (reviewed in Woerther et al., 2013; Bezabih et al., 2021; Bui et al., 2015; Jacquier et al., 2023). Accordingly, in European communities, coming back from South and Southeast Asia is a crucial danger issue for the carriage of ESBL-producing Enterobacterales (Tängdén et al., 2010; Kantele et al., 2015; Ruppé et al., 2015; Birgy et al., 2016; Karanika et al., 2016; Arcilla et al., 2017; Hu et al., 2020; Raffelsberger et al., 2023). Based on this statement, it has been argued that these vacationers might doubtlessly play an vital function in shaping native ranges of resistance (van der Bij and Pitout, 2012; Woerther et al., 2017; Olesen et al., 2020). Additionally, previous antibiotic use was repeatedly recognized as a danger issue for carriage of ESBL-producing E. coli (Birgy et al., 2016; Karanika et al., 2016; Hu et al., 2020; Raffelsberger et al., 2023), as is the case extra typically for antibiotic-resistant micro organism (Chatterjee et al., 2018). In spite of this data, the relative contributions of the totally different mechanisms within the noticed upkeep of a steady low frequency of ESBL in Europe stay unknown (Emons et al., 2025).

Mechanistic mathematical fashions of resistance evolution are wanted to assist perceive these contributions and ultimately develop higher public well being insurance policies to scale back resistances. Models are highly effective instruments to disentangle the function of interacting mechanisms, whether or not they’re instantly noticed or not, on the dynamics of resistance, and assess the influence of administration methods. The influence of a discount in antibiotic use (Lipsitch, 2001; Sundqvist et al., 2010; Blanquart et al., 2017; Emons et al., 2025) and the influence of a discount in transmission (Davies et al., 2019; Jacopin et al., 2020) on resistance frequency ranges each rely on modelling assumptions. For instance, within the case of the influence of transmission, transmission can favor the resistant pressure by enabling it to transmit to empty handled hosts. But it may well additionally favor the delicate pressure by permitting it to supercolonize and displace the resistant one. The finish results of these two processes differ relying on the assumptions on co-colonization. Thus, mechanistic fashions with sturdy assumptions are wanted to anticipate each future evolution and impacts of interventions.

Intriguingly, there are few mechanistic mathematical fashions of ESBL resistance evolution in E. coli (Rahbé et al., 2024). Morevoer, most of those fashions endure both from assumptions that aren’t supported by organic information, and/or from predicted behaviors that aren’t in step with the noticed resistance dynamics. One central set of assumptions regards the outline of competitors between strains. Competition between strains profoundly shapes resistance dynamics (Lipsitch et al., 2009; Blanquart, 2019; Davies et al., 2019). It is biologically intuitive that strains of the identical species that solely differ of their resistance standing ought to occupy the identical area of interest of their host, and thus compete for nutrient or house. This has been just lately proven for E. coli in a number of datasets (Morel-Journel et al., 2025): E. coli strains are cleared quicker when a number of strains compete concurrently within the host. Moreover, ESBL resistant strains are most frequently coexisting with delicate strains at low density (Ruppé et al., 2013; de Lastours et al., 2016). In spite of those observations, half of present fashions of antibiotic-resistance in Enterobacterales describe solely the resistant pressure (Rahbé et al., 2024) which questions the plausibility and applicability of their predictions. Studies that appropriately embrace the competitors between delicate and resistant strains typically don’t reproduce the noticed coexistence of those strains–the so-called “coexistence problem” (Lipsitch et al., 2009). Instead, fashions predict that both the delicate or resistant pressure replaces the opposite in most circumstances (that is the case for fashions in Sundqvist et al. 2010; MacFadden et al. 2019; Godijk et al. 2022). A notable exception is the work of Kachalov et al. (2021) on Okay. pneumoniae; they modelled the dynamics of ESBL resistance in hospitals and the group and reproduced the frequency of resistance throughout eleven European international locations.

The coexistence of resistant and delicate strains on the group scale, as noticed in E. coli, poses a extra basic theoretical drawback (Blanquart, 2019). Several theoretical options have been proposed (Colijn et al., 2010), together with host inhabitants construction (Cobey et al., 2017; Blanquart et al., 2018), variability within the period of carriage of bacterial strains (Lehtinen et al., 2017), sluggish dynamics of exclusion of the resistant strains by the delicate pressure inside hosts (Davies et al., 2019). However, these theoretical fashions are both not exactly parameterized with information, or the parameterization of those fashions with present information fails to completely clarify coexistence. This failure will be defined in two methods. A primary chance is that but unknown components play a serious function within the stabilization of resistance frequencies. A second chance is {that a} mixture of the aforementioned options collectively would clarify coexistence. Parameterizing complicated fashions is troublesome and requires sufficiently detailed information on a system. This might need hindered progress in the direction of a passable answer combining a number of mechanisms.

In this research, we use complete information on the danger components for carriage of ESBL-producing E. coli within the French group, alongside detailed microbiological characterization of each resistant and general E. coli, to develop a biologically believable mathematical mannequin of ESBL resistance unfold in France combining a number of mechanisms stabilizing coexistence. Earlier analyses recognized journey to South-East Asia and prior use of third-generation cephalosporins as key danger components for ESBL E. coli carriage, in step with different research (Birgy et al., 2016). Here, we use this dataset prolonged to a few extra years, and moreover quantify the full Enterobacterales density and the density of ESBL-producing Enterobacterales in a subset of samples. This permits us to exactly mannequin competitors by inferring the prevalence of various colonization states: resistant-only, sensitive-only, and co-colonization. We mannequin each the influence of journey to international locations with a excessive prevalence of ESBL, and the influence of remedy clearing delicate strains, with the intention to breed the inferred danger components for ESBL carriage. We additionally mannequin the identified variability in carriage period throughout E. coli strains (Östblom et al., 2011; Martinson et al., 2019; Morel-Journel et al., 2023), and believable within-host dynamics of resistant and delicate strains (Cotto et al., 2023). Thus, all in all, our research consists of the principle theoretical explanations for coexistence: inhabitants construction, variability within the period of carriage, and within-host dynamics. We mannequin and infer in a Bayesian framework a number of health prices related to epidemiological traits akin to transmission and clearance of resistant strains. We confirm the goodness-of-fit of our mannequin and eventually predict how modifications within the charge of journey, the transmission charge and antibiotic use affect resistance ranges in the neighborhood.

2.1 Data and mannequin construction

Our fundamental information set is an epidemiological research of ESBL-producing E. coli in rectal samples from N = 3443 French kids aged 6-24 months from 2010 to 2018. Rectal samples have been offered throughout visits to pediatricians on a voluntary foundation. Sampling was related to a questionnaire offering data on behaviors previous to sampling, together with journey and antibiotic remedy. Children present process antibiotic remedy inside 7 days earlier than sampling have been excluded from the survey. All samples have been screened for the ESBL-producing Enterobacterales for a complete of N = 257 resistant strains discovered. The particulars of the survey will be present in Birgy et al. (2016).

We centered on kids to learn from a longtime community of pediatricians for the gathering of rectal samples. The literature on ESBL resistance in Europe means that the prevalence of this resistance doesn’t differ a lot between age courses. Indeed, imply estimates of ESBL prevalence vary from 3.5 to 7.6% in preschool kids (Van den Bunt et al., 2016; Birgy et al., 2016), just like these estimated in staff and retired age courses (6-8.6 %, Nicolas-Chanoine et al. 2013; Reuland et al. 2016).

2.1.1 Risk components related to ESBL carriage

A earlier statistical evaluation of a subset of the info (from 2010 to 2015, Birgy et al. 2016) revealed that kids who’ve traveled to South-East Asia (SEA) or who’ve acquired betalactam antibiotics inside a three-months interval previous the sampling usually tend to carry ESBL-producing strains. We replicated the analyses of (Birgy et al., 2016) to our prolonged information set (2010-2018) and located related outcomes (Supp.Mat. S1).

Following the results of the statistical evaluation (Birgy et al. 2016 and Supp.Mat. S1) and former outcomes on danger components for ESBL carriage, we modeled 5 compartments to explain antibiotic remedy and journey to SEA (Figure 1A). The first compartment, “untreated” U counts people who neither traveled nor used antibiotics. Individuals obtain antibiotic remedy with charge τ and subsequently enter compartment “treated” T. Antibiotic remedy consists of all sorts of beta-lactams (see Supp. Mat. S1). Treatment stops at charge ω and people enter compartment “three months post-treatment” T₃ the place they continue to be on common three months (they go away T₃ at charge ω₃ = 1/3 month⁻¹). Travel to SEA was modeled equally to antibiotic remedy. Individuals journey to SEA at charge ν. While touring, people are in compartment “South-East Asia” SEA the place they continue to be for a interval measured by ψ⁻¹ the place ψ is the speed of return from journey. After journey, these people stay in compartment “three months post South-East Asia” SEA₃ for 3 months (they go away SEA₃ to compartment U at charge ψ₃ = 1/3 month⁻¹). For mathematical comfort, we assumed an ordinary exponential distribution residency time in T₃ and SEA₃. For simplicity, people will be both in compartment SEA or T, such that we didn’t take into account people that each traveled and used antibiotics.

We used the frequencies of ESBL carriers within the (noticed) compartments U, SEA₃ and T₃ for parameter inference (Figure 1).

2.1.2 Multiple carriage

We developed a structurally impartial, mixed-carriage mannequin (sensu Davies et al., 2019) to explain the epidemiological dynamics within the system of compartments described above (Figure 1C). Individuals will be colonized by as much as two strains. We assumed that strains will be both delicate (S) or resistant (R) to beta-lactams.

In the preliminary mannequin, we thought-about two pressure sorts, R and S. After inference of the parameters (see under), the mannequin was capable of reproduce the noticed equilibrium, however coexistence between R and S strains occurred for a slender vary of parameters and the dynamics of equilibration was a lot slower than the noticed one (introduced in Supp. Mat. S7). To enhance the mannequin, we launched additional genetic structuring by modelling a locus underlying within-host persistence, as principle confirmed that variability in period of carriage throughout totally different bacterial genotypes generally is a issue sustaining coexistence (Lehtinen et al., 2017). At this locus, we thought-about colonizer (C) and chronic (P) sorts. Including this genetic structuring within the mannequin was additional supported by latest findings in E. coli displaying that pressure sorts find alongside a colonization-persistence trade-off (Morel-Journel et al., 2025). Further, ESBL-carriage seem like extra frequent in phylogroups B2, D, F and significantly in sequence kind ST131 (Birgy et al., 2016), all related to lengthy carriage time (Östblom et al., 2011; Johnson et al., 2022; Martinson et al., 2019). Overall, we thought-about 4 sorts of strains, that are {S_C, S_P, R_C, R_P}, the place the capital letter and index characterize resistance to beta-lactams and intestine persistence, respectively. It follows that people will be both uncolonized (1 chance), single colonized by one of many 4 strains (4 prospects), or co-colonized (absolutely delicate S_okS_l, absolutely resistant R_okR_l, or combined R_okS_l, ok and l in {C, P}, 10 prospects), which all collectively correspond to fifteen doable colonization statuses. Resistance corresponds to ESBL-producing strains

To inform the outline of co-colonizations in our mannequin, we inferred from our information what fraction of hosts have been uncolonized by Enterobacterales, colonized solely by ESBL strains, or by ESBL along with delicate strains (Fig. 1C). To achieve this, among the many N = 3443 samples, we randomly chosen 301 samples to measure the full density of Enterobacterales (EB) along with the density of ESBL-carrying EB (Supp.Mat.S2). Samples the place no EB have been discovered have been thought-about uncolonized. Samples containing each ESBL and delicate strains have been thought-about as mixed-carriage people (Fig. 1D). Lastly, samples containing solely ESBL strains have been thought-about as R people, with both just one or two R strains. The frequencies of uncolonized and mixed-carriage hosts have been used for parameter inference.

Previous analyses confirmed that clearance of resistant strains happens as an abrupt course of following a doubtlessly lengthy interval of carriage (Cotto et al., 2023). Change of within-host densities of resistant and delicate strains was not observable on the time scale of that research. Such dynamics are a key determinant of the between-host epidemiological dynamics (Davies et al., 2019). In accordance with our outcomes, we modeled competitors between resistant and delicate strains as a clearance-recolonization course of (Figure 1C). Strains are often cleared from the intestine of their hosts, thus opening alternatives for re-colonization. Strains compete to colonize new hosts and reside in them (competitors on the between-host scale).

All in all, the epidemiological dynamics is described by a system of bizarre differential equations (ODE) defining the modifications within the frequencies X_j,i of the 75 sorts of hosts, the place the index j is a pair within the set {∅, S_C, S_P, R_C, R_P} indicating the colonization standing as outlined above (15 doable pairs), and the index i corresponds to one of many 5 epidemiological compartments {U, T, T₃, SEA, SEA₃}. The full system of equations is obtainable as a web based supplementary materials. The mannequin was simulated till equilibrium was reached. Comparisons between the frequencies predicted by the mannequin at equilibrium and the 5 statistics within the information (frequencies of ESBL in U, SEA₃, T₃ and frequencies of uncolonized and mixed-carriage hosts) in a chance framework have been used to deduce the parameters of curiosity (see under).

2.2 Parameters

2.2.1 Models for the transmission and clearance charges

Transmission and clearance charges in relation to the colonization standing

Transmission and clearance charges in precept rely each on the pressure and the kind of host. For inference, we diminished the variety of parameters based mostly on biologically believable assumptions (Table 1). First, we modeled the transmission charge of a given pressure as relying solely on whether or not the recipient host is already colonized. We notice and , the colonization by pressure j_ok of uncolonized and single-colonized hosts, respectively.

We additionally simplified pure clearance charges. The clearance charge of a given pressure will depend on whether or not the host is single- or co-colonized. The clearance charge from a bunch single-colonized by pressure j_ok is . For co-colonized hosts, we assumed area of interest differentiation: clearance of 1 pressure is slower when it co-occurs within the host with a pressure of the opposite kind (C or P). Niche differentiation between colonizer and chronic strains is measured by parameter ϵ ∈ [0, 1]. The clearance charge of pressure j_ok from a bunch co-colonized by strains j_ok and i_l (with ok, l ∈ C, P) is if l = ok, and in any other case (i.e., clearance is diminished by an element ϵ if the co-occurring strains are of a unique kind). We additional assumed that clearance charges are the identical in all epidemiological compartments.

We didn’t explicitly mannequin the epidemiological dynamics overseas. Instead, we modeled a continuing power of colonization whereas touring, equal to λ_y, the place y is the colonizing pressure.

Transmission charge from co-colonized hosts

We modeled hosts as two “spaces”, every of which will be colonized by a single pressure. This assumption might outcome, for instance, from the fine-scale spatial construction of the area of interest occupied by E. coli within the intestine (e.g. Pereira and Berry 2017). We assumed that co-colonized hosts transmit twice as a lot as single-colonized hosts, such that the transmission charge from {y, y}-hosts is 2X_y,yβ_y,i (i ∈ 1, 2). We carried out an evaluation of sensitivity to this assumption and located that it didn’t have an effect on our fundamental conclusions (Supp.Mat.S6).

For simplicity, we uncared for the likelihood of co-transmission by co-colonized hosts. Further, we modeled that mixed-carriage hosts carrying each a resistant (R) and a delicate (S) pressure can shed these strains at totally different charges, scaled by the parameter m_x. Precisely, {R_ok, S_l}-hosts transmit R strains at charge and S pressure at charge , the place j ∈ {1, 2} and ok, l ∈ {C, P}. Given that we anticipate R strains to happen at low frequency relative to S strains (Ruppé et al., 2013; de Lastours et al., 2016), we assume that m_x ∈ [0, 1]. The worth m_x = 1 corresponds to R strains in mixed-carriage transmitting as a lot as in the event that they have been alone within the host.

Transmission and clearance charges of resistant strains

The epidemiological parameters of resistant strains are outlined relative to these of delicate strains via the associated fee parameters δs performing on the transmission and clearance charges. The prices of resistance didn’t rely on whether or not it was related to a persistent (P) or colonizer (C) kind. Given and , we distinguish the price of resistance on colonization and super-colonization as

the place δ_β,R,1 and δ_β,R,2 are each included in [0, 1] and measure the price of resistance on colonization and super-colonization, respectively. Similarly, we assumed that resistance can scale back persistence via an elevated clearance charge that will depend on whether or not the host is single- or co-colonized.

We modeled the clearance charges of resistant strains in comparison with that of delicate strains as

the place δ_γ,R,1 and δ_γ,R,2 are the prices of resistance included in [0, ∞].

2.2.2 Parameter estimation

Overall, we assume that the prices of resistance can happen on transmission and clearance and that they will rely on the colonization standing of the hosts. We infered the prices of resistance which might be measured by the set of parameters :

The different parameters have been instantly obtained from the focal information or from information in related communities, as broadly described under. The particulars of the parameter estimation are offered in Supplementary Material S3. Table 1 recapitulates the mannequin parameters.

Rates of journey to SEA and antibiotic use

These charges have been instantly estimated from the imply per-year fraction of kids touring to SEA and having used antibiotics [including Cefpodoxime N=144 (11%), Amoxicillin N=681(54%), Amoxicillin and clavulanic acid N=370 (29%), others N=75 (6%)] within the three months earlier than sampling in our epidemiological dataset.

Force of an infection in South-East Asia

We assumed a set power of recent colonization by delicate and resistant strains in South-East Asia, denoted λ_S and λ_R, respectively. The worth of λ_R was chosen such that about 50% of vacationers could be colonized by R-strains throughout a 20-days journey (see Supp.Mat.S3). Without data on the colonization charge by S-strains throughout journey, we assumed λ_S = λ_R.

Epidemiological parameters of delicate strains

Our focal information set didn’t enable inference of the colonization and clearance charges of delicate strains. To achieve this, we used information on the turnover of E. coli strains in Swedish kids from start to 12 months outdated from Östblom et al. (2011). We assumed this Swedish kids group had related traits to our focal one. The evaluation of those information is offered in Supplementary Material S3. These information allowed to estimate the clearance charges of delicate strains from single- and from co-colonized hosts and for each the colonizer and chronic sorts (ok ∈ {C, P}), and the area of interest differentiation parameter ϵ (Table 1). The persistent (P) and colonizer (C) sorts have been recognized as belonging and never belonging to the phylogroup B2, respectively, in step with the outcomes of Östblom et al. (2011). We then computed the transmission charges such that strains S_C and S_P had the identical R₀ (health) and will due to this fact readily coexist in the neighborhood.

Inference of the epidemiological value parameters of resistant strains

We used a Bayesian strategy to deduce the posterior distribution of the associated fee parameters in P_R. We formulated the chance with 5 unbiased elements, utilizing two items of knowledge derived from our focal dataset. For every element, the chance of the noticed numbers (n_R,i) is their possibilities beneath the regulation B(N_i, p_i) with N_i the full variety of observations in element i and p_i the corresponding frequency predicted by the mannequin. First, our epidemiological information inform on the frequency of resistance in compartments U (n_R,U = 105, N_U = 1632), and SEA₃ . Note that we didn’t take into account people each returned from journey and who used antibiotics. We moreover measured the density of Enterobacterales and ESBL-producing E. coli on a sub-sample of the info (N = 301, Supp.Mat S2) to tell on the general frequency of mixed-carriage hosts X_RS (n_RS = 15) and uncolonized hosts X_∅ (n_∅ = 5).

Posterior distributions of the parameters have been estimated utilizing a Bayesian strategy with flat priors (see Table 1). The posterior distributions have been sampled by MCMC utilizing a Metropolis-Hastings algorithm with adaptive sampling (Supp.Mat.S4). We ran 5 unbiased MCMC chains of 250000 iterations to examine convergence to the identical posterior distributions. For every chain, the preliminary circumstances have been drawn randomly throughout the prior distribution (Table 1). The efficient pattern dimension and the convergence of every chain have been verified utilizing the Gelman-Rubin diagnostic carried out within the R-package coda (Plummer et al. 2006). Statistics on the MCMC chains are offered in Supplementary Material S5.

We first current the epidemiological information, earlier than turning to parameter inference with the mannequin and its sensitivity evaluation. The frequency of resistance in the neighborhood was outlined because the frequency of hosts who carry a minimum of one resistant pressure.

3.1 ESBL carriage

Over 2010-2018, ESBL have been present in 257 of the 3443 assessed kids (7.5%, 95% CI=6.7-8.4). There was no pattern within the change in ESBL through the research interval (black on Fig. 2), as the rise in ESBL frequency in the neighborhood occurred earlier than that interval (blue on Fig.2). The ESBL frequency was significantly steady over the interval 2015-2019. Among the assessed kids, 0.6 % had travelled to SEA and 37% had acquired antibiotics inside seven to a few months earlier than enrollment. Further particulars are offered in tables S1 and S2.

3.2 Inference of the only and combined carriage of EBSL

To inform the mixed-carriage mannequin, we inferred the full variety of Enterobacterales (largely and hereafter E. coli) and the variety of these carrying ESBL in a subset of 301 samples (particulars in Supp. Mat. S2). We discovered that, normally, ESBL and non-ESBL E. coli co-occurred within the samples (Fig. 1D). Indeed, in most (N = 15) of the samples containing ESBL-carrying E. coli (N = 18, see Tab. S3), the estimated variety of ESBL-carrying E. coli was decrease than the estimated complete variety of E. coli. In these samples, ESBL-carrying E. coli represented on common 4.8 % of the full variety of E. coli. In the remaining ESBL-carrying samples (N = 3), the ESBL-carrying E. coli fashioned nearly all of E. coli (Tab. S3). Lastly, we discovered 5 samples that have been apparently not colonized by any E. coli.

3.3 Inference of the prices of resistance

Goodness of match

Overall, after inference, the mannequin was capable of reproduce the frequencies of the several types of host colonization noticed within the focal information set. The distribution of the host-type frequencies predicted by the mannequin after sampling the posterior distribution falls primarily throughout the 95% confidence interval of the noticed frequencies (Fig.S10).

Costs of resistance on transmission and clearance

We discovered that resistance is expensive: R-strains transmit much less effectively and are cleared quicker than S-strains (Fig. 3A-D). The prices of resistance relied on the colonization standing of the hosts. Our mannequin was capable of infer comparatively exact estimates of the prices of resistance on colonization of already colonized hosts and on persistence in co-colonized hosts (i.e., extra clearance). These prices should be low to clarify the info. Precisely, we inferred the associated fee on the flexibility to colonize already-colonized hosts (δ_β,R,2) at 0.14 (median = 0.12, 95% CI = 0.006-0.38, Fig. 3B); and the associated fee on the persistence potential in co-colonized hosts (δ_γR,2) at 0.23 (median = 0.20, 95% CI = 0.009-0.62, Fig. 3D). In different phrases, resistant strains colonize occupied hosts at a charge diminished by −14%, and are cleared +23% quicker when in competitors with different strains. These two prices are probably the most exactly inferred, as a result of they concern the most typical ecological contexts through which resistant strains are encountered.

In distinction, the posterior distributions of the prices on transmission to uncolonized hosts and clearance from single-colonized hosts have been a lot wider. The 95% CI of the posterior distribution span many of the prior distribution for the parameters δ_β,R,1 and δ_γ,R,1 (shadowed space in Fig. 3A and C). A variety of resistance prices on colonization of uncolonized hosts and persistence in single-colonized hosts are suitable with the info.

Contributions of mixed-carriage to transmission

We discovered that the transmission charge of a R-strain is comparable when it’s in combined carriage with a S-strain and when it’s alone. Indeed, we estimated that the imply transmission charge of a R-strain whereas in combined carriage (m_x parameter) is 88% (median 90, 95% CI = 68-99) of that when it’s alone (Fig. 3E). This is in step with the evaluation of the rely information displaying that the density of ESBL-carrying E. coli is comparable in hosts colonized solely by these strains and in these carrying each ESBL and non-ESBL E. coli (Fig. 1D).

Overall, given the inferred imply values of the totally different health prices of resistance, our mannequin predicts that after stopping journey and antibiotic remedy, it might take roughly 30 (resp. 60) months for the frequency of antibiotic resistance in the neighborhood to say no under 1% (resp. 1/1000).

3.4 Predictions of the mannequin

Next, we investigated the dynamical penalties of modifications in key parameters of the mannequin on the frequency of resistance in the neighborhood. The reference parameter values have been these of Table 1 accomplished by the technique of the posterior distributions for the inferred parameters (Fig. 3). Resistance frequency was primarily pushed by the remedy charge and the transmission charge of E. coli (Fig. 4A and B), however not by journey (Fig. 4A and C).

Impact of the remedy charge

We predicted {that a} two-fold improve within the charge of antibiotic remedy would end in resistance growing from the reference equilibrium worth, 0.067 (predicted by the mannequin), to 0.34 (a five-fold improve). A two-fold discount in remedy would scale back resistance to 0.0031 (Fig. 4A, C, D).

Impact of the transmission charge

A two-fold improve within the transmission of E. coli will increase resistance to 0.41 (a six-fold improve in comparison with 0.067), whereas a two-fold discount reduces resistance to 0.0026 (a 26-fold lower in comparison with 0.067) (Fig. 4B, D). Interestingly, the saturating shapes of the treatment-resistance and transmission-resistance relationships (Fig. 4A, B) suggest that for a similar baseline degree of resistance, lowering transmission or remedy charge might have a small or massive influence on the frequency of resistance (arrows on Fig. 4A, B).

Impact of the journey charge

In distinction, various the speed of journey had little or no influence on the equilibrium frequency of resistance (Fig. 4A, C). The solely discernible influence was when the remedy charge may be very small, through which case resistance is counter-selected and the equilibrium frequency of resistance is due to this fact proportional to journey (Fig. 4A inset). In inhabitants genetics phrases, on this parameter area resistance is maintained at a selection-migration stability.

Negative frequency-dependent choice on resistance

We investigated the frequency-dependent choice (Harmand et al., 2019) on resistance generated by our mannequin that features a mixture of mechanisms stabilizing resistance at an intermediate frequency. Our mannequin generates adverse frequency-dependent choice (NFDS), as proven by the slope of the choice coefficient on resistance as a operate of its frequency in Figure 5, which ends up in the coexistence of resistant and delicate strains. The equilibrium frequency of resistance is the place the curve crosses the x-axis. The choice coefficient on resistance first quickly declines, earlier than reducing because the frequency of resistance will increase. The preliminary section of robust constructive “selection” is definitely pushed by journey and importation of resistance from overseas. This preliminary improve is quicker when the journey charge is excessive. The second section of decline will depend on the remedy and transmission charges. Increasing the remedy or the transmission charge strengthen choice in favor of resistance at massive resistance frequency and consequently will increase the equilibrium frequency of resistance (Fig. 5A,C).

Modeling each a colonizer and a persistent technique strengthens NFDS in comparison with a mannequin assuming a single generalist technique with the identical common health (Fig. 6A). We additional discovered that when each a colonizer and a persistent technique are carried out, resistance is related to the persistent technique (Fig. 6B), whereas sensitivity is related to the colonizer technique (not proven). These associations improve when differentiation alongside the persistence-colonization commerce off will increase (Fig. 6B). Overall, the NFDS generated by the prevalence of differentiated methods on the colonization-persistence trade-off, and the non-random affiliation of resistance with these methods, favor the coexistence of resistant and delicate strains in the neighborhood.

In Europe, the stabilization of the frequency of resistant E. coli at a low degree after a speedy improve within the early 2000s stays largely unexplained (Emons et al., 2025). Empirical and theoretical research have pointed to a number of mechanisms that favor resistance carriage and coexistence of resistant and delicate strains in the neighborhood (Blanquart, 2019; Rahbé et al., 2024). However, how these mechanisms contribute to the noticed dynamics stay largely unknown. Here,we proposed a data-informed epidemiological mannequin that leverages i) a big dataset on ESBL-driven antibiotic resistance in wholesome infants ii) an in depth microbiological characterization of samples and iii) our present information of the ecology and epidemiology of E. coli, to realize a mechanistic understanding of the evolution of ESBL-driven antibiotic resistance in the neighborhood in France.

It has repeatedly been reported that coming back from South and Southeast Asia is a danger issue for carriage of resistant strains of Enterobacterales. Although journey has been introduced as an vital contributor to resistance in high-income international locations, and measures to restrict the importation of resistance by vacationers have been proposed (e.g., Reuland et al., 2016; Arcilla et al., 2017), the precise influence of vacationers on native resistance had not been quantified. This influence not solely will depend on the speed of journey, but additionally on the dynamics of onward transmission of resistance by vacationers and the eventual clearance of resistant strains. In our mannequin together with these phenomena, we present that importations have little impact on the native equilibrium ESBL frequency, besides initially or when the native remedy charge may be very low. At the present or greater remedy charges, a 10-fold improve within the charge of journey would barely change the equilibrium frequency of resistance within the focal group. Our outcomes are in step with one other latest mathematical mannequin (Rahbe et al., 2025).

ESBL resistance should include prices that counterbalance its benefit beneath remedy. Costs of antibiotic resistance have typically been present in in vitro development assays (e.g. Melnyk et al., 2015). However, experimental proof on whether or not and the way antibiotic resistance alters transmission remains to be lacking (Andersson, 2006) and the prices when it comes to epidemiological parameters haven’t beforehand been quantified. Here, we modeled and inferred 4 prices of resistance: two prices on transmission to empty and colonized hosts, and two prices on clearance in single- and co-colonized hosts. We might solely exactly estimate the prices of transmission to colonized hosts and clearance in co-colonized hosts, which corresponds to probably the most frequent ecological context through which ESBL E. coli are discovered. These prices have been discovered to be comparatively low (−14% and +23% on transmission and clearance, respectively).

We additionally discovered that resistant strains transmit as a lot from single-colonized hosts as from mixed-carriage hosts (m_x near 1), which is in step with our information, the place resistant strains reached related density when they’re alone of their host as when they’re co-carried with delicate strains. The mannequin additional implies that ESBL strains can’t persist in the neighborhood if resistance transmission from mixed-carriage hosts is decrease than from hosts carrying the resistant strains solely.

Overall, the comparatively small value of resistant strains in transmission despite considerably decrease density could possibly be defined if the connection between within-host density and transmission charge is sublinear, as is the case for different pathogens the place this relation was characterised (Fraser et al., 2007; Duong et al., 2015; Marc et al., 2021). Such relationship could possibly be decided for ESBL resistance from family research, however this has not but been completed to our information (Arcilla et al., 2017; Perez et al., 2025).

In our mannequin, the frequency of ESBL is strongly decided by the speed of remedy with beta-lactams. The relationship between the speed of remedy and the equilibrium degree of resistance was fairly progressive, in distinction to most earlier theoretical work (Blanquart, 2019; Davies et al., 2019). Negative-frequency dependent choice (NDFS) is vital to advertise coexistence between resistant and delicate strains (Davies et al., 2019). Our mannequin with biologically believable construction and parameters was capable of generate NFDS and reproduce the progressive use-resistance relationship.

An vital discovering is that the variability in carriage period, right here modeled with a twice longer carriage period for strains of phylogroup B2, was a very powerful issue favoring coexistence. Resistance preferentially evolves on the genotype with lengthy period of carriage in our mannequin (Fig. S8) (Lehtinen et al., 2017). This prediction was additionally verified to some extent in our information: typing among the resistant strains isolates from these information confirmed that ESBL resistance was largely confined to ST131 (phylogroup B2) (Birgy et al., 2016), a sequence kind with remarkably lengthy period of carriage (Johnson et al., 2022). Thus, the heterogeneity in period of carriage should be modeled to correctly reproduce qualitative patterns in information. Other components don’t generate a lot coexistence. The clearance-recolonization dynamics, which we modelled in step with accessible information (Cotto et al., 2023), is poorly conducive to coexistence (Fig. S6). Lastly, journey had a really restricted function in stabilising coexistence: despite robust differentiation between France and Southeast Asia, the speed of journey was too small to considerably influence equilibrium ranges of resistance.

Nevertheless, the noticed coexistence and the NFDS have been much less robust in our mannequin than in information. Among the components that we didn’t embrace within the mannequin, the construction in hospital-community and the acquisition of ESBL strains in hospitals could not play an vital function: in broad phrases, hospitalization has analogous penalties to journey. In France, there are an estimated 10M hospital stays per yr. If about half of people return from hospital stays colonized by ESBL, the influence of hospitalization is equal to the influence of journey at a charge of 0.02 per thirty days (10-fold higher charge), which is negligible. Moreover, in our information, infants with a latest hospital keep didn’t have significantly elevated ranges of resistance (Table S1). Other unobserved sources of inhabitants construction might play a job in coexistence, however this could suggest {that a} main danger issue for ESBL carriage has been neglected within the many earlier research. The final and most believable rationalization in our view is that the distribution of carriage period is variable at a finer phylogenetic unit (i.e. throughout ST). For lack of higher information, we modelled a quite simple distribution of carriage period reflecting a two-fold longer carriage for B2 phylogroup vs. non-B2 phylogroup as in Swedish infants (Östblom et al., 2011). The true worth of this ratio was bigger (4-fold) in one other research (Johnson et al., 2022). Modelling this distribution in additional particulars could result in a extra progressive use-resistance relationship (Lehtinen et al., 2017), stronger NFDS, and will due to this fact be an vital focus for future work.

We additionally discovered that growing transmission will increase the prevalence of resistance in the neighborhood. Importantly, the impact of transmission is almost as robust as that of the speed of antibiotic remedy. This discovering is in step with a worldwide evaluation that discovered that components related to poor sanitation, however not antibiotic use, are related to antibiotic resistance in E. coli (Collignon et al., 2018), and different latest modelling work (Rahbe et al., 2025). Transmission favors colonization by resistant strains throughout and instantly after remedy, when competitors by delicate strains is launched (Blanquart et al., 2018; Davies et al., 2019). In untreated hosts, growing transmission equally favors resistant and delicate strains with out affecting their relative health (assuming a set value of resistance). Without remedy, delicate strains improve in frequency by out-competing resistant strains within the clearance-recolonization dynamic of untreated hosts. Explicitly modeling remedy construction is important to seize how transmission favors resistance (Blanquart et al., 2018) (a mannequin with out such construction predicts that transmission disfavors resistance− not proven).

Our dataset, despite its very massive dimension, give attention to wholesome people, and detailed data, has some limits. We centered on wholesome infants aged lower than two years and use these information to mannequin the group as an entire. We notice that the majority research specializing in resistance within the intestine should resort to comfort sampling, e.g. infants, pregnant ladies, hospital sufferers. Several outcomes help that the epidemiology of E. coli resistance and sensitivity weakly will depend on the age of the hosts. Children are very quickly colonized by E. coli after start (Palmer et al., 2007). The equilibrium ESBL frequency in our information was just like that present in different age teams within the Paris space (Nicolas-Chanoine et al., 2013). Data from invasive infections collected by the European Center for Disease Prevention and Control (ECDC) additionally doesn’t help a powerful and constant construction throughout age courses for E. coli ESBL resistance (Fig.S11). Lastly, the patterns within the information that inform our inferences are sturdy to analyzing infants, kids or adults. History of remedy and journey as fundamental danger components for resistance are nicely established (e.g. Birgy et al., 2016), as is the truth that resistant more often than not co-occurs as a minority strains inside hosts (Ruppé et al., 2013; de Lastours et al., 2016). Another potential limitation of our information is that kids present process antibiotic remedy inside 7 days previous to the sampling have been excluded from the survey. These kids doubtlessly carry ESBL at greater frequency and density (Ruppé et al., 2013; de Lastours et al., 2016), which might result in underestimating the frequency of ESBL in our focal group and the prevalence of people colonized solely by resistant strains. This underestimation is more likely to be weak, as given the noticed charge of remedy we’d miss solely 9 infants with this exclusion criterion. Accordingly, the ESBL frequency in our information is in step with different research (e.g. Gagliotti et al., 2011; Bezabih et al., 2021).

Secondly, whereas we fastidiously crafted our mannequin to incorporate all believable processes affecting antibiotic resistance, among the assumptions could possibly be mentioned. We didn’t particularly mannequin people who each travelled and used antibiotics (N=7). It is unlikely that modeling these people would have impacted our outcomes, given the restricted function of journey that we discovered and the mannequin constrains some vacationers to return again with ESBL. We didn’t embrace hospitalization as a danger consider our mannequin, however as mentioned above hospitalization was not a danger consider our information, and in principle can’t play an important function. We assumed that co-colonized hosts carry and transmit twice as a lot micro organism than single-colonized ones. Models typically alternatively assume that the transmission charge is similar for single- and co-colonized hosts (Alizon et al., 2013). The predictions of our mannequin weren’t affected by this assumption (Fig.S3). Lastly, our mannequin permits solely weak aggressive launch of the resistant pressure: remedy results in clearance of the delicate pressure, and the resistant pressure doesn’t transmit extra when it’s alone within the host. Empirical information counsel that remedy clears a big fraction of the entire microbiota. This might enable a stronger improve within the density of resistant strains than modelled right here (Ruppé et al., 2013; Niehus et al., 2020). Our predictions stay unchanged when assuming a 5-fold improve in transmission of resistance upon remedy (Fig.S4).

Our findings have a number of implications. The most vital drivers of native resistance frequency have been antibiotic use in the neighborhood and transmission, not the importation of resistant strains from Southeast Asia. Coexistence was primarily stabilized by the interplay with one other locus figuring out the period of carriage (Lehtinen et al., 2017). This affiliation has a number of vital penalties. First, it’s troublesome to measure a price of resistance on pressure clearance as a result of resistance is related to strains with lengthy period of carriage. To instantly measure this value, it’s mandatory to regulate for the influence of the genetic background on carriage period (Krishna et al., 2025). Second, an intervention to decolonize resistant strains of E. coli ought to be very environment friendly: certainly, if resistant strains are related to “slow-turnover” strains with lengthy period of carriage and low colonization charges (Morel-Journel et al., 2023), then eradicating these strains would severely scale back their health. Thirdly, novel resistances may primarily emerge first on travellers, second on long-duration of carriage genomic backgrounds. Strains with lengthy period of carriage could possibly be significantly focused for surveillance. It would thus be essential to quantify the distribution of residence instances of bacterial strains in commensalism, and perceive what components keep this range, to completely perceive resistance evolution.

To conclude, utilizing one of many largest cohorts learning ESBL resistance in the neighborhood in wholesome people, along with detailed and knowledgeable modeling of competitors between delicate and resistant strains–a essential level that has been neglected by some earlier research–we understood mechanistically the evolution of resistance in the neighborhood in France. This factors to round 10-20% prices of resistance on transmission and clearance, a serious function of choice by native antibiotic use and transmission, and variability within the period of carriage as the principle issue stabilizing delicate and resistant strains. This opens views for the quantitative research of resistance evolution, most apparently testing our prediction on the sublinear relationship between pressure density and transmission (for instance in family research); and improved characterization of the period of carriage of commensal bacterial strains.

S1 Statistical evaluation

Summary of the complete information set

3443 rectal samples from French kids aged 6-24 months have been collected from 2010 to 2018, similar to 3417 totally different people. Sampling was related to a questionnaire. Information so gathered is absolutely described in (Birgy et al., 2016).

Birgy et al. (2016) additional analyzed the danger components related to ESBL carriage (based mostly on samples obtained through the 2010-2015 interval). In specific, a multivariate evaluation recognized dwelling care, the use antibiotics and the journey historical past inside three months previous to sampling as the principle danger components associated to ESBL carriage. Furthermore, univariate analyses recognized journey to Oceania/Asia as the one area related to a major improve within the danger of ESBL carriage (Birgy et al., 2016).

In the current research, we centered on the danger components associated to ESBL carriage similar to i) journey to Asia/Oceania, hereafter Southeast Asia [SEA], and ii) using antibiotics, each inside three months previous to sampling (Table S1). We prolonged the analyses of Birgy et al. (2016) to the interval 2010-2018 and located related outcomes (Table S2).

S2 Density of resistant and delicate E. coli strains in mixed-carriage hosts

Among the above samples (years 2014-2016), 301 have been additional analyzed to judge the full density of Enterobacterales (EB) and the density of ESBL-carrying E. coli strains. Density has been estimated utilizing dilution counts (by an element 10⁻¹ to 10⁻⁶) on selective media (Drigalski for Enterobacterales and ESBL-selective). There was 5 samples the place no EB have been detected.

We additional assumed that the variety of micro organism in every pattern adopted a Poisson distribution. The Maximum Likelihood estimator for the variety of micro organism is then

the place d_i is the dilution issue and ok_i is the variety of micro organism counted at dilution d_i.

For every pattern the place EB have been discovered, we then in contrast two fashions utilizing a AIC criterion (i.e. ΔAIC ≥ 2): i) η_EB = η_ESBL (all EB carry ESBL genes), ii) η_EB ≠ η_ESBL (some EB carry ESBL genes). The outcomes are introduced on Table S3.

We discovered 5 samples with out EB cells (Tab. S3). In most samples with ESBL strains, ESBL strains co-occurred with delicate strains. We discovered three samples the place all EB carried ESBL, together with two samples through which the variety of EB cells carrying ESBL was estimated to be bigger than the full variety of EB cells, most likely as a consequence of measurement error. Figure S1 exhibits the distribution of the relative ESBL density. We discovered that normally, ESBL density represents greater than 1% of the full EB density (30% quantile at 1% and imply at 17%). In our chance calculation, we thought-about that hosts with belonged to kind “RS”, and we used the variety of uncolonized hosts η_EB = 0 (see fundamental textual content).

S3 Parameters

We present right here the strategies used to acquire the parameter values that aren’t inferred with the mannequin.

Treatment charge

In the focal information set, on common 37% (min 33%, max 43% relying of the yr) of kids acquired antibiotics inside three months earlier than sampling. The variety of kids beginning an antibiotic remedy throughout a given time interval follows a Poisson course of. The elapsed time T between the top of the remedy and the sampling follows an exponential regulation of parameter τ. The likelihood that the sampling happens throughout a three-month interval after remedy is P (T ≤ 3) = 1 − e^−3τ = 0.37. Solving for this equation, we discover τ = 0.15 m⁻¹. The charge at which the occasion “sampling occurs within three months after treatment” happens is the same as the speed of prevalence of “starting an antibiotic treatment”, such that τ is the remedy charge.

Duration of the remedy

The typical period of antibiotic remedies is one week (Spellberg and Rice, 2019), similar to a month-to-month charge of 4 m⁻¹.

Antibiotic effectivity

We used outcomes from Paterson et al. (2016) who investigated the optimum antibiotic focus to remedy bacterial an infection. Using their typical values for antibiotic focus (23 µg.L⁻¹) and delicate strains of their equation 5, we discover an approximate dying charge as a consequence of antibiotics of 8.6 d⁻¹ of their unique scaling, similar to 266 m⁻¹.

Post-treatment interval

By selecting ω₃ = 1/3 m⁻¹ the imply time spent within the compartment T3 is three months.

Travel charge

In the focal information set, the proportion of kids touring to Southeast Asia diversified between 0 and a pair of% per yr, with imply 0.7% per yr. The journey charge is calculated because the remedy charge. It is the answer of P (V ≤ 3) = 1 − e^−3ν = 0.007, such that ν = 0.002 m⁻¹.

Duration of journey

Most research estimated the imply journey time to SEA is about 20 days (e.g. Ruppé et al., 2015; Arcilla et al., 2017), similar to a charge of return from journey of 1.5 m⁻¹.

Post-travel interval

As for the compartment T3, we selected ψ₃ = 1/3 m⁻¹ such that the imply time spent on this compartment is three months.

Force of an infection throughout journey

Previous research confirmed that 30 to 70% of vacationers to SEA are colonized by ESBL strains throughout journey (Paltansing et al., 2013; Ruppé et al., 2015; Kantele et al., 2015; Reuland et al., 2016; Arcilla et al., 2017; Woerther et al., 2017). We assumed that about 50% of the vacationers are colonized throughout a 20-day journey (Ruppé et al., 2015). We additional assumed that the power of colonization of resistant strains didn’t rely on their colonization kind: .

A easy mannequin of an infection can inform on the power of colonization whereas touring. Consider a cohort of people that journey to Asia throughout 20 days. None of them initially carry MRE. The dynamics of MRE carriage throughout journey is given by: dX_R[t]/dt = λ_R(1 − X_R[t]) (neglecting clearance throughout journey) the place X_R is the density of vacationers carrying MRE throughout journey, λ_R is the power of colonization throughout journey (assumed to be unbiased of X_R and t). Solving for the differential equation with X_R[0] = 0, we get hold of . Further utilizing t = 20 and searching for X_R[20] = 0.5N and fixing for λ_R, we get hold of λ_R ≈ 0.035 ≈ d⁻¹ 1.1 m⁻¹

No data is obtainable regarding the power of colonization by delicate strains throughout journey. As a baseline, we assumed that λ_S ≈ λ_R.

Clearance and colonization charges of delicate strains

We estimated these parameters utilizing an exterior dataset on the turnover of E. coli strains in 127 Swedish infants adopted longitudinally from start (Nowrouzian et al., 2003, 2005; Östblom et al., 2011). The infants have been adopted at 3 days, 1 week, 2 weeks, 4 weeks, 2 months, 6 months, 12 months after start. At every go to, E. coli colonies have been remoted on Drigalski agar from faecal samples. As detailed in Östblom et al. (2011), one to 6 colonies with totally different morphologies have been remoted, typed with Random Amplified Polymorphic DNA (RAPD) profiling and genotyped with a number of PCRs. Here, we used the phylogroup of those strains obtained from Clermont typing (Clermont et al., 2000). Phylogroup B2 was related to an extended period of carriage than others (Nowrouzian et al., 2005; Östblom et al., 2011). Thus, we recognized “colonizer” and “persistent” sorts to non-B2 and B2 strains. We used a Bayesian strategy to estimate the colonization and clearance charges relying on each the colonization standing of the host and on the phylogroup of the strains.

Overview of the mannequin fitted to longitudinal information

We described the pressure turnover inside infants by a steady time, discrete-state Markov chain mannequin. We outlined the strains as 0 (0 ≡ C ≡ non-B2) or 1 (1 ≡ P ≡ non-B2). We thought-about as variables the frequency of empty hosts, hosts colonized by a single pressure, and hosts colonized by two strains. Colonization by greater than two strains solely occurred twice, and these two timepoints have been faraway from our evaluation. There have been due to this fact six variables representing the system: 1 for empty hosts, 2 for the only colonizations, 3 for the co- colonizations. Transitions between colonization states occurred in 3 ways:

• colonization by one of many strains occurring at charges λ₀, λ₁, λ₀₀, λ₀₁, λ₁₀, λ₁₁. These charges account for each the transmission charge and the frequency of the colonizing pressure in the neighborhood. For co-colonization, the primary index represents the pressure already current within the host, and the second index represents the colonizing pressure.

• clearance of one of many strains occurring at charges µ₀, µ₁, µ₀₀, µ₀₁, µ₁₀, µ₁₁. For clearance of co-colonized people, the primary index represents the pressure that continues to be, and the second index represents the pressure that’s cleared.

• direct substitute of one of many strains by one other occurring at charges ν₀₁, ν₁₀. The second index represents the pressure that replaces.

A 6 × 6 transition matrix A described the charges of transitions between host colonization states. The likelihood of a selected noticed change from state i to state j in an toddler, over an interval of period Δt, was given by exp (AΔt)_i,j. The likelihood of noticed succession of states in an toddler was the product of those possibilities. Finally, the chance–the likelihood of the dataset given the parameters–was given by the product of all toddler possibilities.

Our intention was to make use of MCMC to estimate the posterior distribution of the 14 parameters:

After preliminary matches of the mannequin the place strains have been thought-about as all equal, we used exponential priors with means (in months⁻¹):

• 0.18 m⁻¹ for easy colonizations λ₀, λ₁

• 0.066 m⁻¹ for co-colonizations λ₀₀, λ₀₁, λ₁₀, λ₁₁

• 0.078 m⁻¹ for clearance in single colonization µ₀, µ₁

• 0.234 m⁻¹ for clearance in co-colonizations µ₀₀, µ₀₁, µ₁₀, µ₁₁

• 0.00192 m⁻¹ for replacements ν₀₁, ν₁₀

We ran an MCMC algorithm for two × 10⁶ iterations and retained solely the final 1 × 10⁶ iterations to estimate the posterior distribution of parameters, after visually checking convergence of the algorithm.

Estimation of clearance charges

We used the imply posterior distribution of the six clearance charges (see under for the transmission charges), which have been estimated to:

These parameters are associated to the parameters of our mannequin following:

the place d measures the extra clearance from co-colonized hosts (such that γ_ok,2 = γ_ok,1(1 − d)) (ok ∈ {S, P}) and ϵ measures area of interest differentiation (see fundamental textual content). We used a linear mannequin to estimate the and ϵ (the 4 mannequin parameters) from the imply of the posterior distributions utilizing

the place is the vector of the imply of the posterior for every of the clearance parameters, the X_i,ok are binary vectors with worth relying on whether or not comprises the corresponding issue ok. The parameters and ϵ will be instantly obtained from the estimated coefficients of the linear mannequin. Solving this linear mannequin, we thus obtained and ϵ = 0.65. These estimates outcome within the clearance parameters introduced in Table 1.

Transmission charges to uncolonized hosts

We used the imply posterior distribution of the six transmission charges, divided by the frequency of pressure 0 (colonizers) or 1 (persisters) in the neighborhood, to estimate the transmission charges:

These parameters are associated to the parameters of our mannequin following:

the place d_β measures the diminished transmission to already colonized hosts and ϵ_β measures area of interest differentiation performing on transmission. We used a linear mannequin to estimate the and ϵ_β (the 4 mannequin parameters) from the imply of the posterior distributions utilizing

the place is the vector of the imply of the posterior for every of the transmission parameters, the X_i,ok are binary vectors with worth relying on whether or not comprises the corresponding issue ok. The parameters and ϵ_β will be instantly obtained from the estimated coefficients of the linear mannequin. Solving this linear mannequin, we thus obtained . As the time period similar to area of interest differentiation on transmission was not important within the linear mannequin, and in accordance with one other research, we thought-about there was no area of interest differentiation on transmission (ϵ_β = 0.).

The inferred transmission charges within the Swedish cohort weren’t massive sufficient to achieve the noticed prevalence of E. coli in our French cohort, maybe due to variations in epidemiology between Sweden and France. Furthermore, these transmission charges along with the inferred clearance charges led to massive distinction in aggressive potential between persisters and colonizers. We due to this fact modified the transmission charges to higher match our information. The transmission charges to uncolonized hosts have been as an alternative computed to make sure that the P- and C-types (roughly) have the identical R₀ and that the anticipated frequency of uncolonized hosts was (roughly) S^∗ = 1.7 % as noticed in our information. Precisely, we solved:

The second equation corresponds to a tough approximation for the anticipated frequency of un-colonized hosts. Solving these two equations for , and , ends in and as introduced in Table 1. We lastly used the inferred impact of colonization on transmission charges worth of, estimated at d_β = 0.6 to compute the values of and . The closing transmission parameters are introduced in Table 1.

Finally, notice that the info from the 127 Swedish infants didn’t help the mechanism of direct pressure substitute: we estimated the imply posterior of those parameters to ν₀₁ = 0.0055 m⁻¹ and ν₁₀ = 0.0055 m⁻¹.

S4 Metropolis-Hastings (MH) algorithm with adaptive sampling

In the next, we describe the MH algorithm used to acquire the MCMC chains.

Let be:

• P_R,i the vector of parameter values at iteration i

• L_i the chance of P_R,i

• Σ_P the covariance matrix between parameters

• U the variety of iterations to be carried out between two updates of the covariance matrix. We used U = 500.

Initialization

• P_R,0 a set of parameters drawn randomly within the respective prior distributions

• Σ_P a diagonal matrix with diagonal parts similar to the preliminary variances of the parameters (we used 10⁻⁵)

The algorithm then proceeds as comply with

S5 MCMC convergence evaluation

We ran 5 unbiased MCMC chains with preliminary circumstances sampled randomly within the prior distributions. The MCMC chains have been ran for 250,000 iterations. The convergence of the MCMC chains was verified utilizing capabilities (see under) offered within the coda R-package (Plummer et al., 2006).

S6 Sensitivity of the inference to the mannequin assumptions

We investigated the sensitivity of our inference to 2 different situations. For brevity, we current solely the posterior distributions obtained for every different state of affairs. Mostly, we discovered that our fundamental outcomes qualitatively maintain. The fundamental results of the choice situations thought-about listed here are on the estimate of the price of resistance on transmission to uncolonized hosts, as we current under.

First, we assumed that the transmission charge of R-strains will increase when hosts are beneath antibiotic remedy (Fig.S3). This state of affairs fashions robust aggressive launch throughout antibiotic remedy (e.g. Smith et al., 2021). Precisely, we assumed that, within the compartment T (hosts beneath remedy), , the place L ∈ {U, T₃, SEA, SEA₃} and the place i ∈ {1, 2} and ok ∈ {P, C}. In this state of affairs, we obtained a comparatively sharp estimate of a powerful value of resistance on transmission to uncolonized hosts (Fig.S3A), in distinction to the reference state of affairs (Fig. 3). Thus, assuming robust aggressive launch was counterbalanced by inferring a powerful value of resistance on transmission to uncolonized hosts.

Second, we modelled the case the place transmission and clearance didn’t rely on density. Precisely, transmission and clearance charges from co-colonized hosts have been the identical as these from single-colonized hosts: β_xx = β_x and γ_xx = γ_x, with x ∈ {R_ok, S_l} and ok, l ∈ {C, P}. The posterior distributions have been just like these obtained within the reference mannequin (Fig.S4).

S7 Model dynamics and comparability of the two-strain mannequin with the four-strain mannequin

Two-strain mannequin

We initially thought-about the dynamics of resistant (R) and delicate (S) pressure, with out genetic structuring. This mannequin will be retrieved from the mannequin introduced in the principle textual content (contemplating extra genetic structuring with persistent and colonizer strains) by assuming and . The outcomes of the inference have been just like these we obtained with the principle mannequin (Fig. S5). However, the mannequin allowed coexistence in a narrower vary of parameters, particularly for remedy charge (Fig. S6). Moreover, within the two-strain mannequin the dynamics to equilibrium was very sluggish, contrasting with the noticed dynamics (Fig. S7).

label of figure S7 : time in years

Dynamics of R frequency

Expanding the mannequin to incorporate genetic structuring alongside the persistence-colonization trade-off (Morel-Journel et al., 2025) partially resolved the above points. Coexistence between delicate and resistant strains was allowed for a variety of remedy charges (Fig. S6). Assuming that the R pressure was initially introduced by vacationers in a resistance-free group, the dynamics of R frequency to equilibrium was additionally quicker within the four-strain mannequin (evaluate pink and pink in Fig. S7), although nonetheless slower than the noticed one (Fig. S7). Adding genetic structuring elevated negative-frequency dependence choice on the resistance locus (via linkage with the locus underlying within-host persistence, Fig. S8). Indeed, we discovered a powerful affiliation between the loci figuring out antibiotic resistance and within-host persistence. Resistance is extra useful in persistent genotypes, which usually tend to be uncovered to antibiotics (Lehtinen et al., 2017). Resistance was strongly related to persistence, whereas sensitivity was strongly related to colonization (Fig. S8).

S8 Supplementary figures