PLoS ONE plos plosone PLOS ONE 1932-6203 Public Library of Science San Francisco, CA USA PONE-D-19-25350 10.1371/journal.pone.0238692 Research Article Medicine and health sciencesPharmacologyDrugsAntimicrobialsAntibiotics Biology and life sciencesMicrobiologyMicrobial controlAntimicrobialsAntibiotics Biology and life sciencesMicrobiologyMicrobial controlAntimicrobial resistanceAntibiotic resistance Medicine and health sciencesPharmacologyAntimicrobial resistanceAntibiotic resistance Physical sciencesPhysicsThermodynamicsEntropy Biology and life sciencesEcologyEcological metricsSpecies diversity Ecology and environmental sciencesEcologyEcological metricsSpecies diversity Biology and life sciencesEcologyBiodiversity Ecology and environmental sciencesEcologyBiodiversity Medicine and health sciencesHealth careHealth care facilitiesHospitalsIntensive care units Biology and life sciencesMicrobiologyMicrobial controlAntimicrobial resistance Medicine and health sciencesPharmacologyAntimicrobial resistance Research and analysis methodsMathematical and statistical techniquesStatistical methodsTime series analysis Physical sciencesMathematicsStatisticsStatistical methodsTime series analysis Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategies Assessment of antibiotic resistance http://orcid.org/0000-0002-7138-6372 Diebner Hans H. Conceptualization Formal analysis Methodology Visualization Writing – original draft Writing – review & editing 1 ¤ * Kather Anna Data curation Project administration Writing – review & editing 2 Roeder Ingo Supervision Validation Writing – review & editing 1 * de With Katja Conceptualization Data curation Project administration Validation Writing – review & editing 2 Carl Gustav Carus Faculty of Medicine, Institute for Medical Informatics and Biometry, Technische Universität Dresden, Dresden, Germany University Hospital Carl Gustav Carus Dresden at the TU Dresden, Division of Infectious Diseases, Dresden, Germany Martinez-Garcia Ricardo Editor International Center for Theoretical Physics - South American Institute for Fundamental Research, BRAZIL

The authors have declared that no competing interests exist.

Current address: Department of Medical Informatics, Biometry and Epidemiology, Ruhr-Universität Bochum, Bochum, Germany

 * E-mail: hans.diebner@tu-dresden.de (HHD); ingo.roeder@tu-dresden.de (IR) 2020 3 9 2020 15 9 e0238692 10 10 2019 21 8 2020 2020 Diebner et al This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Diversity as well as temporal and spatial changes of the proportional abundances of different antibiotics (cycling, mixing or combinations thereof) have been hypothesised to be an effective administrative control strategy in hospitals to reduce the prevalence of antibiotic-resistant pathogens in nosocomial or community-acquired infections. However, a rigorous assessment of the efficacy of these control strategies is lacking. The main purpose here is to present a mathematical framework for the assessment of control stategies from a processual stance. To this end, we adopt diverse measures of heterogeneity and diversity of proportional abundances based on the concept of entropy from other fields and adapt them to the needs in assessing the impact of variations in antibiotic consumption on antibiotic resistance. Thereby, we derive a family of diversity measures whose members exhibit different degrees of complexity. Most important, we extent these measures such that they account for the assessment of temporal changes in heterogeneity including otherwise undetected diversity-invariant permutations of antibiotics consumption and prevalence of resistant pathogens. We apply a correlation analysis for the assessment of associations between changes of heterogeneities on the antibiotics and on the pathogen side. As a showcase, which serves as a proof-of-principle, we apply the derived methods to records of antibiotic consumption and prevalence of antibiotic-resistant germs from University Hospital Dresden (cf. supplement “DiebnerEtAl_Data-Supplement”). Besides the quantification of heterogeneities of antibiotics consumption and antibiotic resistance, we show that a reduction of prevalence of antibiotic-resistant germs correlates with a temporal change of similarity with respect to the first observation of antibiotics consumption, although heterogeneity remains approximately constant. Although an interventional study is pending, our mathematical framework turns out to be a viable concept for the assessment and optimisation of control strategies intended to reduce antibiotic resistance.

 Open Access Funding by the Publication Fund of the TU Dresden. Data Availability All relevant data are within the manuscript and its Supporting Information files. Introduction

The drastic increase of antimicrobial resistance worldwide resulting in an alarming increase in morbidity and mortality from clinical infections urges scientists and clinicians to develop counter-strategies [1]. Designing new antiinfective agents is an option. However, the creation of new drugs is time-consuming and success is not guaranteed. Therefore, the control of consumption of available antibiotics or, more general, of antiinfectives, is obligatory.

Since a quick replacement of existing antibiotics is not feasible, antibiotic strategies as “antibiotic mixing”, “antibiotic cycling”, “antibiotic switching”, and “rotation protocols” gained evermore attention in the recent decades [2–6] (for a review cf. [7]). All these concepts refer to heterogeneity of antibiotic usage and non-constant prescription rates. For example, antibiotic cycling means to extract one or a subset of classes of antibiotics from administration in a temporarily alternating way whereas other strategies refer to a scheduled change of the dominantly used class of antibiotics. Frequently, mixing refers to a strategy where a group of patients receives drug (class) A and another group receives drug (class) B in an alternating way. Hereby, the permutation usually takes place between wards leading to a spatial heterogeneity of antibiotic consumption. In clinical reality, empirical antibiotic therapy rarely follows strictly defined control schemes but rather adjusted forms of cycling and mixing in consequence of clinical requirements, thus “clinical cycling” (notion adopted from [4]).

Although there is some evidence that the temporal and spatial permutations of rates of consumption of different antiinfectives are able to reduce prevalence of resistant pathogens [2–4, 8], there is a lack of rigorous evaluations of the ongoing processes, which prevents optimisation. The nonlinear time series analysis presented by Lopez-Lozano et al. 2019 is indicative for an approach from a processual stance, but needs to be generalised. Usually, the impact of cycling and mixing strategies (for a review and meta analysis cf. [7]) is investigated by means of prospective (randomized clinical trials (RCTs), controlled clinical trials (CCT), controlled before-after, cross-over) based on well-defined rotation/switch protocols, thereby neglecting (continuous real-world) processes. Interrupted time series analyses with at least three observations before and after intervention exist, but once more, a precise time point of intervention is preconditioned, thereby excluding adjusted strategies of cycling and mixing. Of note, other researchers as e.g. Karam et al. [5] could not confirm a significant reduction of antimicrobial resistance through cycling or mixing protocols.

The main purpose of this work is to present an analytical framework to quantify heterogeneity of both, antibiotic consumption as well as prevalence of antibiotic-resistant pathogens as a function of observation time. This enables the assessment of associations between consumption and resistance and, potentially, to optimise control strategies based on diversity, mixing and cycling either in their adjusted versions, i.e. “semi-controlled” field-like or observational studies, or in their extremes. The analytic framework consists of adapted methods known in other areas like ecosystems [9, 10]. For the purpose of illustrating the mathematical framework, the method is applied to real observational data, i.e., to records of antibiotic consumption and prevalences of antibiotic-resistant pathogens of University Hospital Dresden in Germany (cf. supplement “S1 File”). Although these data have been recorded in the absence of a clearcut study design and without genuine applications of common cycling or mixing strategies, we nevertheless opted to use these data instead of simulated data for illustration purpose and to preserve a “real world” character of our methodological approach. Thus, the application of the proposed mathematical framework has to be understood as a proof-of-principle. Rigorous controlled interventional studies are planned, however, we regard the immediate provision of the analytical framework to have utmost priority.

 Materials and methods Records of antibiotic consumption

The available data set (cf. supplement “S1 File”) contains 25 consecutive quarterly records of antibiotic consumption starting from the first quarter 2012. Consumption has been recorded per administrative units (cost centres, typically wards). However, for the elementary stage of illustrating the proposed method, a grouping of these small subunits into functional units is considered to be sufficient. Therefore, the departments are grouped into the three top-level units: surgical/OP units, intensive care units, and medical/normal care units. Consumption has also been recorded per active agent, which is nested within antibiotic class. In total, 49 active agents have been observed, which are pooled to 12 antibiotic classes.

The consumption of an antibiotic is measured in standardised units according to the ATC/WHO definition of defined daily doses, DDD, in order to allow for comparisons of different active agents. In addition, the number of cases as well as patient days have been recorded on a quarterly basis. This allows to compute the consumption density DDD per 100 patient days in hospitals: D D D d e n s i t y = D D D 100 patient days . Please note, for the sake of completeness, consumption density sometimes refers to DDD per 100 or per 1000 cases, respectively. In the following, we use DDD only since DDDdensity gives virtually the same results (data not shown). Moreover, some measures of diversity are functions of proportions of “species” within an “ecosystem”, which is why we make use of proportions of consumption. If DDDi(t) denotes the consumption of antibiotics within the antibiotic class i ∈ {1, …, n} at time t, the proportion is given by dddi(t)= DDDi (t) ∑ i=1 n DDDi (t) Depending on the context, index i may also refer to the active agent.

 Coefficient of variation

The coefficient of variation, V ( t ) = S D ( D D D ( t ) ) M e a n ( D D D ( t ) ) , where the mean is taken over the antibiotic classes and SD(DDD(t)) denotes the corresponding standard deviation, can be used as a rough estimate of “homogeneity” in a properly defined sense. Unfortunately, analogous to the notion of “dispersion,” “homogeneity” is an ambiguous term which deserves clarification. In ecological analyses, the concept of “maximisation of statistical heterogeneity” refers to an approach by means of an entropy or a related diversity measure as discussed in the following section (cf. [11]). In this context, an ecosystem is maximally heterogeneous, thus has maximum entropy, if all species are equally abundant, at least in the absence of specific weights of the species abundances [9]. In the latter case, V would be zero, i.e., the system has no variability and is thus without (statistical) dispersion. In contrast, physicists prefer to speak of a perfect dispersion, aka a perfect mixture, in such a situation. Thus, an ecosystem or a society close to a monoculture is “homogeneous” whereas a multi-cultural society/ecosystem is called “heterogeneous” which is used synonymously to “diversity.”

In economics, to the contrary, the distribution of incomes is called homogeneous in the case of equal incomes of all individuals in accordance with the idea of a dispersion-free perfect mixture [12]. Due to compatibility, we stick with the ecological approach in the sequel. In this regard, the coefficient of variation is an inverse measure of ecological heterogeneity. Arguably, heterogeneity in the ecological sense is better captured using the concept of diversity, as introduced in the following section. In order to provide compatibility with the terminology based on the notion of “heterogeneity” suggested in relevant publications on antibiotics resistance [2, 13], we cannot completely drop this term.

 Heterogeneity and entropy

Although the coefficient of variation can be interpreted as a rough measure of (inverse) heterogeneity, it has several inadequacies including the lack of uniquely capturing temporal changes. In the sequel, we harness methods known in ecological population modelling and other fields of research for an adequate quantification and assessment of antibiotic mixing behaviour and strategies as well as temporal patterns of prevalence of antibiotic resistance.

Let ai and bi with i = 1, …, n be the proportions of species of two n-species populations composed of the same set of species with ∑ i = 1 n a i = ∑ i = 1 n b i = 1. Similarity of these two populations in terms of species’ abundances can be quantified by the similarity index S I = 1 - 0 . 5 ∑ i = 1 n | a i - b i | . If ai = bi, ∀i = 1, …, n, then SI = 1, i.e., the populations are identical in terms of their species distributions. If, on the contrary, the populations consist of disjoint sets of species, then SI = 0. Similarity index SI scales between 0 and 1. If we now fix say the first population to a i = 1 n , ∀ i = 1 , … , n, which means maximum heterogeneity for this reference population, then, for the other population a heterogeneity index can be defined by H I = 1 - n 2 ( n - 1 ) ∑ i = 1 n | 1 n - b i | . Hereby, the slightly adapted factor n 2 ( n - 1 ) compared to 0.5 in SI (Eq 4) ensures HI ∈ [0, 1] independent from the concrete value of n. As far as we know, HI defined by Eq 5 has been used by Sandiumenge et al. [2] for the first time in the context of assessing antibiotic resistance and reapplied by Plüss-Suard et al. [13]. Please note that the typesettings of the formulas for HI in refs. [2, 13] are incorrect. Abel zur Wiesch and collaborators [4] explicitly call this measure “antibiotic heterogeneity index (AHI)”, although it originated in other fields of research.

Diversity, a notion frequently used in ecology, is a more general concept than heterogeneity [9, 14]. However, diversity is not uniquely defined. Thus, it depends on the specific context to which particular definition of diversity should be drawn on. Even within the field of antibiotic consumption and related antibiotic resistance, the concrete context can vary considerably. Since we here aim at presenting a general mathematical framework, a family of measures whose members are characterised by exhibiting different levels of complexity is presented.

To start with, an obvious somewhat simplistic way to quantify diversity is given by the so called richness, which is merely the number of species in a multi-species population (e.g. ecosystem). In terms of richness, a heterogeneous n-species population with equally frequent species has the same diversity as an n-species population with a minority of dominating and a majority of very rare species, that is to say n. It follows that an expedient diversity measure should be based on the distribution of species’ abundances in some way.

A meaningful definition of diversity Da is based on an “effective number of species” (cf. [9, 14]) given by the species’ proportions pi and a weight parameter a by means of: Da= ( ∑n p i a ) 1 1−a . Setting a = 0 yields D0 = n independently from pi, i.e. richness. Other special cases are: D 1 = e R 1 D 2 = 1 ∑ i = 1 n p i 2 D ∞ = 1 max ( p i ) with R 1 = - ∑ i = 1 n p i ln ( p i ) being the so called “Shannon entropy”, sometimes also called “Shannon index.” A unique name for D1 itself, i.e. the exponential of the Shannon entropy, does not exist, however, in information science it is sometimes called “perplexity.” Diversity measure D2 is called “inverse Simpson index.” The frequently used Gini-Simpson-Index derived from D2 is given by: G S = 1 - 1 D 2. The inverse of D∞ is found in the literature named “Berger–Parker index,” which is simply the proportional abundance of the most abundant type.

The Shannon entropy is a special case of a Renyi entropy defined by R a = 1 1 - a ln ( ∑ i = 1 n p i ) , thus we have D a = e R a. In other words, Ra is a monotonous function of Da, thus, the two measures can be used interchangeably without loss of information since diversity has only a relative meaning, anyway. The same holds for GS and other possible monotonous functions of Da. Entropy Ra, thus Da, independently of a reach their maximum for the fully heterogeneous situation p i = 1 n ( ∀ i = 1 , … , n ) and it then follows that Da = n. From the latter result we conclude that richness might be a sufficient diversity measure for populations close to full heterogeneity. Having said that, heterogeneity HI itself, although it cannot be derived as a special case of a Renyi entropy, shares features of an entropy and is thus a legitimate measure of diversity.

Finally, the frequently used Gini coefficient deserves to be mentioned: G = 1 - 1 2 ( n - 1 ) ∑ i = 1 n ∑ j = 1 n | p i - p j | . The Gini coefficient in this form (for this and other variants see [14]) is an interesting variant of the similarity index SI insofar as it can be interpreted as kind of a self-similarity. Once more, full heterogeneity (equal proportions) p i = 1 n , ∀ i = 1 , … , n, implies G = 1, and maximally unequal proportions (e.g. pi = 1, pj ≠ i = 0) implies G = 0.

For what follows, it is important to bring to mind that both heterogeneity, HI, as well as measures of diversity, Da, GS, and G, are invariant under permutations of indices. In other words, if the proportions of two species are exchanged, diversity (heterogeneity) does not change. However, similarity index SI is capable to account for such a change after some adaptations, as shown in the Results section. In order to assess both (temporal) cycling as well as (spatial) mixing behaviour of antibiotic administration or consumption, respectively, the proper similarity index extends to S I = 1 - 0 . 25 ∑ i = 1 n ∑ j = 1 m | a i j - b i j | where ∑ i = 1 n | a i j | = ∑ i = 1 n | b i j | = 1 ∀ j = 1 … m. Hereby, the inner sum is taken over the m wards or groups of patients between which antibiotic mixing takes place. In this case, aij and bij refer to the relative abundances before and after the swap of antibiotics administrations between the wards, respectively. In practice, i.e. in the absence of a properly defined study protocol, an allocation of both antibiotic consumption as well as the prevalence of resistant pathogens to precisely defined wards or groups of patients is hampered by the reality of adjusted clinical cycling/mixing. In the following, due to lack of appropriate information on mixing strategies, we present definitions of similarity tailored to our needs without taking strict mixing into account.

 Statistical analysis

Main purpose of this article is to apply a diversity analysis to records of both clinical consumption of antibiotics as well as prevalence of pathogens exhibiting antimicrobial resistance. We refer to the previous section for a detailed introduction of the applied diversity measures. It remains to mention that the expected impact of diversity of antimicrobial consumption on the prevalence of resistant pathogens is analysed by means of Pearson’s correlations of the corresponding time series.

Specifically, slopes along with their significance of differing from zero taken from linear regression quantify the temporal changes of both diversity as well as differential diversity. Of particular interest is the comparison between the time courses of differential diversities of antibiotic consumption and prevalence of resistant pathogens. Such a comparison can be achieved by testing of whether the two corresponding slopes differ significantly or not, or, equivalently, by calculating Pearson’s correlation coefficient along with the corresponding significance test. In the same line, the time series of the relative abundance of resistant pathogens is tested for correlations with the time course of the differential diversity of antibiotic consumption using, as before, Pearson’s product moment correlation analysis.

Of note, each of the three time series, i.e. differential diversity of consumption, differential diversity of prevalence, and relative abundance of resistant pathogens, are expected to exhibit autocorrelations. This is a common feature of time series analyses. Nevertheless, Pearson’s correlation is commonly used as a standard technique to quantify the co-variation of two correlated time series and we here regard it to be sufficient for raising hypotheses based on the available preliminary dataset. A more appropriate application of a cross-correlation function with one of two time series being subject to a time lag is not applicable in our case due to the insufficient lengths (and low sampling frequencies) of the time series.

Numerical calculations, statistics, and graphics have been performed with R [15].

 Results Preliminary note

Before presenting the results of applying the derived diversity measures to observational clinical data, we reemphasise that this work mainly aims in presenting the mathematical framework. The following application to real data has an illustrative purpose and outlines future applications to adequately collected data from a controlled trial. Therefore, in terms of clinically relevant results, the following explanations remain provisional.

 Descriptive analysis

Fig 1a shows the 12 time courses of antibiotic consumption per antibiotic class, DDDi(t). The consumptions of 9 classes largely remain constant on a moderate level. One class, that is to say “second-generation cephalosporins”, is characterised by a high consumption at the outset but declines approximately monotonously by more than half towards the end of the observation period. The consumptions of two other classes, in contrast, i.e. “aminopenicillin/beta-lactamase inhibitors” and “narrow-spectrum penicillins” increase approximately monotonously and roughly compensate for the aforementioned decline.

10.1371/journal.pone.0238692.g001 Time course of antibiotic consumption by antibiotic class.

Time courses of a) consumption DDD per antibiotic class, b) proportions of consumption ddd per antibiotic class, c) mean consumption averaged over the antibiotic class, d) coefficient of variation with respect to the antibiotic classes.

Calculations based on DDD are hardly distinguishable from calculations based on the corresponding densities, 100 × DDDi/patient days. Henceforth, due to these minor differences, we skip to report our results with respect to consumption densities since we here primarily deal with an introduction of a methodological concept. The distinction between DDD and the corresponding densities might become important in other contexts, though. The proportions, dddi(t), however, depicted in Fig 1b, will be used in later sections where we calculate measures of diversity.

Fig 1c shows the time course of the quarterly sampled mean antibiotic consumption, Mean(DDD(t)), averaged over the 12 observed antibiotic classes (cf. Fig 1a). The corresponding coefficient of variation, Eq 3, is depicted in Fig 1d.

In the present case, variability thus homogeneity in the ecological sense is rather high during the first 4 to 6 quarters compared with the remaining time course. After an approximately monotonous decline until 2016, V(t) slightly increases again during the final quarters. These results are consistent with the visual impressions from Fig 1a. Starting with a rather homogeneous distribution at the outset with an outstandingly large proportion of a single antibiotic class, we observe a trend towards a narrow distribution around the mean that starts to weakly widen towards the end. Equal proportions, thus V(t) = 0, means perfect heterogeneity, therefore, V(t) can be interpreted as an inverse measure of heterogeneity.

In the same line, the descriptive analysis can be applied to consumption with respect to active agents. Fig 2a shows the 49 time courses of consumption per active agent. We observe that a single active agent, viz. “cefuroxime”, dominates consumption at the outset but declines approximately monotonously towards the end to a level still significantly above the bulk. This decline is compensated by an increase in consumption mainly of “amoxicillin + clavulanic acid” but also some other agents. The time courses of mean DDD and the coefficient of variation with respect to the active agents shown in Fig 2b and 2c reveal that variability remains on a high level during the time course. Thus, it turns out that pooling agents into antibiotic classes has a damping effect with respect to variability or homogeneity, respectively.

10.1371/journal.pone.0238692.g002 Time course of antibiotic consumption by active agent.

Time courses of a) consumption DDD per active agent, b) mean consumption averaged over the active agents, c) coefficient of variation with respect to the active agents.

Next step is to account for the Hospital’s functional units. The three panels of Fig 3 show the time courses of antibiotic consumption per antibiotic class stratified by the three functional units: unit 1 = intensive care units, unit 2 = medical/normal care units, unit 3 = surgical/OP units. Unit 1 consumed antibiotics out of 11 classes, whereas unit 3 consumed antibiotics out of only 8 different classes in at least one quarter during the whole observation period. Only unit 2 has non-zero consumption of antibiotics out of all 12 classes in at least one quarter.

10.1371/journal.pone.0238692.g003 Time courses of antibiotic consumption by antibiotic class separated by functional units.

Time courses of antibiotic consumption per antibiotic class shown separately for the functional units a) unit 1 = intensive care units, b) unit 2 = medical/normal care units, c) unit 3 = surgical/OP units. Please note, the different scales of the y-axes reflect the different total amounts of consumption within each unit due to their different sizes. Important are the relative abundances within each unit.

Fig 4 shows the time courses of mean consumption averaged over the antibiotic classes per functional unit (Fig 4a) as well as the three corresponding coefficients of variation (Fig 4b). Unit 3 (the surgical/OP units) exhibits the lowest mean consumption but by far the highest coefficient of variation, both of which remain approximately constant over the time course. The explanation follows by throwing a glance on Fig 3c: Unit 3 has one absolutely dominating consumption of antibiotics out of the class “second-generation cephalosporins.” Units 1 and 2 both exhibit approximately temporarily constant mean consumption, however, unit 2 on a roughly 5-fold higher magnitude. Noteworthy, the variation of consumption of unit 2 approximately follows the variation for the whole clinic with a more or less monotonous decline during the first half of the observation period, whereas the coefficient of variation for unit 1 is approximately constant over the time course, with the exception of a marked rise at the final observation (first quarter of 2018), which can be explained by the sudden rise of consumption of “narrow-spectrum penicillins” antibiotics (cf. Fig 3a).

10.1371/journal.pone.0238692.g004 Mean antibiotic consumptions and coefficient of variation by functional unit.

Time courses per functional unit of a) mean antibiotic consumption averaged over antibiotic classes, b) corresponding coefficient of variation, with unit 1 = intensive care units, unit 2 = medical/normal care units, unit 3 = surgical/OP units.

 Heterogeneity of antibiotic consumption

The diversity measures introduced in the “Materials and methods” section are now calculated using the observed proportions pi of antibiotic consumption with respect to antibiotic classes, thus i = 1, …, 12 refers to the 12 antibiotic classes. Fig 5 shows time courses of a) Renyi entropies, Ra, for 7 different values of weight parameter a (cf. figure legend), b) the corresponding diversities, D a = e R a for the same set of parameters a, c) the Gini-Simpson diversity, GS, and d) the Gini coefficient, G. Specifically, a = 0 yields D0 = n = 12 (Fig 5b) in agreement with what we expected from theory.

10.1371/journal.pone.0238692.g005 Measures of diversity with respect to antibiotic classes.

a) Renyi entropies Ra for a = 0, 0.5, 0.99, 1.5, 2, 3, 100, b) Diversities Da for a = 0, 0.5, 0.99, 1.5, 2, 3, 100, c) Gini-Simpson index GS, d) Gini coefficient G.

Apparently, for all a > 0 the curves exhibit the same shape, i.e., they differ at each time point by a factor that seems to be a monotonous function of the values of a reference curve at these time points. A unique rule which specifies the best value of a does not exist. However, some researchers invoke a rule of thumb (e.g. [10]) which states that if one exactly observes such a monotonous relation between the different curves as we did, then the Renyi entropy is a robust measure and a > 0 can be chosen arbitrarily but consistently, since entropy does not have an absolute meaning, anyway. Apparently, the rule also holds for the Gini-Simpson diversity, GS, and the Gini coefficient G.

Fig 5 reveals that all diversity measures increase from 2012 until approximately 2016. Thereafter, this tendency is stopped and the data even suggest the initiation of a decrease in diversity. This behaviour coincides with the time course of the coefficient of variation (Fig 1).

Compared to the Renyi entropy, heterogeneity HI, defined in Eq 5, is easier to comprehend, which might explain that its application in the context of antibiotic resistance is unparalleled up to date [2, 4, 13]. However, the invariance under species permutations has not been taken into account thus far. A cycling strategy that dictates an occasional temporal swap of consumption of two antibiotics with respect to the hospital unit under consideration leads to a temporarily constant heterogeneity and, therefore, to wrong conclusions if the assessment is merely based on HI. Likewise, a mixing strategy, i.e. switching the consumption of two antibiotics between two units would remain unnoticed unless the diversity of the two units are not separately calculated (cf. Eq 10). Since heterogeneity is nothing but a special case of similarity SI, we can now make use of SI to introduce a differential measure. Similarity with respect to the initial observation at the outset of a study S I 0 ( t ) = 1 - n 2 ( n - 1 ) ∑ i = 1 n | p i ( t = 0 ) - p i ( t ) | , with t = 0, 1, …, 24 being the number of elapsed quarters, captures changes with respect to the first observation. In the same line, S I Δ ( t ) = 1 - n 2 ( n - 1 ) ∑ i = 1 n | p i ( t - 1 ) - p i ( t ) | , for t > 0 . defines changes with respect to consecutive quarters, thus defines an approximation to a differential measure of similarity. The time courses of HI, SI0, and SIΔ are depicted in Fig 6a–6c. Heterogeneity HI varies only within a small range from 0.63 to 0.69 (Fig 6a). However, we observe a monotonous, almost linearly increasing displacement from the first observation (Fig 6b). Due to this linear increase, the magnitude of the differential displacement SIΔ(t) is more or less constant and is approximately one minus the slope of SI0(t) (Fig 6c). Obviously, the difference of the distribution of antibiotic abundances accumulates over the time course, where SIΔ(t) measures the intensity of the change as a function of time. Thus, we now have a sound basis for the evaluation of changing abundances and possibly related switching strategies.

10.1371/journal.pone.0238692.g006 Measures of heterogeneity and similarity with respect to antibiotic classes.

a) Heterogeneity index HI, b) Similarity index SI0 with respect to proportions of the first observation, c) Similarity index SIΔ with respect to proportions of the preceding observation, d) Kullback-Leibler heterogeneity KL e) Kullback-Leibler difference KL0 with respect to proportions of the first observation, f) Kullback-Leibler difference KLΔ with respect to proportions of the preceding observation. Confer text for definitions of these measures.

Within the scope of physics and information sciences, it is common to base measures of heterogeneity and diversity, respectively, on the Shannon entropy because it can be interpreted as the negative mean information that arises from averaging over the individual contributions ln(pi) to information. In this context, the similarity index between two distributions given by ai and bi corresponds to the Kullback-Leibler divergence [11] K L D ( a i , b i ) = ∑ i = 1 n a i log ( a i b i ) . However, KLD is asymmetric, which is why the symmetric variant KLD(ai, bi) + KLD(bi, ai), known as Kullback-Leibler difference, has been introduced. KLD is the mean information difference taken over the individual information differences log(ai) − log(bi).

In order to harness KLD for our needs, we define the Kullback-Leibler heterogeneity K L ( t ) = 1 - 1 2 ln ( 2 ) ( ∑ i = 1 n a i ( t ) log ( a i ( t ) + 1 1 n + 1 ) + ∑ i = 1 n 1 n log ( 1 n + 1 a i ( t ) + 1 ) ) . Furthermore, the Kullback-Leibler similarity KL0(t) of distribution ai(t) at time t with the distribution at t = 0 (first observation) can be defined by K L 0 ( t ) = ∑ i = 1 n a i ( t ) log ( a i ( t ) + 1 a i ( t = 0 ) + 1 ) + ∑ i = 1 n a i ( t = 0 ) log ( a i ( t = 0 ) + 1 a i ( t ) + 1 ) . Finally, the Kullback-Leibler similarity KLΔ(t) between two distributions observed at subsequent time points (here quarters) is given by K L Δ ( t ) = ∑ i = 1 n a i ( t ) log ( a i ( t ) + 1 a i ( t - 1 ) + 1 ) + ∑ i = 1 n a i ( t - 1 ) log ( a i ( t - 1 ) + 1 a i ( t ) + 1 ) . Hereby, the + 1 terms within the arguments of the logarithms ensure that situations with pi = 0 remain well-defined.

Fig 6d–6f show time courses of KL, KL0 and KLΔ, respectively. A comparison with the ordinary heterogeneity and similarity measures of Fig 6a–6c reveals that the values of KL, KL0 and KLΔ are located within narrower intervals. In addition, KL appears to be smoother than HI and, noteworthy, the decreasing curve of KL0(t) has a concave shape. From these differences we conclude that measures based on information differences, due to their logarithmic dependence, weight larger differences in proportions stronger than small differences, whereas the ordinary measures exhibit a proportional weight.

In the same line, the time courses of measures of diversity and heterogeneity, respectively, with respect to active agents are depicted in Figs 7 and 8. Qualitatively, similar results as for the antibiotic classes are obtained. Once more, as already observed for the coefficient of variation, we see a damping effect of pooling the active agents into antibiotic classes. The variations of temporal changes of heterogeneity and related measures are larger for the active agents than for the more coarse grained antibiotic classes. No need to mention, the question of which stratification level should be prioritised is a matter of the concrete studies’ objectives and the availability of adequate data. The usage of the more fine-grained level of active agents is advisable if records of prevalence of antibiotic resistance are available at the same fine-grained level.

10.1371/journal.pone.0238692.g007 Diversity measures with respect to active agents.

a) Renyi entropies Ra for a = 0, 0.5, 0.99, 1.5, 2, 3, 100, b) Diversities Da for a = 0, 0.5, 0.99, 1.5, 2, 3, 100, c) Gini-Simpson index GS, d) Gini coefficient G.

 10.1371/journal.pone.0238692.g008 Measures of heterogeneity and similarity with respect to active agents.

a) Heterogeneity index HI, b) Similarity index SI0 with respect to proportions of the first observation, c) Similarity index SIΔ with respect to proportions of the preceding observation. Confer text for the definitions of these measures.

Finally, we briefly report on the results obtained when the hospital’s functional units are included as a second factor in addition to the antibiotic classes. Firstly, Fig 9 shows time courses of diversity measures stratified by the three functional units as previously defined. Secondly, heterogeneity HI and the similarity indexes SI0 and SIΔ are shown in Fig 10. Strikingly, diversities Da and GS as well as heterogeneity HI remain approximately constant in the course of time for functional unit 3 and varies only very slightly for unit 1. To the contrary, the diversities for functional unit 2 resemble the corresponding curves for the whole hospital as shown in Figs 5 and 6, i.e., they exhibit an increase in the course of time. Apparently, the functional units have different policies of antibiotic administration. This becomes even more obvious when throwing a glance onto the curve SI0(t) shown in Fig 10b. The administration in unit 3 essentially remains constant with respect to the first observed administration in 2012. The administrations in unit 1 and unit 2, in contrast, show a cumulative difference with respect to the first observation.

10.1371/journal.pone.0238692.g009 Measures of diversity with respect to antibiotic classes stratified by functional units.

a) Diversities Da for unit 1 with a = 0, 0.5, 0.99, 1.5, 2, 3, 100, b) Diversities Da for unit 2 with a = 0, 0.5, 0.99, 1.5, 2, 3, 100, c) Diversities Da for unit 3 with a = 0, 0.5, 0.99, 1.5, 2, 3, 100, d) Gini-Simpson Index GS. See text for definitions.

 10.1371/journal.pone.0238692.g010 Heterogeneity and similarities with respect to antibiotic classes stratified by functional units.

a) Heterogeneity HI. b) Similarity index SI0 with respect to proportions of the first observation, c) Similarity index SIΔ with respect to proportions of the preceding observation. Confer text for the definitions of these measures.

So far, we conclude that neither of the diversity or heterogeneity measures Da, GS, G, HI, and KL are capable to catch suspected policies of clinical cycling without a concomitant assessment based on the newly introduced similarity indexes SI0 and SIΔ or, alternatively, KL0 and KLΔ. A very weak long-term clinical cycling, as actually observed for the data under investigation, can leave diversity more or less invariant. To be specific, the constant non-vanishing slope of SI0(t), thus the constant SIΔ(t) < 1 reflects a long-term “cycling-like” change of antibiotic abundances. Assuming a full cycle in case of a rigorously applied cycling protocol, SI0(t) would also exhibit a full cycle in the course of time.

 Correlation of antibiotic administration and prevalence of antibiotic resistance

The most important and at the same time most challenging question, in the given context, is whether the clinical cycling of antibiotic administrations correlates or even causally relates to the prevalence of antibiotic resistances. Only sufficient knowledge about existence and structure of such an association renders the design of administration policies that aim in minimising resistances meaningful. Unfortunately, recorded data on prevalence of antibiotic resistance are rare and often collected in a non-systematic way. Therefore, the following analysis should be viewed as paradigmatic rather than taking the results as credible.

Infections have been recorded on a yearly basis within intensive care units and medical/normal care units, however, not in a controlled and regular way as may be required by a controlled study design. Fig 11 shows the time courses of the number of registered cases per pathogen stratified by resistance. Resistance, hereby, has been dichotomised in a yes/no-variable although for some cases a more detailed information on the type of resistance (the corresponding antibiotic agent, multiresistance, etc.) is available. The time courses of infection frequencies suggest a rising prevalence. However, the awareness of the problem of antibiotic resistance and the adherence to diagnostic and therapeutic guidelines increased over time. We suspect that the rigorous diagnostic of pathogens is responsible for the added detection of more (resistant) pathogens. The proportions of infections with resistant infectious agents per type of pathogen is perhaps more reliable than the total number of infections. Fig 12 shows the time courses of these proportions and it no longer appears as drastic as before. It appears natural to apply the measures of heterogeneity and similarity introduced above to the proportions of resistant pathogens with respect to the total population of resistant pathogens.

10.1371/journal.pone.0238692.g011 Pathogen prevalence.

Sum of yearly registered number of cases of 9 observed pathogens stratified by resistance.

 10.1371/journal.pone.0238692.g012 Fraction of antibiotic-resistant germs.

Time courses of the fractions of resistance per pathogen plotted separately for units 1 (intensive care) and unit 2 (normal care unit).

Fig 13 depicts the time courses of HI, SI0, and SIΔ both for the proportions of antibiotic consumption and for the proportions of resistant pathogens in order to allow for a direct comparison. Heterogeneity hardly changes in the time’s course both for antibiotic consumption and resistant pathogens (Fig 13, left panel). The slopes of heterogeneity HI(t) (denoted by mean (2.5%CI; 97.5%CI) in the following) resulting from linear regression are essentially zero (which is the null hypothesis of the linear model) with − 0.004 (− 0.012; 0.004) and p = 0.300 (antibiotics, unit 1), with 0.001 (− 0.003; 0.005) and p = 0.705 (pathogens, unit 1), with − 0.003 (− 0.021; 0.016) and p = 0.629 (antibiotics, unit 2), and with − 0.006 (− 0.019; 0.008) and p = 0.300 (pathogens, unit 2), respectively. However, heterogeneity of the population of resistant pathogens is slightly lower compared to antibiotic consumption. Thus, we have constant heterogeneity but pathogens may, as observed for antibiotics, exhibit a “cycling-like” characteristic in form of exchanges of prevalences of pathogens which leave heterogeneity invariant.

10.1371/journal.pone.0238692.g013 Association of heterogeneities of antibiotic consumption and pathogen prevalence.

Time courses of heterogeneity, HI and similarity indexes, SI0 and SIΔ for antibiotic consumption with respect to antibiotic classes and prevalence of resistant pathogens stratified by the hospital’s units 1 (intensive care) and 2 (medical/normal care).

The differential (quarter-by-quarter or year-by-year) changes of distributions with time averages 1 − SIΔ of 0.1 (0.08, 0.11) (antibiotics, unit 1), 0.15 (0.12, 0.18) (pathogens, unit 1), 0.053(0.05, 0.06) (antibiotics, unit 2), and 0.11(0.07, 0.14) (pathogens, unit 2), are slightly greater for the distribution of resistant pathogens compared to antibiotics consumption (Fig 13, right panel), however, both are constant in essence for both units with slopes (derived from a linear model) 0.0002 (− 0.0078; 0.0081) and p = 0.968 (antibiotics, unit 1), with − 0.0091 (− 0.0460; 0.0279) and p = 0.492 (pathogens, unit 1), with − 0.0010 (− 0.0053; 0.0034) and p = 0.653 (antibiotics, unit 2), and with 0.021 (− 0.002; 0.045) and p = 0.0637 (pathogens, unit 2). Therefore, we expect that SI0(t) exhibits a linear decline with constant slope approximately given by SIΔ − 1, as shown in the following.

Most strikingly, the time series of the accumulated similarity index SI0 for the resistant pathogens strongly correlates with the corresponding time series of antibiotic consumption, in fact in both units. A Pearson product moment correlation analysis gives correlation coefficients 0.92 with p = 0.009 (unit 1) and 0.89 with p = 0.02 (unit 2). The slopes significantly differ from zero with the concrete values − 0.055 (− 0.062; − 0.047) per year (antibiotics, unit 1, p < 10−3), − 0.049 (− 0.078; − 0.021) per year (pathogens, unit 1, p = 0.009), − 0.067 (− 0.072; − 0.063) per year (antibiotics, unit 2, p < 10−3), and − 0.044 (− 0.088; − 7.3e − 04) per year (pathogens, unit 2, p = 0.048), respectively. It is appealing to speculate whether these coinciding changes are a result of correlations or even causal relations. For the time being, this speculation has to be treated with caution. However, this analysis gives directions to a proper controlled observational or experimental study design.

A further observation underpins our speculation. The total percentage of resistant pathogens reduces significantly from roughly 20% to 10% in functional unit 1. A linear regression gives a slope of − 0.017 (− 0.025; − 0.008) per year for the proportion (significantly different from zero with p = 0.005). In functional unit 2 the proportion of resistant pathogens reduces non-significantly by − 0.007 (− 0.015; 7.1e − 05) per year, however, at the edge of significance (p = 0.051). An additional support for our hypothesis is given by explicitly calculating the correlation coefficients between SI0 and the approximately linear decline of the ratio of resistant pathogens: 0.92 (p = 0.009) for unit 1 and 0.86 (p = 0.03) for unit 2.

To conclude, although a genuine control strategy in the common sense of antimicrobial cycling or mixing was not applied, we observe that a rather long-term temporal change (clinical cycling) of consumption of different antibiotics (SI0 applied to antibiotics consumption) correlates with a change of prevalence of antibiotic-resistant bacteria (SI0 applied to prevalence of resistant pathogens). This correlation is expressed by means of almost equal slopes as well as a corresponding large correlation coefficient, where the slopes are significantly different from zero and approximately equal, of the two similarity indexes SI0 for antibiotics and pathogens, respectively. Whether these temporal changes in antibiotic consumption have a direct causal impact is still speculative but gains additional evidence through the observed reduction of prevalent resistant bacteria. Controlled studies that allow comparisons with more or less static “control strategies” and other types of switching behaviours (including mixing) are needed to draw reliable inferences. It is the main intention of this work to supply an appropriate mathematical framework for such studies.

 Discussion

Applications of measures of heterogeneity and diversity are rare and unsatisfactory in the context of assessing antibiotic resistance. This is somewhat surprising since antibiotic administration policies that rely on cycling or mixing strategies in order to reduce antibiotic resistances have been promoted for quite some time [2, 4, 6, 7, 13] (for a counter example see [5]). Cycling strategies, this is our claim, are best characterised by means of differential measures of heterogeneity and diversity, respectively. This approach can, in principle, be extended to capture mixing by introducing a spatial stratification of the diversity measures. Although some attempts to tackle antibiotic resistance by means of heterogeneity analyses exist [2, 4, 13], a satisfactory mathematical framework is due.

We adopted diversity measures known in other fields of research [9, 11] and adapted them to the needs within the scope of analysing antibiotic resistance. It is natural to seek for dependencies between the heterogeneity of consumption of antibiotics and the heterogeneity of the pattern of prevalence of antibiotic-resistant pathogens.

In order to provide a flexible methodological basis for the analysis of antibiotic resistance, we introduced and discussed a simple measure of heterogeneity as well as a general family of diversity measures, i.e., the so called family of Renyi diversities and derivatives thereof. It should be noted that the notions of “heterogeneity” and “diversity” do not refer to conceptually different measures, they merely reflect their emergence in different fields of application. As a novel aspect within the given context, we derived differential measures of similarity which are needed to capture temporal changes due to swapping proportions which leave moments like heterogeneity and diversity invariant.

For many real-world applications, the simple heterogeneity measure HI and differential measures of similarity SI0 and SIΔ will suffice. However, showing simultaneously that the whole family of diversity measures leads to the same conclusions supplies additional evidence (“We can regard a sample more diverse if all of its Renyi diversities are higher than in another samples.”, [10]). Moreover, the smoothing and non-linear weighting effect of higher order measures like Shannon entropy and derivatives (Kullback-Leibler heterogeneity, etc.) might become important for damping spurious fluctuations by weighting larger deviations. A solid reason for the choice of entropies is the straightforward application of a maximum entropy method. Maximum entropy proved as the method of choice when it comes to learn dynamics of biological systems (e.g. [16], see also [11]). With the aid of such an optimisation tool we expect that an optimal cycling and arguably also an optimal mixing schedule can be learned from the observed correlation patterns between antibiotics consumption and prevalences of antibiotic-resistance.

The presented inclusion of covariates and factors like clinical units and groupings of active agents has exemplary character. The concrete choice of covariates depends on their availability and, most important, on the specific questions that are raised. In the case of mixing with two (or more) subpopulations of patients that receive different drugs in a temporarily alternating way, it might be better to stratify for these subpopulations instead of functional units, unless these strata coincide. Furthermore, since transmission occurs at the microlevel, it would certainly be of advantage to include individual-level administration data instead of aggregated dispensing data. Such microlevel data have not been available for our elementary methodological approach. However, our analytical framework is flexible enough to account for such peculiarities. In addition, we point to the possibility to expand measures of heterogeneity and similarity to be applicable to joint probabilities of antibiotic consumption and resistance. This is beyond the scope of the present work, however, we paved the way for doing so.

It deserves to be mentioned, that some authors approached the problem by means of extended SIR-like epidemiological models [4, 17]. From a theoretical point of view, these models have benchmark character. However, the validation of these models necessitates recording of data on antibiotics consumption and pathogen load on an individual basis which is not feasible for most hospitals. As opposed to this, so called composite indices as “summary measures of the net impact of antibiotic resistance on empiric therapy” [18] are much more coarse-grained epidemiological measures based on the cumulative antibiogram [19], which reside on a higher population level. Our approach is compatible to both sides and bridges the gap.

In addition, our method complements time series analyses (e.g. [8]) that pointed to thresholds in associations between population antibiotic use and prevalence of resistant pathogens. Within the scope of the time series analyses as discussed by López-Lozano et al. [8], the correlation of the time series of the prevalence of a specific pathogen with the time series of the amount of corresponding administered antibiotics is calculated. Commonly, such a correlation of two time series is given by a mutual entropy. Thus, our approach is a generalisation in that it treats diversities of both antibiotic consumption as well as pathogen prevalence and correlates these diversities.

The time series analyses [8] supplied evidence that prevalence rates increase in a nonlinear fashion when exceeding a prevalence threshold after a sufficiently long duration of administration of certain antibiotics. The existence of such a threshold indicates that a switch to an alternative antibiotic agent is due. Our approach goes a step further by including the dynamics of switching in the analysis to allow, eventually, for an optimised (temporal and/or spatial) switching strategy. Results of stochastic simulations of microbial populations subjected to a periodic presence of antimicrobials [20] boost our confidence.

Moreover, due to its intermediate complexity it is able to serve as a performative boundary object [21], thence constituting a clinically relevant basis for a modelling for policy [22]. This holds all the more if implemented on a boundary infrastructure [23] as, for example, the modelling platform MAGPIE [24] that enables experts with different expertises to dock on. In other words, the proposed method has the potentiality to be translated to the point of decision making as a monitoring system.

 Conclusion

To conclude, the presented analysis has paradigm character. We focused on setting up a methodological framework because the available data do not allow to assess cycling or mixing strategies in a controlled way. To be exact, a control strategy in a genuinely defined sense of cycling or mixing appears not to have been applied. In other words, we have a purely observational situation exhibiting a weak long-term “clinical cycling” but without ample background information particularly on individual-level administration. However, the performed applications of the suggested analytic methods to records of antibiotic consumption and prevalence of antibiotic-resistant bacteria definitely go beyond mere illustrations. That is to say, the results allow to raise hypotheses or at least to formulate conjectures. Specifically, we observe a strong positive correlation of time courses of similarity with respect to the initial observation of antibiotic consumption and prevalence of antibiotic-resistant pathogens. In addition, clinical cycling correlates with a decreasing ratio of resistant pathogens. These correlations have to be confirmed in an experimental/interventional study. We are convinced that the derived mathematical framework provides a sound basis to substantially improve the determination of a viable roll back administration policy to defeat antibiotic resistance.

 Supporting information <p>(GZ)</p> </caption> </supplementary-material> <supplementary-material id="pone.0238692.s002" mimetype="application/pdf" position="float" xlink:href="info:doi/10.1371/journal.pone.0238692.s002" xlink:type="simple"> <label>S1 File</label> <caption> <title/> <p>(PDF)</p> </caption> </supplementary-material> </sec> </body> <back> <ref-list> <title>References UN Interagency Coordination Group (IACG) on Antimicrobial Resistance. No Time to Wait: Securing the future from drug-resistant infections. Report to the Secretary-General of the United Nations. Geneva: United Nations; 2019. Sandiumenge A, Diaz E, Rodriguez A, Vidaur L, Canadell L, Olona M, et al. Impact of diversity of antibiotic use on the development of antimicrobial resistance. Journal of Antimicrobial Chemotherapy. 2006;57:1197–1204. doi: 10.1093/jac/dkl097 16565158 Bennett KM, Scarborough JE, Sharpe M, Dodds-Ashley E, Kaye KS, Hayward TZ, et al. Implementation of antibiotic rotation protocol improves antibiotic susceptibility profile in a surgical intensive care unit. The Journal of trauma. 2007;63:307–311. doi: 10.1097/TA.0b013e318120595e 17693828 Abel zur Wiesch P, Kouyos R, Abel S, Viechtbauer W, Bonhoeffer S. Cycling Empirical Antibiotic Therapy in Hospitals: Meta-Analysis and Models. PLOS Pathogens. 2014;10(6):1–13. Karam G, Chastre J, Wilcox MH, Vincent JL. Antibiotic strategies in the era of multidrug resistance. Critical Care. 2016;20(1):136. doi: 10.1186/s13054-016-1320-7 Beardmore RE, Peña Miller R, Gori F, Iredell J. Antibiotic Cycling and Antibiotic Mixing: Which One Best Mitigates Antibiotic Resistance? Molecular Biology and Evolution. 2017;34(4):802–817. Davey P, Brown E, Charani E abd Fenelon L, Gould IM, Holmes A, Ramsay CR, et al. Interventions to improve antibiotic prescribing practices for hospital inpatients. Cochrane Database of Systematic Reviews. 2013;(4). doi: 10.1002/14651858.CD003543.pub3 López-Lozano J, Lawes T, Nebot C, Beyaert A, Bertrand X, Hocquet D, et al. A nonlinear time-series analysis approach to identify thresholds in associations between population antibiotic use and rates of resistance. Nature Microbiology. 2019;4(7):1160–1172. doi: 10.1038/s41564-019-0410-0 30962570 Tuomisto H. A consistent terminology for quantifying species diversity? Yes, it does exist. Oecologia. 2010;164:853–860. doi: 10.1007/s00442-010-1812-0 Zhang L, Cham J, Paciorek A, Trager J, Sheikh N, Fong L. 3D: diversity, dynamics, differential testing—a proposed pipeline for analysis of next-generation sequencing T cell repertoire data. BMC Bioinformatics. 2017;18(1):129. doi: 10.1186/s12859-017-1544-9 Gorban AN. Maxallent: Maximizers of all entropies and uncertainty of uncertainty. Computers and Mathematics with Applications. 2013;65(10):1438 – 1456. doi: 10.1016/j.camwa.2013.01.004 Allison PD. 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Mathematical Biosciences. 2017;294:85–91. doi: 10.1016/j.mbs.2017.09.002 Hughes JS, Hurford A, Finley RL, Patrick DM, Wu J, Morris AM. How to measure the impacts of antibiotic resistance and antibiotic development on empiric therapy: new composite indices. BMJ Open. 2016;6(12). doi: 10.1136/bmjopen-2016-012040 Hindler JF, Stelling J. Analysis and Presentation of Cumulative Antibiograms: A New Consensus Guideline from the Clinical and Laboratory Standards Institute. Clinical Infectious Diseases. 2007;44(6):867–873. doi: 10.1086/511864 Marrec L, Bitbol AF. Resist or perish: Fate of a microbial population subjected to a periodic presence of antimicrobial. PLOS Computational Biology. 2020;16(4):1–19. Star SL, Griesemer JR. Institutional Ecology, ‘Translations’ and Boundary Objects: Amateurs and Professionals in Berkeley’s Museum of Vertebrate Zoology, 1907-39. Social Studies of Science. 1989;19(3):387–420. doi: 10.1177/030631289019003001 Zeiss R, van Egmond S. Modeling for Policy—Science-based models as performative boundary objects for Dutch policy making. Science Studies. 2010;23(1):58–78. Star SL. This is Not a Boundary Object: Reflections on the Origin of a Concept. Science, Technology, & Human Values. 2010;35(5):601–617. doi: 10.1177/0162243910377624 25309997 Baldow C, Salentin S, Schroeder M, Roeder I, Glauche I. MAGPIE: Simplifying access and execution of computational models in the life sciences. PLOS Computational Biology. 2017;13(12):1–11. 10.1371/journal.pone.0238692.r001 Decision Letter 0 Martinez-Garcia Ricardo Academic Editor 2020 Ricardo Martinez-Garcia This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 0

27 Feb 2020

PONE-D-19-25350

Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategies

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Reviewer #1: Authors perform a rigorous mathematical analysis on the observational data, use multiple measures to asses the diversity and the heterogeneity of antibiotic consumption, and connect their output to the resistance prevalence observed in the dataset.

Although I think the approach is very valuable, I think there are major points that need to be modified before the publication of this manuscript. My comments are below.

Major Points

1) I don't understand how measures of diversity or heterogeneity are connected to mixing (spatial heterogeneity) or cycling (temporal heterogeneity) strategies here. Mixing refers to giving different antibiotics to the different members of the cohort, but that can also end up having a constant DDD over time (say you keep 50% of the population on one antibiotic and the other 50% on the other, but the point is to alternate between specific individuals.). Similarly, cycling - the temporal approach - should have way more variation over time, such as excluding certain antibiotics completely from the hospital or the cohort. So I cannot relate the antibiotic consumption observed in this dataset with the common strategies used.

2) The authors' most important conclusion is only expressed in one sentence, and not fully clear (lines 362-364). Do they mean almost a no slope by saying that " means of parallel slopes "? Is this really a good way to back their argument up? Given that this is the message of the paper, I really expect more elaboration on this point.

3) In general, I understand why authors use so many different measures to asses diversity or heterogeneity, and while reading the manuscript, it was obvious that this is the way they explored the whole research question. But is it necessary to include all these measures in the main text? There is a lengthy section about the explanation and use of so many different measures, but the conclusion and the connection to resistance prevalence is relatively really short, which is supposed to be the main point. I think the manuscript needs a bit of structural revision in this regard.

Minor Points

1) Figures 4-10 : please also use dots for the data points, like you did for figures 1-3.

2) line 14 : these strategies are not really "recent" anymore.

3) lines 19-21 : please refer to the key publications that includes the "evidence" you have mentioned.

Reviewer #2: Diebner and collaborators present data about antibiotic consumption and pathogen resistance in three administrative units of a hospital over several years. They aim at using these data to test hypotheses related to the efficiency of heterogeneous antibiotic treatments strategies to mitigate the emergence of resistance in hospitals. For this they suggest the use of various measures of heterogeneity (that are classically used in theoretical ecology) in the context of antibiotic treatment and resistance.

I am not convinced of the interest of this framework for the analysis of heterogeneous antibiotic treatment strategies, for 3 reasons:

- The measures that go beyond the coefficient of variation are interesting when strictly more than 2 antibiotics are in use. In the "antibiotic mixing" literature, to my knowledge the vast majority of the literature considers the case of 2 antibiotics (the question being whether treating half of the patients with AB1 and half with AB2 is better than treating everybody with AB1, or better than cycling, see further below).

- These measures are used in ecology to deal with very heterogeneous data, eg when abundances are very different between different species. This is useful (in ecology) because a species may be several orders of magnitude less abundant than another but still be essential for ecosystem function. But in the case of antibiotic treatments, the point of DDD is that different antibiotics can be compared between them. If the DDD of 2 antibiotics differ by orders of magnitude, it means that the one with the low DDD is virtually absent and has no chance of contributing to the establishment of resistance (or to infection clearance).

- In much of the literature (including the one cited by the authors) on heterogeneous antibiotic treatments, a big question is "mixing vs cycling", but I do not see how the measures suggested by the authors can be useful to study cycling. The authors simply do not ever mention cycling after the introduction. It gives me the impression that the papers they cite do not really match the question they want to study.

Most importantly, I am also not convinced that the presented data support the claimed result ("a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption"), for several reasons:

- Only time series are analysed. In time series, the different time points are not independent from each other, and it is thus not appropriate to decide whether the slope is significantly different from 0 using a linear regression or a correlation test (by the way the authors do not even precise which statistical test they use).

- Regarding panels 1 and 3 of fig 13 (HI and SIdelta), both the curves for heterogeneity / similarity of antibotic use and pathogen resistances are flat. The authors conclude that the curves for treatment and resistance are similar and thus that there methodology permits to establish that heterogeneous treatments cause heterogeneous resistance. There is absolutely no statistical support (nor proper testing) for this finding. And a more parsimonious explanation for the fact that both curves are flat could be that HI and SIdelta do not capture any property of the data and are always constant. Actually on all figures of the papers, all the temporal curves of HI and SIdelta are indeed flat. I think that questions the interest of these measures.

- Regarding pannel 2, the observation (that the slopes of SI0 are the similar for antibiotic use and pathogen resistance) is not uninteresting, but these are only two independent data points (two hospital units) and it is thus impossible to have any statistical support. The authors mention that before pooling all of them into 3 units, they have a larger number of administrative units. Maybe they could test their hypothesis at the level of these smaller units? I do not agree that "a grouping of these small subunits into functional units [is] sufficient" if they want to test this hypothesis.

- I think there is an important difference between the question presented in the introduction and in the literature (do mixing startegies reduce antibiotic resistance?) and the question mainly adressed in the results (do heterogeneity of treatment correlate with heterogeneity of resistance?). It is only in the last paragraph of the results that the authors attempt to adress the former.

So in the current state, I do not see how the methods and the data presented in this article permit to adress what the authors present as the main question, and can not recommend the publication of this article as long as this is not clarified. But I am sure that there is potential interest in the data presented by the authors (time series of antibiotic treatments and pathogen resistance in different units of a hospital).

Below are more detailed comments about the manuscript, in three parts (A: abstract, B: introduction and methods, C: results).

A) I did not find the abstract understandable before reading the rest of the article:

"Temporal changes of the proportional abundances of different antibiotics (e.g. mixing or cycling)": unlike cycling, mixing does not implies temporal changes but spatial heterogeneity

"Although such a mixing strategy appears to be plausible": at this point mixing is not defined, and the previous sentence seems to rather relate to cycling than mixing ("temporal changes").

"We adopt diverse measures of heterogeneity and diversity": the authors should at least precise the variable whose they are trying to quantify heterogeneity of diversity

"We show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I would suggest to precise the direction of the change in the abstract

"we introduce a scheme based on linear regression for the assessment of associations between changes of heterogeneities on the antibiotics and the pathogen side" and "we show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I do not see what is new in the method (what does it mean to "introduce a linear regression scheme"? Linear regression is not new, and here is not really adapted for time-series data as explained above). All the authors do is plotting two curves next to each other and say they look similar (fig 13), without proper statistical analysis.

B) The introduction and methods should be clarified, and need proper references to the literature:

L16 "whereas other strategies refer to a scheduled change of the dominantly used class of antibiotics": in the literature (including the one cited by the authors), antibiotic mixing does not involve scheduled change of antibiotic over time, but treatment of different subgroups of patients with different antibiotics.

L17 "a fraction of patients": I think the authors should introduce the context better, to explain that these strategies are defined at the scale of a group of patients (for example in a hospital). This is important because with the definition provided by the authors, cycling could be wrongly understood as applied to individual patients.

L19 "Although there is some evidence that": citations needed. Overall there is a huge lack of references to the literature in the article.

L23-25 I find the sentence very unclear. What does it mean to "quantify the heterogeneity of [...] time courses of prevalence of antibiotic-resistant pathogens" ?

L57 is "DDDi" referring to the DDD or the "consumption density DDD"? If the former, why mentioning "consumption density DDD"?

L56-57: At this point, it is not clear what is "the antibiotic group i". The previous paragraph explains that the hospital departments are clustered into 3 administrative units, and that the antibiotics are pooled in 12 antibiotic classes. To which kind of group this sentence refers to (are the authors computing antibiotic consumption for a class of antibiotic or for a unit of the hospital)? It only becomes clear much later in the article.

L60: Is the coefficient of variation computed for the vector of DDDi(t) for all i? Maybe clarify the formula?

The paragraph "Coefficient of variation" must include references. It raises an interesting but rather simple point, and could be shortened.

The paragraph "Heterogeneity and entropy" is mostly paraphrasing the literature without proper citations. The authors do not explain how these theoretical ecology measures will be applied to quantify antibiotic resistance and antibiotic uses. I understand that they want to stay general because they will apply the same measures to different data, but everything would be more clear if the "antibiotic groups" had been properly defined (see my previous comment)

L90, The definition of the "proportions" are not clear at this point. Do a1+a2+a3+...+an=1, or do ai+bi=1 for all i? Said otherwise, is "ai" the proportion of individuals of population a that belong to species i, or is it the proportion of individuals of species i that belong to environment a ? It only becomes clear much later in the text.

C) The data presented in the results (antibiotic consumption and resistance in a hospital) have some potential interest, but at this stage the authors do not show anything convincing.

Figure 1 : If I correctly understand, the fact that fig 1a and fig 1b look very similar suggest that the sum of DDD for all the antibiotic groups is almost constant over time. This should be made visible on Fig 1c by making the y-axis start at 0, otherwise a quick look at this panel conveys an inaccurate message.

L167-174 I do not see how what it presented in this paragraph is a result. The authors are mostly saying that the coefficient of variation is a good measure of variation.

L 192-293 "Please note, the different scales of the y-axes reflect the different total amounts of consumption within each unit due to their different sizes": if this really comes from difference in size of the units (number of patients), Isn't it a strong argument for normalising DDD per patient days? I do not understand why the authors describe this normalisation in the methods but do not use it later in the results while it seems needed and appropriate.

L305 "The most important and at the same time most challenging question, in the given context, is whether the mixing behaviour of antibiotic administrations correlates or even causally relates to the prevalence of antibiotic resistances.": the authors do not show that their methods are useful for such a test. Do they really bring something compared to the coefficient of variation?

Figure 12 mentions "intensive care" and "normal care" unit, while everywhere else in the article the units are labeled "unit 1", "unit 2" and "unit 3"

I do not understand the sentence "We conclude that mixing of antibiotic consumption correlates with the prevalence of antibiotic-resistant bacteria by means of parallel slopes of similarity indexes SI0"

Dataset: the authors made the effort of making the full dataset and analysis programs available. Just a minor suggestion: the variable names in R data sheets and the text in the R markdown file are currently in german, I think they would be more useful if translated to english.

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 10.1371/journal.pone.0238692.r002 Author response to Decision Letter 0 Submission Version 1

18 Jun 2020

Response to Reviewer #1:

Major Points

1) I don't understand how measures of diversity or heterogeneity are connected to mixing (spatial heterogeneity) or cycling (temporal heterogeneity) strategies here. Mixing refers to giving different antibiotics to the different members of the cohort, but that can also end up having a constant DDD over time (say you keep 50% of the population on one antibiotic and the other 50% on the other, but the point is to alternate between specific individuals.). Similarly, cycling - the temporal approach - should have way more variation over time, such as excluding certain antibiotics completely from the hospital or the cohort. So I cannot relate the antibiotic consumption observed in this dataset with the common strategies used.

Answer: Yes, it is true that cycling or mixing can end up in having the same diversity. This is one of the major aspects of our discussion and lead us to the introduction of a differential measure SI_0. In the revised version, we additionally derived Eq 10 which is appropriate to capture mixing. In fact, the discussion of SI_0 is the most crucial point of our analysis and elaborated on that in the revised version.

A general remark with respect to the dataset seems to be due. Main purpose of our work is to derive a sound mathematical framework for the analysis of antibiotic control strategies. The application to the data available for us has to be understood mainly as illustration of how the mathematical methods are applied. We emphasise the fact, that a control in a genuinely defined way either of cycling or mixing, respectively, has obviously not been applied in the Hospital from which we draw data. However, a long-term change of antibiotic administration can be observed which we call „clinical cycling“ following Abel Zur Wiesch et al. 2014. In the light of this rather weak temporal changing behaviour we feel inclined to formulate our (secondary!) finding with respect to the concrete observation in a positive way: Despite the relatively weak differential change of diversity in antibiotic consumption, we observe a positive correlation with the corresponding differential change of prevalent resistant pathogens. We strengthend this point in the revised version.

This point is strongly related to the following objection of Reviewer #1

2) The authors' most important conclusion is only expressed in one sentence, and not fully clear (lines 362-364). Do they mean almost a no slope by saying that " means of parallel slopes "? Is this really a good way to back their argument up? Given that this is the message of the paper, I really expect more elaboration on this point.

Answer: Throughout the revised manuscript, we clarified this point. In addition, with respect to the paragraph mentioned by Reviewer #1 (lines 362-364 of the first submission), we explicitly added/revised:

„To conclude, although a genuine control strategy in the common sense of antimicrobial cycling or mixing was not applied, we observe that a rather long-term temporal change (clinical cycling) of consumption of different antibiotics ($SI_0$ applied to antibiotics consumption) correlates with a change of prevalence of antibiotic-resistant bacteria ($SI_0$ applied to prevalence of resistant pathogens). This correlation is expressed by means of almost equal slopes as well as a corresponding large correlation coefficient, where the slopes are significantly different from zero and approximately equal, of the two similarity indexes $SI_0$ for antibiotics and pathogens, respectively. Whether these temporal changes in antibiotic consumption have a direct causal impact is still speculative but gains additional evidence through the observed reduction of prevalent resistant bacteria. Controlled studies that allow comparisons with more or less static ``control strategies'' and other types of switching behaviours (including mixing) are needed to draw reliable inferences. It is the main intention of this work to supply an appropriate mathematical framework for such studies.“

Please also note that we strengthened our arguments in the Results section where we discussed the corresponding findings of how SI_0 behaves.

3) In general, I understand why authors use so many different measures to asses diversity or heterogeneity, and while reading the manuscript, it was obvious that this is the way they explored the whole research question. But is it necessary to include all these measures in the main text? There is a lengthy section about the explanation and use of so many different measures, but the conclusion and the connection to resistance prevalence is relatively really short, which is supposed to be the main point. I think the manuscript needs a bit of structural revision in this regard.

Answer: It is indeed important for us to introduce and discuss the whole family of diversity measures within the main text and compare it with the antibiotic homogeneity index previously used by other authors. Although all members of the family share common features, the concrete choice nevertheless depends on the concrete structure of studies and related research questions to which the method will be applied. The discussion of features of the proposed measures and their comparisons are indeed crucial points of our manuscript.

Minor Points

1) Figures 4-10 : please also use dots for the data points, like you did for figures 1-3.

Answer: We would like to stick with the rule to only represent actual data points and summary statistics/moments like mean, median, variance etc. by dots but not nonlinear functions like entropies etc.

2) line 14 : these strategies are not really "recent" anymore.

Answer: Yes, we agree and changed it accordingly.

3) lines 19-21 : please refer to the key publications that includes the "evidence" you have mentioned.

Answer: We added references that show the mentioned evidence including

Sandiumenge et al. 2006

Bennett et al. 2007

AbelZurWiesch et al. 2014

Lopez-Lozano et al. 2019

Davey et al. 2013

and also one reference for a counter-example:

Karam et al. 2016

Response to Reviewer #2:

Reviewer #2: Diebner and collaborators present data about antibiotic consumption and pathogen resistance in three administrative units of a hospital over several years. They aim at using these data to test hypotheses related to the efficiency of heterogeneous antibiotic treatments strategies to mitigate the emergence of resistance in hospitals. For this they suggest the use of various measures of heterogeneity (that are classically used in theoretical ecology) in the context of antibiotic treatment and resistance.

General answer that applies to several objections raised by Reviewer #2:

Main purpose of our work is to derive a sound mathematical framework for the analysis of antibiotic control strategies (cf. the title „Mathematical basis for the assessment ...“ ). The application to data available to us has to be understood mainly as illustration of how the mathematical methods are applied. We emphasise the fact, that a control in a genuinely defined way either of cycling or mixing, respectively, has obviously not been applied in the Hospital from which we draw data. However, a long-term change of antibiotic administration can be observed which we call „clinical cycling“ following Abel Zur Wiesch et al. 2014. In the light of this rather weak temporal changing behaviour we feel inclined to formulate our finding with respect to the concrete observation in a positive way: Despite the relatively weak change of diversity in antibiotic consumption, we observe a positive correlation with the decline of prevalent resistant pathogens. Throughout the revised manuscript, wherever appropriate, we more strictly pointed out our main intention to present a math framework rather than presenting the results of a clinical study. In addition, we strengthend the provisional and somewhat speculative results with respect to data analysis. Even in the virtual absence of „heterogeneous antibiotic treatments“ there is a weak but noticeable impact of the increase of diversity and, more important, of the changes with respect to the initial observation time (captured by SI_0) on the reduction of antibiotic resistance.

We clarified and strengthened this point throughout the revised manuscript wherever appropriate. Also confer the following answers.

I am not convinced of the interest of this framework for the analysis of heterogeneous antibiotic treatment strategies, for 3 reasons:

- The measures that go beyond the coefficient of variation are interesting when strictly more than 2 antibiotics are in use. In the "antibiotic mixing" literature, to my knowledge the vast majority of the literature considers the case of 2 antibiotics (the question being whether treating half of the patients with AB1 and half with AB2 is better than treating everybody with AB1, or better than cycling, see further below).

Answer:

Cycling and mixing in their extreme forms are rarely met, which is why Abel zur Wiesch et al. 2014 introduced the notion of „clinical cycling“ for the more frequently met adjusted forms. In a real clinical situation, in the absence of a strict mixing plan (with only 2 AB involved), usually more than 2 AB are involved.

Surely, in the case of a given strict mixing with only 2 AB and a single given type of infection (pathogen), one would switch to prospective trials (randomized clinical trials (RCTs), controlled clinical trials (CCT), controlled before-after, cross-over) based on well-defined rotation/switch protocols, thereby, however, excluding continuous real-world processes. Alternatively, one can perform a time series analysis as discussed in Lopez-Lozano et al. 2019. Here the prevalence time series of the resistant pathogen is correlated with the time series of the amount of administered AB. Usually, in such a correlation analysis of two time series, a mutal entropy is used to quantify the correlation. In fact, we see in our approach a generalisation to more than 2 involved AB but this does not exclude the application to only 2 AB.

We agree that we used the notions of cycling and mixing in a somewhat vague way. We added clarifying explanations and, most important, we added Eq. 10 in the revised version which explicitly accounts for mixing in the strict sense. Even in the case of only 2 AB these measure results in an interpretable output, although in such a case one would prefer to switch to the time series correlation analysis of 2 time series, as mentioned above.

- These measures are used in ecology to deal with very heterogeneous data, eg when abundances are very different between different species. This is useful (in ecology) because a species may be several orders of magnitude less abundant than another but still be essential for ecosystem function. But in the case of antibiotic treatments, the point of DDD is that different antibiotics can be compared between them. If the DDD of 2 antibiotics differ by orders of magnitude, it means that the one with the low DDD is virtually absent and has no chance of contributing to the establishment of resistance (or to infection clearance).

We have problems in fully understanding this objection. A typical application in ecology is the impact of crop diversity on the vulnerability of the system to pests. One may hypothesise, that a monoculture (a low diversity) may lead to high vulnerability. This hypothesis is driven by real observations. This vulnerability might be reduced either by implementing a diverse crop culture or by annually switching the type of crop (including spatial switching=mixing) or a combination of that. We hope that this gives a vivid illustration that there are indeed many analogues between ecosystem and antibiotic stewardship.

To quantify the impact of diversity as well as the impact of temporal and spatial changes that might leave overall diversity invariant, an appropriate measure is needed, which we presented (SI_0). We do not see why such measures should not be applicable to situations where a diverse (>2 species) system is compared to a 2-species-system including strategies of cycling and mixing. Furthermore, we do not see why this should not be translated into the area of antibiotic stewardship. By the way, with respect to DDD, even the reverse direction might be of interest, namely to introduce an abundance measure that renders crops comparable with each other by means of their contribution to vulnerability. This is of course much beyond the scope of our approach and we are lucky to already have a working definition of DDD.

More specific, given a fixed number of different species, diversity (thus heterogeneity in the ecological sense) reaches is maximum if all species are equally abundant, no matter of whether the whole ecosystem consists of only 2 or more different species. Whether or not diversity correlates positive or negative or not at all with a target parameter as, e.g., prevalence of resistant germs, has to be shown. This is the first of our objectives of presenting diversity measures and we are not quite sure what should be wrong with this approach.

Relating the prevalence of resistant pathogens to diversity is what other authors approached earlier, e.g. Sandiumenge et al. 2006 or Pluess-Suard et al. 2013, however, using the antibiobtic heterogeneity index (HI in our notation) only. Two things should be mentioned in this context. Firstly, looking at the mere diversity may not suffice, but one should also treat the effect of changes of the amount of AB even if these changes leave diversity invariant. This is what we did when introducing differential measure SI_0. Whether such effects exist or not is still open, but we supply the proper method to test it in the future, and, importantly, we supplied initial evidence through our analysis. Secondly, we used other (better in certain contexts) measures in form of Renyi entropies and compared them to HI. HI fails to capture weak changes of diversity. If low DDD of a given AB really has no impact, than it should be reflected in the impact of a low diversity.

- In much of the literature (including the one cited by the authors) on heterogeneous antibiotic treatments, a big question is "mixing vs cycling", but I do not see how the measures suggested by the authors can be useful to study cycling. The authors simply do not ever mention cycling after the introduction. It gives me the impression that the papers they cite do not really match the question they want to study.

We agree that it has not always been clearly formulated of whether „only“ diversity is addressed or whether changes in administration (cycling, mixing) are addressed that might in principle leave overall diversity invariant. We addressed this lack of clear formulation in the revised version of our manuscript. Particularly, we introduced Eq 10 in the revised version that explicitly captures mixing strategies. Most important, SI_0 captures cycling very well and we put more effort into explaining this fact in the revised version.

Most importantly, I am also not convinced that the presented data support the claimed result ("a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption"), for several reasons:

- Only time series are analysed. In time series, the different time points are not independent from each other, and it is thus not appropriate to decide whether the slope is significantly different from 0 using a linear regression or a correlation test (by the way the authors do not even precise which statistical test they use).

Answer: Yes, time series are often autocorrelated, but also non-autocorrelated time series exist. More important beyond this general remark, two time series may be mutually correlated. Why is the use of time series analyses formulated as an objection or as a limitation („ONLY time series are analysed“)? It is the very essence and the strength of our processual approach to evaluate whether such a mutual information (correlation) of two time series (antibiotic consumption and pathogen prevalence) exists. For an exquisite example that points in this direction see Lopez-Lozano et al. 2019.

We have quarterly data over several years and why should we not use this information on the observed time courses? Moreover, the processual approach is applicable to field-like real-world observational studies. If sequential observations at several observation time points are given, why should one not use this time series? The crucial point is to aggregate the time series of many abundances into a manageable and meaningful proxy time series: diversity and/or differential similarity.

Unfortunately, for the data available to us, we cannot attribute time series of specific antibiotic treatment to the emergence of a specific resistant pathogen as in Lopez-Lozano et al. What we have is the time series of diversity (HI or Renyi entropies) and (auto-) similarity measures (e.g. SI_0) and we relate these time series for antibiotics and pathogens with each other. We observe a correlation. Moreover, SI_0 is a measure which captures adjusted administration of AB including cycling even if diversity is unchanged. Unfortunately, for our data, we do not observe a full cycle but rather a long-term change which can be interpreted as the onset of cycling (weak clinical cycling). In our case, the adjusted administration appears in form of a linear change of SI_0 and we think it is legitimate to ask whether this change differs significantly from zero (by means of inference based on confidence).

As long as we do not have experimental (prospective) data but have to rely on an observational study only, we cannot conclude that this correlation is due to causality. The only thing we can do is raising a hypothesis, which is what we did.

Perhaps, using a more complex multivariate statistical approach to time series would be the ultimate improvement. However, using diversity and similarity measures, as we did, as surrogate measures to capture essential aspects of the time series appears to be a legitimate approach with medium (i.e. comprehensible) complexity.

In the revised version, we elaborated on our processual stance mostly within the introduction but also throughout the manuscript. In addition, we emphasised the relevance of SI_0 and we extended the according analysis.

- Regarding panels 1 and 3 of fig 13 (HI and SIdelta), both the curves for heterogeneity / similarity of antibotic use and pathogen resistances are flat. The authors conclude that the curves for treatment and resistance are similar and thus that there methodology permits to establish that heterogeneous treatments cause heterogeneous resistance. There is absolutely no statistical support (nor proper testing) for this finding. And a more parsimonious explanation for the fact that both curves are flat could be that HI and SIdelta do not capture any property of the data and are always constant. Actually on all figures of the papers, all the temporal curves of HI and SIdelta are indeed flat. I think that questions the interest of these measures.

Answer: HI (and likewise the Renyi entropy) measures per definition diversity. It remains constant for the observed data. This is, in our opinion, a relevant and most illuminating result. We do not understand why diversity should not capture a relevant property of the data.

If the abundances of a low-abundant and a highly-abundant species are exactly swapped, diversity/heterogeneity does not change. This means, diversity/heterogeneity (particularly HI) does not capture the swap of species abundances. The very strength of our approach is, that a measure for the strength of cycling can indeed by derived: SI_0.

So far, to the best of our knowledge, only HI has been used in the literature to describe antibiotic diversity which does not capture such a swap. For our data, HI is almost constant over time, although throwing a glance onto the time courses of the consumptions confirms that indeed the amount of essentially three antibiotics switched in the course of time.

Thus, we think that it is one of the strengthes of our work having shown that HI or any other pure diversity does not suffice to capture relevant changes in consumption. To capture changes in abundances – INCLUDING those which leave diversity invariant – we introduced SI_0 which detects whether a change took place or not.

With respect to the analysed data, there is a more or less constant shift of some of the antibiotics abundances, which is captured by a linear decline of similarity with respect to the first observation. SI_0. The slope is a measure of the intensity of these shift. SI_Delta is an approximation to this slope and the magnitude of 1-SI_Delta is an exquisite measure of the strength of cycling. It is weak and constant in our case, but this is a particular and relevant finding for this case.

Thus, HI (i.e. diversity) might in concrete real observations, as is the case for our data, turn out to exhibit no change at all, although there have been changes in consumption. Cycling and mixing might leave heterogeneity/diversity invariant, therefore, we introduced a „differential“ measure to account for such changes. In a sense it is a nice coincidence that our data exhibits an almost constant diversity, although changes in administration happened, because this supplies additional evidence (in form of a proof-of-principle) for the need of our newly introduced „differential diversity measure“ SI_0. SI_Delta is in a certain sense redundant, since it captures the slope of SI_0 which is constant in case of linear SI_0. However, it is arguably a good index that quantifies the strength of cycling in a general sense, including adjusted forms of AB prescriptions.

- Regarding pannel 2, the observation (that the slopes of SI0 are the similar for antibiotic use and pathogen resistance) is not uninteresting, but these are only two independent data points (two hospital units) and it is thus impossible to have any statistical support. The authors mention that before pooling all of them into 3 units, they have a larger number of administrative units. Maybe they could test their hypothesis at the level of these smaller units? I do not agree that "a grouping of these small subunits into functional units [is] sufficient" if they want to test this hypothesis.

Answer: We completely agree, for a statistical support particularly of the impact of mixing, it is necessary to look at a more fine-grained level of administrative units. „Sufficient“ here refers to the sufficiency for the illustrative purpose with respect to the math framework. We added Eq 10 in the revised version that clarifies this point. However, we would like to keep the focus on the math framework instead of going into depth with the analysis of a dataset that has been recorded in the absence of a well-defined or documented administration strategy. Particularly, information on unit-specific prevalences is missing. We think that making the math framework public even without the analysis of an adequate controlled study has priority.

Moreover, our quantification will become most important when being compared to the outcome of other hospitals. Nevertheless, the fact that the „cycling-like“ changes of prevalences of pathogens measured by SI_0 follows the (weak) clinical cycling is, in our opinion, quite striking. In this respect, what we presented is within-unit statistical significance. As mentioned, between-unit variability should be the next step. Our result is, as we mentioned in our manuscript, provisional. Our intention is to supply researchers with our method and see it applied to other records of antibiotic consumptions in order to see how other hospitals perform.

- I think there is an important difference between the question presented in the introduction and in the literature (do mixing startegies reduce antibiotic resistance?) and the question mainly adressed in the results (do heterogeneity of treatment correlate with heterogeneity of resistance?). It is only in the last paragraph of the results that the authors attempt to adress the former.

Answer: We agree, the notations for different strategies have been used in a somewhat confusing way. Our aim is to provide a method that is capable to deal with diversity and with administrative strategies including cycling and mixing that may leave diversity invariant but are nevertheless relevant. Although mixing refers to a spatial permutation, this permutation takes place at a certain instance of time. This is why we carelessly generalised to refer to temporal changes. In our revised version we addressed the different strategies in a much more careful way.

So in the current state, I do not see how the methods and the data presented in this article permit to adress what the authors present as the main question, and can not recommend the publication of this article as long as this is not clarified. But I am sure that there is potential interest in the data presented by the authors (time series of antibiotic treatments and pathogen resistance in different units of a hospital).

Below are more detailed comments about the manuscript, in three parts (A: abstract, B: introduction and methods, C: results).

A) I did not find the abstract understandable before reading the rest of the article:

"Temporal changes of the proportional abundances of different antibiotics (e.g. mixing or cycling)": unlike cycling, mixing does not implies temporal changes but spatial heterogeneity

"Although such a mixing strategy appears to be plausible": at this point mixing is not defined, and the previous sentence seems to rather relate to cycling than mixing ("temporal changes").

"We adopt diverse measures of heterogeneity and diversity": the authors should at least precise the variable whose they are trying to quantify heterogeneity of diversity

"We show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I would suggest to precise the direction of the change in the abstract

"we introduce a scheme based on linear regression for the assessment of associations between changes of heterogeneities on the antibiotics and the pathogen side" and "we show that a reduction of prevalence of antibiotic-resistant germs correlates with a change of heterogeneity of antibiotics consumption" -> I do not see what is new in the method (what does it mean to "introduce a linear regression scheme"? Linear regression is not new, and here is not really adapted for time-series data as explained above). All the authors do is plotting two curves next to each other and say they look similar (fig 13), without proper statistical analysis.

Answer: We addressed all points and substantially revised the abstract. With respect to objection „All the authors do is plotting two curves next to each other and say they look similar (fig 13), without proper statistical analysis.“ , our answer is that it depends a bit to which school of statisticians you belong. We did not just plot the curves next to each other, but we did a legitimate proper statistical inference based on confidence (overlapping CIs of slopes) rather than trusting too much in p-values. Anyhow, we now added a correlation analysis, but as expected, the inference does not change because the correlation is essentially the same as properly comparing the slopes in this case.

B) The introduction and methods should be clarified, and need proper references to the literature:

L16 "whereas other strategies refer to a scheduled change of the dominantly used class of antibiotics": in the literature (including the one cited by the authors), antibiotic mixing does not involve scheduled change of antibiotic over time, but treatment of different subgroups of patients with different antibiotics.

Answer: We agree, as mentioned above, that the notations have been used in a confusing way. We revised accordingly. See also the above answers.

L17 "a fraction of patients": I think the authors should introduce the context better, to explain that these strategies are defined at the scale of a group of patients (for example in a hospital). This is important because with the definition provided by the authors, cycling could be wrongly understood as applied to individual patients.

Answer: We had „groups of patients“ in mind and revised the text accordingly.

L19 "Although there is some evidence that": citations needed. Overall there is a huge lack of references to the literature in the article.

Answer: We added references.

L23-25 I find the sentence very unclear. What does it mean to "quantify the heterogeneity of [...] time courses of prevalence of antibiotic-resistant pathogens" ?

Answer: We changed to

„… quantify heterogeneity of both, antibiotic consumption as well as prevalence of antibiotic-resistant pathogens as a function of observation time.“

L57 is "DDDi" referring to the DDD or the "consumption density DDD"? If the former, why mentioning "consumption density DDD"?

Answer: We used DDD and added a corresponding clarification.

L56-57: At this point, it is not clear what is "the antibiotic group i". The previous paragraph explains that the hospital departments are clustered into 3 administrative units, and that the antibiotics are pooled in 12 antibiotic classes. To which kind of group this sentence refers to (are the authors computing antibiotic consumption for a class of antibiotic or for a unit of the hospital)? It only becomes clear much later in the article.

Answer: Antibiotic class was meant and we revised accordingly.

L60: Is the coefficient of variation computed for the vector of DDDi(t) for all i? Maybe clarify the formula?

Answer: We clarified that point: „… the mean is taken over the antibiotic classes and SD(DDD(t)) denotes the corresponding standard deviation.“

The paragraph "Coefficient of variation" must include references. It raises an interesting but rather simple point, and could be shortened.

Answer: We considerably reduced redundancy and added references to that paragraph.

The paragraph "Heterogeneity and entropy" is mostly paraphrasing the literature without proper citations. The authors do not explain how these theoretical ecology measures will be applied to quantify antibiotic resistance and antibiotic uses. I understand that they want to stay general because they will apply the same measures to different data, but everything would be more clear if the "antibiotic groups" had been properly defined (see my previous comment)

Answer: Intention was to address antibiotic classes and we revised accordingly. Please also see the related previous answer.

L90, The definition of the "proportions" are not clear at this point. Do a1+a2+a3+...+an=1, or do ai+bi=1 for all i? Said otherwise, is "ai" the proportion of individuals of population a that belong to species i, or is it the proportion of individuals of species i that belong to environment a ? It only becomes clear much later in the text.

Answer: a_i is the proportion of individuals of population a that belong to species i (b analogues). We revised accordingly.

C) The data presented in the results (antibiotic consumption and resistance in a hospital) have some potential interest, but at this stage the authors do not show anything convincing.

Figure 1 : If I correctly understand, the fact that fig 1a and fig 1b look very similar suggest that the sum of DDD for all the antibiotic groups is almost constant over time. This should be made visible on Fig 1c by making the y-axis start at 0, otherwise a quick look at this panel conveys an inaccurate message.

Answer: We changed the scale of the y-axis accordingly.

L167-174 I do not see how what it presented in this paragraph is a result. The authors are mostly saying that the coefficient of variation is a good measure of variation.

Answer: With respect to the coefficient of variation (CV), the application belongs to a descriptive part. We discussed the application of CV mainly for the sake of completeness, for comparisons with other measures, and because CV is found in the literature as a measure of homogeneity. We occasionally refer to CV in the text and we think that it might be beneficial for some readers to see how the application of CV in such a case behaves.

L 192-293 "Please note, the different scales of the y-axes reflect the different total amounts of consumption within each unit due to their different sizes": if this really comes from difference in size of the units (number of patients), Isn't it a strong argument for normalising DDD per patient days? I do not understand why the authors describe this normalisation in the methods but do not use it later in the results while it seems needed and appropriate.

Answer: All essential measures we discuss use proportions (relative abundances) as entry argument with the exception of CV. However, we find it illuminating to see the absolute DDDs. With respect to the results based on the diversity measures, it does not matter whether absolute DDDs or normalised DDDs are used. Relative abundances turn out to be almost identical for the two cases. We mentioned this but we did not show data due to redundancy. Moreover, the recorded patient days in the dataset may not be as reliable as the reports on antibiotic consumption. In addition, the surgical unit plays a uniqe role of treating mostly prophylactically. Residency in this unit per patient is short and patient days are not really meaningful. An absolute consumption of AB is arguably more informative.

L305 "The most important and at the same time most challenging question, in the given context, is whether the mixing behaviour of antibiotic administrations correlates or even causally relates to the prevalence of antibiotic resistances.": the authors do not show that their methods are useful for such a test. Do they really bring something compared to the coefficient of variation?

Answer: I do not see how the coefficient of variation can be modified such that a differential measure turns out comparable to SI_0. CV may reflect to some extent heterogeneity/diversity but it does – very much like HI - neither capture cycling nor mixing in an adequate way. However, HI (entropies) CAN be modified accordingly to yield SI_0.

Figure 12 mentions "intensive care" and "normal care" unit, while everywhere else in the article the units are labeled "unit 1", "unit 2" and "unit 3"

Answer: Within the text we usually used the clear text in combination with the code to get an intuition of which unit we talk of. We introduced the code for labeling figs and the clear text for the fig legends, but unfortunately not in a consistent way. We revised the figures and text accordingly.

I do not understand the sentence "We conclude that mixing of antibiotic consumption correlates with the prevalence of antibiotic-resistant bacteria by means of parallel slopes of similarity indexes SI0"

Answer: This objection is perhaps due to our confusing usage of „mixing“ and „cycling“ (see previous answers). What the correlation of SI_0 for AB and SI_0 for pathogens show is, that the magnitude of effective (rather weak) long-term clinical cycling given by the slope of SI_0 is the same as the magnitude of effective cycling-like changes in pathogen prevalences. This might be trival, because reducing a certain AB may reduce the prevalence of pathogens resistant to that AB and vice versa. But the almost perfect correlation is nevertheless quite striking, in our opinion. Moreover, we have shown that the amount of „cycling“ expressed by SI_0 correlates strongly with a reduction of the prevalence of resistant bacteria. Please see previous answers as well as the substantially improved explanations throughout the revised text.

Dataset: the authors made the effort of making the full dataset and analysis programs available. Just a minor suggestion: the variable names in R data sheets and the text in the R markdown file are currently in german, I think they would be more useful if translated to english.

Answer: We plan to upload the revised R-script to github upon acceptance of the manuscript.

Submitted filename: Response to Reviewers.docx

 10.1371/journal.pone.0238692.r003 Decision Letter 1 Martinez-Garcia Ricardo Academic Editor 2020 Ricardo Martinez-Garcia This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 1

11 Aug 2020

PONE-D-19-25350R1

Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategies

PLOS ONE

Dear Dr. Diebner,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

I appreciate your efforts in carefully addressing most of the points raised by both Reviewers. However, as you may see in their detailed comments below, although most of the points raised by Reviewer 2 have now been clarified, some concerns still remain about your time series statistical analysis. As a potential reader of your study, I also miss some more details on the specific statistical analysis that you conducted. This point is especially important for PLOS One publication criteria, available in the journal webpage.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: (No Response)

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The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Partly

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: I Don't Know

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Reviewer #2: Yes

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Reviewer #2: Yes

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Reviewer #2: Both in their answer and in the updated manuscript, Diebner and collaborators clarified that they see the mathematical framework as the main contribution of their article, and the application to the hospital dataset as a mere example of what can be achieved using this framework.

While most of my objections regarding the analysis of this hospital dataset and the interpretation of the findings remain, I guess they should thus not be seen as a blocking point. Besides, I think the authors clarified many points regarding their methodology and the context of their work. Many of my methodological questions are now addressed, in the manuscript or in the answer. As a biologist, it is still hard for me to judge the interest of the measures proposed by Diebner and colleagues without seeing a convincing biological application. So all I can say is that the methods and the context are clearly described, especially with the clarifications made in this revision, and that the figures are easily understandable.

Still, I am surprised that the authors do not see the problem with correlating two time-series which are both highly autocorrelated (L400-408). The statistical analysis is not described in details so I may be wrong, but I looked at the code and it seems that the authors directly compute a Pearson correlation between the two vectors (each vector being a time series). If so, I think this is quite inappropriate (because the data points of each vector are not independently sampled from a given distribution). This is what I was trying to point in my initial review by stating that ``only time series are analysed''. I have nothing per se against the analysis of time series, but it should be done correctly with the appropriate statistical tools (for example analysing the residuals).

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Reviewer #2: No

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 10.1371/journal.pone.0238692.r004 Author response to Decision Letter 1 Submission Version 2

14 Aug 2020

Response to Reviewers

Manuscript PONE-D-19-25350R1

Reviewer #2:

Both in their answer and in the updated manuscript, Diebner and collaborators clarified that they see the mathematical framework as the main contribution of their article, and the application to the hospital dataset as a mere example of what can be achieved using this framework.

While most of my objections regarding the analysis of this hospital dataset and the interpretation of the findings remain, I guess they should thus not be seen as a blocking point. Besides, I think the authors clarified many points regarding their methodology and the context of their work. Many of my methodological questions are now addressed, in the manuscript or in the answer. As a biologist, it is still hard for me to judge the interest of the measures proposed by Diebner and colleagues without seeing a convincing biological application. So all I can say is that the methods and the context are clearly described, especially with the clarifications made in this revision, and that the figures are easily understandable.

Still, I am surprised that the authors do not see the problem with correlating two time-series which are both highly autocorrelated (L400-408). The statistical analysis is not described in details so I may be wrong, but I looked at the code and it seems that the authors directly compute a Pearson correlation between the two vectors (each vector being a time series). If so, I think this is quite inappropriate (because the data points of each vector are not independently sampled from a given distribution). This is what I was trying to point in my initial review by stating that ``only time series are analysed''. I have nothing per se against the analysis of time series, but it should be done correctly with the appropriate statistical tools (for example analysing the residuals).

Answer:

The reviewer raised an important point regarding the correlation of time series. Yes, strictly speaking, Pearson’s correlation requires independent sampling. Usually, the correlation of two time series is analysed by applying a cross-correlation-function which uses a time-lagged version of one of the time series. Unfortunately, this is not applicable here due to the low sampling frequency and very short time series. Thus, our argument here is that Pearson’s correlation is sufficient to at least raise hypotheses. Arguably, this supports the idea of a proof-of-principle. Based on the proposed analysis using diversity measures it is planned to design a proper study and we think that also other researchers dealing with antibiotic stewardship will benefit from our mathematical framework. We added the subsection “Statistical analysis” at the end of the “Materials and methods” section to elaborate on these points.

Submitted filename: Response to Reviewers.docx

 10.1371/journal.pone.0238692.r005 Decision Letter 2 Martinez-Garcia Ricardo Academic Editor 2020 Ricardo Martinez-Garcia This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Submission Version 2

24 Aug 2020

Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategies

PONE-D-19-25350R2

Dear Dr. Diebner,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Kind regards,

Ricardo Martinez-Garcia

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

 10.1371/journal.pone.0238692.r006 Acceptance letter Martinez-Garcia Ricardo Academic Editor 2020 Ricardo Martinez-Garcia This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

25 Aug 2020

PONE-D-19-25350R2

Mathematical basis for the assessment of antibiotic resistance and administrative counter-strategies

Dear Dr. Diebner:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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on behalf of

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PLOS ONE