Incentivizing Narrow-Spectrum Antibiotic Development with Refunding
IIncentivizing Narrow-Spectrum Antibiotic Development with Refunding
Lucas B¨ottcher
1, 2, 3 and Hans Gersbach Computational Medicine, UCLA, 90095-1766, Los Angeles, United States ∗ Institute for Theoretical Physics, ETH Zurich, 8093, Zurich, Switzerland Center of Economic Research, ETH Zurich, 8092, Zurich † (Dated: July 14, 2020)The rapid rise of antibiotic resistance is a serious threat to global public health. Without furtherincentives, pharmaceutical companies have little interest in developing antibiotics, since the successprobability is low and development costs are huge. The situation is exacerbated by the “antibioticsdilemma”: Developing narrow-spectrum antibiotics against resistant bacteria is most beneficial forsociety, but least attractive for companies since their usage is more limited than for broad-spectrumdrugs and thus sales are low. Starting from a general mathematical framework for the study ofantibiotic-resistance dynamics with an arbitrary number of antibiotics, we identify efficient treat-ment protocols and introduce a market-based refunding scheme that incentivizes pharmaceuticalcompanies to develop narrow-spectrum antibiotics: Successful companies can claim a refund from anewly established antibiotics fund that partially covers their development costs. The proposed re-fund involves a fixed and variable part. The latter (i) increases with the use of the new antibiotic forcurrently resistant strains in comparison with other newly developed antibiotics for this purpose—the resistance premium—and (ii) decreases with the use of this antibiotic for non-resistant bacteria.We outline how such a refunding scheme can solve the antibiotics dilemma and cope with varioussources of uncertainty inherent in antibiotic R&D. Finally, connecting our refunding approach tothe recently established antimicrobial resistance (AMR) action fund, we discuss how the antibioticsfund can be financed. ∗ [email protected] † [email protected] a r X i v : . [ q - b i o . P E ] J u l I. INTRODUCTION
According to the World Health Organization (WHO), antibiotic resistances are a serious threat to global publichealth [1]. Some studies see in the emergence of antimicrobial resistances (AMR) the beginning of a postantibioticera and a threat similar to the one posed by climate change [2]. In the European Union, more than 33,000 people dieevery year due to infections caused by drug-resistant microbes and the corresponding yearly AMR-related healthcarecosts and productivity losses are estimated to be more than 1.5 billion Euros [3].Antibiotic resistances result from mutations in microbes and from evolutionary pressure, which selects those mu-tations that are resistant against certain antibiotics. The large-scale use of antibiotics in medical and agriculturalsettings in high-income countries led to the emergence of various multi-resistant strains. Recent findings indicatethat certain strains of Enterobacteriaceae even developed resistances against the usually highly-effective class of car-bapenems [4]. Carbapenems are so-called drugs of last resort and only used if other antibiotic agents fail to stop thepropagation of microbes.The reasons for the development of bacterial resistances and the decline in effective treatment possibilities aremultifaceted. Historically, pharmaceutical companies focused on the development of broad-spectrum antibiotics thattarget various strains as is the case for certain β -lactam antibiotics, whose second and third generation compoundswere intentionally developed to target a broader spectrum of microbes [5, 6]. The rationale behind this developmentis that it offers pharmaceutical companies a higher return than the development of antibiotics that only target specificstrains. In addition, broad-spectrum agents can be prescribed fast and without—or only with limited—diagnosiseffort. On the downside, the use of broad-spectrum antibiotics seems to be correlated with an increase in antibioticresistances [6–10]. Another disadvantage of broad-spectrum agents is that they are associated with outbreaks of C. difficile infections that result from antibiotic-induced disturbances of the gut microbiome [6, 11, 12].These examples illustrate that the use of narrow-spectrum antibiotics may lead to a slower development of antibioticresistance and to lower risks of
C. difficile infections. However, the treatment of microbial infections with narrow-spectrum agents requires efficient diagnostic techniques to quickly and precisely determine the type of bacterial strainthat causes a certain infection [6]; still, the clinical diagnosis of bacterial strains is often based on traditional and slowmicrobial-culture methods [13]. Only more recently, progress in the development of advanced diagnostic techniquesmade it possible to reduce the diagnosis time from a few days to only a few hours only [14]. The notions of “narrow” and “broad” spectrum antibiotics are only loosely defined. Some studies distinguishbetween antibiotics that are applicable to Gram-positive and Gram-negative strains of microbes. The distinctionbetween these two categories is based on the so-called Gram strain test, which categorizes bacteria into these categoriesaccording to physiological properties of their cell walls. As in Ref. [6], we will reserve the term “narrow-spectrum”for antibiotics that only affect one strain or a small number of strains when given to a patient.Treatment strategies involving narrow-spectrum antibiotics have been implemented by some northern-Europeancountries such as Norway and Sweden [17, 18]. The Norwegian strategy is based on penicillin G and aminoglycosideas initial treatment substances [17] and it avoids broad-spectrum β -lactam antibiotics. However, further studies arenecessary to better understand the influence of such narrow-spectrum treatment approaches on the population levelover time.The objective of policy is to devise strategies that help incentivizing pharmaceutical companies to focus on thedevelopment of narrow-spectrum antibiotics. According to a recent report of the European Court of Auditors [19], “the antimicrobials market lacks commercial incentives to develop new treatments” . It suggests to reallocate someof the EU AMR research budget to create economic incentives for pharmaceutical companies [20]. In the US, theGenerating Antibiotic Incentives Now (GAIN) Act from 2012 pursues similar goals by “stimulating the developmentand approval of new antibacterial and antifungal drugs” [21].In this paper, we develop a complementary approach by constructing a refunding scheme for successful developmentsof antibiotics. The development costs of new antibiotics can amount to several billion USD and the probability of asuccessful development might be only a few percent [22, 23]. Our refunding scheme aims at making the developmentof new narrow-spectrum antibiotics nevertheless commercially viable. This can be achieved by creating an antibioticsfund and implementing a refunding scheme for it. The refunding scheme rewards companies that have successfullydeveloped a new antibiotic (narrow or broad) as follows. A successful company can claim a refund from the antibioticsfund to partially cover its development costs. The proposed refund involves a fixed and variable part. The variablepart increases with the use of the new antibiotic for currently resistant strains in comparison with other newlydeveloped antibiotics for the same purpose—the resistance premium—and (ii) decreases with the use of this antibioticfor non-resistant bacteria. With an appropriate choice of refunding parameters, it becomes commercially attractiveto develop a narrow-spectrum antibiotic, or to switch to such an antibiotic if it becomes feasible in the R&D process, For further information on tailored antibiotic treatment approaches (i.e., personalized medicine), see also Refs. [15, 16].
Figure 1.
Number of registered antibiotics.
We show the number of registered or patented antibiotics from 1890-2020 [28]. while developing broad-spectrum antibiotics becomes less attractive. The antibiotics fund, in turn, is continuouslyfinanced by fees levied on the use of existing antibiotics, and should be started by initial contributions from theindustry and public institutions like the recent AMR Action fund. In Sec. II, we provide further details on the rationale of developing incentives for the development of narrow-spectrumantibiotics. To formulate our refunding approach, we first introduce a mathematical framework of antibiotic-resistancedynamics in Sec. III. Our framework is able to account for an arbitrary number of different antibiotics, whereas previousmodels [24–26] only considered two to three distinct antibiotics and compared different treatment protocols such astemporal variation and combination therapy. Similar “low-dimensional” descriptions of antibiotic resistance have beenused to study the economic problem of optimal antibiotic use [27].In Sec. IV, we use our general framework to derive a model variant that allows us to study the “antibiotics dilemma”:Developing narrow-spectrum antibiotics, which are only effective against specific bacterial strains, is most beneficialfor society, but least attractive for pharmaceutical companies due to their limited usage and sales volumes. We couplethis variant of the general antibiotics model to our refunding scheme in Sec. V and illustrate how refunding can lead tobetter treatment protocols and a lower share of resistant strains. The refund for the development of a certain antibioticcovers part of the development costs and satisfies the following properties: (i) The refund is strongly increasing withthe use of the new antibiotic for currently resistant bacteria in comparison with other newly developed antibiotics forthis purpose. (ii) The refund decreases with the use of this antibiotic for non-resistant bacteria in comparison withother antibiotics used for this purpose. In Sec. VI, we outline possibilities to design refunding schemes in terms of thegeneral antibiotic-resistance model of Sec. III. We conclude our study in Sec. VII
II. BROADER PERSPECTIVE
It is useful to place our proposal into a broader context. First, incentivizing the development of narrow-spectrumantibiotics has to be matched by the development and use of efficient diagnostic techniques to quickly and preciselydetermine the type of bacterial strain that causes a health problem and by collecting information regarding the type ofbacterial strain, the treatment and its outcome. Many OECD countries have already implemented extended reportingsystems (see e.g. the Swiss antibiotic strategy 2015 [29]).In addition to the development of narrow-spectrum agents, it is important to also consider alternative approachessuch as medication that sustains and boosts the human immune system during infections, or improved sterilization andsanitation in hospitals [6]. Other strategies for fighting bacterial infections, such as targeting virulence or treatmentwith antibodies or phage [30–32], are also alternatives to antibiotics.In practice, it will not be easy to encourage pharmaceutical companies to refocus their R&D activities. Thedisappointing finding that genomics did not lead to many new classes of antibiotics caused the close-down of manyantibiotic research laboratories [2, 33]. Currently, the pipeline of new ideas seems to be rather small (see Fig. 1),while the costs of clinical trials are very high. The authors of Ref. [34] analyzed the clinical trial costs of 726 studiesthat were conducted between 2010-2015. In the initial clinical trial phase, the median cost was found to be 3.4 million https://amractionfund.com/, retrieved on July 13, 2020. Figure 2.
Model schematic.
Susceptible individuals ( X ) can be infected with bacterial strain i at rate b i . Infected individualsin state Y i recover spontaneously at rate r i . The corresponding antibiotic-induced recovery rate of antibiotic A is f i B h i B .However, only a fraction 1 − s i B of individuals in state Y i recovers after a treatment with antibiotic B. The remaining fraction s i B becomes resistant against antibiotic B and ends up in a compartment Y j of bacterial strains exhibiting more resistances. Thesets of effective antibiotics in compartments Y i and Y j are A i and A j , respectively. Infection and recovery processes with therespective rates are also present in compartment Y j . For the case of n = 3 antibiotics, we show the possible antibiotic-treatmentclasses A , A , . . . , A . USD and the median cost of phase III in the development process was reported to be more than 20 million USD.High development costs of antibiotic drugs limit the number of players in this area and require major companies tobe involved in the development process. A good research ecosystem for antibiotic development necessarily involveslarge companies, entailing significant in-house efforts, but also collaborations with academia, buying or investing inSMEs, and joint ventures with other large pharmaceutical companies. An appropriately designed refunding schemecan help to foster such a research ecosystem.There are also arguments that the development of new antibiotics is not so critical. The emergence of antibioticresistance, even for new classes of antibiotics, is inevitable and one may conclude that research and development effortsshould mainly focus on antibiotic substances that are effective against highly resistant strains (see e.g. Ref. [2]).A recent report published by the European Observatory on Health Systems and Policies [3] suggests a multifoldR&D approach to combat AMR that includes: (i) push incentives (e.g., direct funding and tax incentives) and pullincentives (e.g., milestone prize and patent buyout) for the development of new antibiotics, (ii) research in diagnostics(e.g., rapid tests to distinguish between bacterial and viral infections), and (iii) vaccine research.Our proposed refunding scheme is a complementary public-private initiative to foster the development of newantibiotics. III. MODELING ANTIBIOTIC TREATMENTA. Treatment of infections with n antibiotics In order to motivate the refunding scheme and model population-level dynamics of antibiotic resistance, we firstintroduce a corresponding mathematical framework that is able to account for an arbitrary number of antibiotics. Wedescribe the interaction between infectious and susceptible individuals in terms of a susceptible-infected-susceptible(SIS) model [36] whose infected compartment is sub-divided into compartments that can each be treated with certainantibiotics. We indicate a susceptible state by X and use Y i to denote infected states that are sensitive to antibioticsin the set A i . If the set A contains two antibiotics A and B (i.e., A = { A , B } ), individuals in state Y can be treatedwith these two antibiotics but not with a potentially available third antibiotic C that can be used to treat individualsin state Y , where A = { A , B , C } (see Fig. 2).We describe antibiotic-resistance dynamics in terms of a mass-action model of multiple antibiotic therapy (see Phase III clinical trials are the last phase of clinical research that has to be satisfactorily completed before regulatory agencies willapprove a new drug. Such trials usually involve large patient groups (ca. 300–3000 volunteers who have the disease or condition) andrequire comparatively long observation periods ranging between 1–4 years [35].
Figure 3.
Antibiotic-resistance network.
For the case of n = 4 antibiotics, we show the antibiotic-resistance network.Nodes correspond to states in which the indicated antibiotics are effective and edges between nodes represent the developmentof resistant strains due to the usage of certain antibiotics. In the shown example, single-antibiotic therapy is being used. Thatis, no combinations of antibiotics are being administered to patients. Fig. 2): d x d t = − x N (cid:88) i =1 b i y i + N (cid:88) i =1 r i y i + N (cid:88) i =1 (cid:88) j ∈A i f ij h ij (1 − s ij ) y i + λ − dx , d y i d t = b i xy i − r i y i − c i y i − (cid:88) j ∈A i f ij h ij y i + (cid:88) k i ∈ { , . . . , N } ) [37]. Additional antibiotic-inducedrecovery from compartment i with antibiotic j ∈ A i occurs with rate f ij h ij , where f ij is the proportion of antibiotic j ∈ A i , relative to other antibiotics, that is used to treat Y i . However, only a fraction 1 − s ij actually recovers,whereas the remaining fraction s ij develops a resistance against antibiotic j ∈ A i . The birth rate of new susceptibleindividuals is λ and the corresponding death rate is d . For infected individuals in state Y i , the death rate is c i . Theset S ( A i ) contains all antibiotics that were used to arrive at the partially or completely resistant compartment Y i from other states Y k ( k < i ) with less resistances. For example, the use of single antibiotics (i.e., one per patient) isdescribed by S ( A i ) = (cid:40)(cid:91) k A k \ A i (cid:12)(cid:12)(cid:12)(cid:12) A k ∈ {A , A , . . . , A N } , |A k | = |A i | + 1 , A k ∩ A i (cid:54) = ∅ (cid:41) (2)and, in this case, each antibiotic is used with proportions f ij = 1 / |A i | , where |A| is the cardinality of set A . Weillustrate an example of a corresponding antibiotic-resistance network for n = 4 antibiotics in Fig. 3. Nodes in sucha resistance network represent states Y i and edges describe treatment strategies. In the example we show in Fig. 3,only single antibiotics (no combinations) are being used for treatment.The general antibiotic-resistance model (see Eq. (1)) has N different compartments, which correspond to N re-sistance states, each accounting for a certain set of effective antibiotics. We denote the total number of antibioticsby n . What is the number of resistance states N that belongs to a certain number of antibiotics n ? Consideringthe antibiotic-resistance network of Fig. 3, we observe that the total number of resistance states N is the sum overall possible combinations of single antibiotics plus one (representing the completely resistant state). For n differentantibiotics, we thus have to consider N = 1 + (cid:80) nk =1 (cid:0) nk (cid:1) = 2 n different states Y i ( i ∈ { , , . . . , N } ). We order themin the following way. We denote by Y the infected state that can be successfully treated with all antibiotics, whereas Y N represents an infection with a completely resistant strain. Let k ≤ n be the number of effective antibiotics. For awild-type strain, the number of effective antibiotics is k = n . In each layer of the antibiotic-resistance network, thereare (cid:0) nk (cid:1) different strains. For a treatment with single antibiotics (see Fig. 3), there are always (cid:0) nk (cid:1) nodes with k edgesin a certain layer that need to be connected to (cid:0) nk − (cid:1) nodes in the following layer. The relative difference is k (cid:0) nk (cid:1)(cid:0) nk − (cid:1) = ( n − k + 1) !( n − k ) ! = n − k + 1 . (3)This equation implies that n − k + 1 elements from the current layer are mapped to one element in the next layer.In the first layer ( k = 4) of the network that we show in Fig. 3, one element from the current layer is mapped to oneelement in the next layer. In the second layer ( k = 3), two elements are mapped to one element in the third layer.Similar considerations apply to other treatment protocols (e.g., combination treatment with multiple antibiotics) andhelp to formulate the corresponding set of rate equations.Previous models only considered the treatment with two and three antibiotics [24–26]. Our generalization to N compartments enables us to provide insights into the higher-dimensional nature of the dynamical developmentof antibiotic resistances. In Appendices A and B we compare the outlined single-antibiotic therapy approach withcombination treatment for different numbers of antibiotics. We also demonstrate in Appendix B that the mathematicalform of the stationary solution of Eq. (1) is unaffected by the number of antibiotics. Still, more antibiotics can beuseful to slow down the development of completely resistant strains, suggesting that rolling out more antibiotics isuseful. However, as we will discuss below, fostering the development of particular types of narrow-spectrum antibioticsis much more powerful to slow down the occurrence of completely resistant strains and reduce the number of deathsthan developing broad-spectrum antibiotics. B. Performance measures
We can compare different treatment protocols in terms of different metrics including the total stationary population P ∗ := x ∗ + N (cid:88) i =1 y ∗ i , (4)where the asterisk denotes the stationary densities of x and y i . Another possible metric is the gain of healthyindividuals G ( T ) := (cid:90) T x ( t ) d t − (cid:90) T x ( t ; h ij = 0)d t , (5)through antibiotic treatment during some time, denoted by T , where x ( t ; h ij = 0) denotes the proportion of susceptibleindividuals in the absence of treatment (i.e., h ij = 0 for all i, j ).Finally, we can calculate the time until half of the infected individuals are infected by bacterial strains that areresistant against any antibiotic. This “half-life” of non-resistance is given by: T / := (cid:40) t (cid:12)(cid:12)(cid:12)(cid:12) y N ( t ) (cid:80) Ni =1 y i ( t ) = 12 (cid:41) . (6)As mentioned above and proven in the Appendix, the long-term stationary population P ∗ is no suitable performancemeasure, since P ∗ is identical for all treatment protocols that we will consider in the following sections. However,both G ( T ) and T / are suitable measures to compare different development strategies for antibiotics. In addition,we will also use G / := G ( T / ) as performance metric. IV. NARROW VERSUS BROAD SPECTRUM ANTIBIOTICSA. Research and development opportunities
To provide a formal representation of both the antibiotics dilemma and the refunding scheme, we consider thesimplest case with n = 2 antibiotics. In Sec. VI, we discuss a more general refunding approach that can be used inconjunction with the general antibiotic-resistance model of Sec. III. Figure 4.
Growth of multi-resistant strains under 50/50 and 100/0 treatment.
The evolution of wild-type y (blacksolid line) and completely resistant infections y (red solid line) under 50/50 treatment with f = f = 0 . f = 1 and f = 0 (b). To obtain the solutions shown, we numerically solve Eqs. (7) with a classicalRunge-Kutta scheme in the time interval [0 , T ] with T = 100 setting λ = 100, d = 1, c = 1 . b = 0 . r i = (2 − k )0 . k is thenumber of effective antibiotics in the respective layer), h = 1, s = 0 . α B = γ B = (cid:15) ≥
0. The initial conditions are x (0) = 50, y (0) = 33 . y (0) = y (0) = y (0) = 0. For the model variant with n = 2 antibiotics, the corresponding sets of antibiotics for the N = 2 = 4 infectedcompartments are A = { A , B } , A = { A } , A = { B } , and A = ∅ .We assume that antibiotic A is already on the market. For the development of a second antibiotic, there are twopossibilities for pharmaceutical companies. • Antibiotic B : This is a broad-spectrum antibiotic that is as effective as antibiotic A against wild-type strains.It is also effective against strains that are resistant against A. • Antibiotic B : This is a narrow-spectrum antibiotic that is, by a factor 1 − α B (0 ≤ α B ≤ B is, by a factor 1 + γ B ( γ B > .Note that we use the terms “narrow” and “broad” to classify antibiotics according to their effectiveness againstcertain bacterial strains .We will later turn to costs and chances to develop such antibiotics, but at the moment, we consider what happensif either B or B is being developed and used for treating patients. B. The model
For both types of antibiotics, we derive the corresponding population-level dynamics according to Eq. (1):d x d t = − bx ( y + y + y + y ) + r y + r y + r y + r y + h (1 − s ) { y [ f + (1 − (cid:15) ) f ] + y + y (1 + (cid:15) ) } + λ − dx , d y d t = [ bx − r − c − h ( f + (1 − (cid:15) ) f )] y , d y d t = ( bx − r − c − h ) y + hs (1 − (cid:15) ) f y , d y d t = ( bx − r − c − h (1 + (cid:15) )) y + hsf y , d y d t = ( bx − r − c ) y + hs [ y + (1 + (cid:15) ) y ] , (7) An alternative mathematical definition of “narrow” and “broad” would be to classify combination treatment as a broad-spectrumapproach and single-antibiotic therapy as narrow. where we have set γ B = α B = (cid:15) ∈ [0 ,
1] (see Sec. IV A). Furthermore, we assumed that the antibiotic-inducedrecovery rate, the infection rate, and the fraction of individuals that develop an antibiotic resistance is constant for allantibiotics, i.e. h ij = h , b i = b and s ij = s . For modeling antibiotic B , we simply set (cid:15) = 0. If antibiotic B is present,we set the values of these parameters to the efficiency disadvantages and advantages of B relative to A. Note thatthe assumption of an equal infection rate b of different strains is justified by corresponding empirical findings [38].We now focus on four different treatment strategies:I. Treatment with antibiotic B and symmetric use of antibiotics (i.e., 50/50):Hence, we have (cid:15) = 0. Moreover, since we consider a symmetric use, f = 0 . f = 0 .
5, this casedescribes a treatment strategy where 50% of patients with a wild-type-strain infection receive antibiotic A andthe remaining 50% receive antibiotic B . Both antibiotics A and B have the same effect on strains 1 and 2 and1 and 3, respectively.II. Treatment with antibiotic B and asymmetric use of antibiotics (i.e., 100/0):Hence, we again have (cid:15) = 0. We only use the new antibiotic B against strains that are resistant against A. Allpatients with a wild-type-strain infection receive antibiotic A, i.e. f = 1 and f = 0.III. Treatment with antibiotic B and symmetric use of antibiotics:The symmetric use implies f /f = 50 /
50 for wild-type-strain infections. The prefactor 1 − (cid:15) accounts for thecorresponding recovery-rate difference in the wild-type compartment. However, antibiotic B is more effectivein compartment 3, where individuals have an antibiotic-induced recovery rate of h (1 + (cid:15) ).IV. Treatment with antibiotic B and asymmetric use of antibiotics:In this scenario, we only use antibiotic A in compartment Y and thus set f = 1 and f = 0. C. Comparisons
We now compare treatment strategies I–IV in terms of P ∗ , G , T / , G / (see Figs. 5 and 6), and the use of theantibiotics A and B and B , respectively. We can keep track of the consumption of antibiotics A and B by integratingd C A d t = f y + y and d C B d t = f y + y (8)over time. We now compare scenarios I–IV, by varying (cid:15) from 0 to 1.We first look for differences between the evolution of the wild-type and fully resistant compartments under 50/50and 100/0 treatment. Based on the simulation data that we show in Fig. 4, we conclude that the variability in y and y is much larger under 50/50 treatment than under 100/0 treatment. Comparing the performance measures P ∗ , G , T / , and G / (see Figs. 5 and 6), we find that the 50 /
50 and 100 / (cid:15) = 0. For larger values of (cid:15) ,the gain G of the 50 /
50 strategy is smaller than the gain of the 100 / C A and C B of antibiotics A and B (see Fig. 7). For (cid:15) (cid:38) .
5, the100 / /
50 protocol.These figures highlight a fundamental dilemma. Developing a narrow-spectrum antibiotic B is highly beneficial forsociety, but then it should only be used very little, namely against the strains which are resistant against antibiotic A.We refer to this issue as the antibiotics dilemma : Developing a narrow-spectrum antibiotic against resistant bacteriais most attractive for society, but least attractive for companies, since usage is should be limited, so that sales arelow. V. REFUNDING SCHEMESA. The basic principles
The situation is further complicated by additional properties of antibiotics development and usage. First, thedevelopment costs are enormous, in the range of several billion USD, and second the chances to succeed are low. Thisis true in general for new drugs [22] but more pronounced for antibiotics, where the success probability may be aslow as 5% [23]. Third, once a narrow-spectrum antibiotics is developed and it is also effective to some degree againstwild-type strains, using it for wild-type strains should not be commercially attractive.To overcome the antibiotics dilemma and associated complications, we suggest to introduce a refunding scheme forthe use of newly developed antibiotics. The refunding scheme works as follows:
Figure 5.
Treatment with two antibiotics (50/50 strategy).
We numerically solve Eqs. (7) with a classical Runge-Kuttascheme in the time interval [0 , T ] with T = 100 setting λ = 100, d = 1, c = 1 . b = 0 . r i = (2 − k )0 . k is the number ofeffective antibiotics in the respective layer), h = 1, s = 0 .
05, and f = f = 0 . α B = γ B = (cid:15) ≥
0. The initial conditionsare x (0) = 50, y (0) = 33 . y (0) = y (0) = y (0) = 0.
1. An antibiotics fund should be started by initial contributions from the industry and public institutions like theestablishment of the recent AMR Action fund. In addition all antibiotic use is charged with a small fee whichis channeled continuously into the antibiotics fund.2. Firms that develop new antibiotics obtain a refund from the fund.3. The refund for a particular antibiotic is calculated with a formula that satisfies the following three properties: • There is a fixed payment for a successful development of an antibiotic, i.e. an antibiotic that is approvedby the public health agency responsible for such approvals (e.g., the U.S. Food and Drug Administration(FDA)). This part is in the spirit of Ref. [39], as it is equivalent to an advanced market commitment.Pharmaceutical companies know that once an approved patent for a new antibiotics is awarded, they willbe reimbursed part of their development costs. • The refund is strongly increasing with the use of the new antibiotic for currently resistant bacteria, com-pared to other newly developed antibiotics for this purpose. This part is the resistance premium . • The refund is weakly or strongly declining in the use of the antibiotics for non-resistant bacteria, comparedto other antibiotics used for this purpose.The objective of our refunding scheme is to financially incentivize pharmaceutical companies to undertake R&D fornew narrow-spectrum antibiotics, by using a minimum-size antibiotics fund. As we will demonstrate below, all threeelements are necessary to achieve this purpose.Several remarks are in order: First, refunding schemes are widely discussed in the environmental literature toprovide incentives for firms to reduce pollution [40]. Second, simple forms of refunding schemes could also be used inother contexts where pharmaceutical companies have only little financial interest to investing in drug research, dueto potentially low sales volumes. This is, for instance, the case for orphan drug development and vaccine research0
Figure 6.
Treatment with two antibiotics (100/0 strategy).
We numerically solve Eqs. (7) with a classical Runge-Kuttascheme in the time interval [0 , T ] with T = 100 setting λ = 100, d = 1, c = 1 . b = 0 . r i = (2 − k )0 . k is the number ofeffective antibiotics in the respective layer), h = 1, s = 0 .
05, and f = 1, f = 0, α B = γ B = (cid:15) ≥
0. The initial conditionsare x (0) = 50, y (0) = 33 . y (0) = y (0) = y (0) = 0. for viral infections including SARS and Ebola or enduring epidemic diseases in the sense of Ref. [41]. However, forsuch cases, refunding schemes are much easier to construct, since they can solely rely on the usage, e.g. the numberof vaccinated individuals. For antibiotics—because of the antibiotics dilemma—one has to construct new types ofrefunding schemes with “sticks and carrots”: The carrot for using the antibiotic against bacterial strains resistantagainst other antibiotics and sticks for using the antibiotics against wild-type strains. Those complications do notarise with the aforementioned (simple) refunding schemes in the environmental sciences. Third, one might also achievesufficiently-strong incentives to develop new antibiotics without a refunding scheme by allowing for very high priceswhen an antibiotic is used against bacterial strains that are resistant against other antibiotics. We do not pursuethis approach since enormously high prices for a treatment would raise ethical and health concerns, since certaintherapies might then not be affordable anymore, which, in turn, would further fuel the spreading of resistant bacteria.Moreover, since the use of an antibiotic against wild-type strains exerts a negative externality on all individuals—dueto the possible emergence of resistant bacteria in response to this use—all cases of antibiotics use contribute to thefinancing of narrow-spectrum antibiotics. Levying a fee on antibiotic use not only fills up the antibiotic fund andincentives the development of narrow-spectrum antibiotics, it also promotes the cautious use of existing antibiotics.Both effects internalize the negative externality caused by antibiotic use. B. Refunding schemes for two antibiotics
The mathematical model we formulate for the proposed refunding and incentivization scheme discussed aboveincludes two elements: • There is a fixed amount, denoted by α , which a pharmaceutical company obtains if it successfully develops anew antibiotic B i , i.e. an antibiotic approved by a public health authority.1 Figure 7.
Antibiotic consumption.
We show the stationary antibiotic consumption lim t →∞ C A and lim t →∞ C B for the 50/50treatment with f = f = 0 . f = 1 and f = 0 (b). We numerically solve Eqs. (7) with aclassical Runge-Kutta scheme in the time interval [0 , T ] with T = 100 setting λ = 100, d = 1, c = 1 . b = 0 . r i = (2 − k )0 . k is the number of effective antibiotics in the respective layer), h = 1, s = 0 .
05, and α B = γ B = (cid:15) ≥
0. The initial conditionsare x (0) = 50, y (0) = 33 . y (0) = y (0) = y (0) = 0. • There is a variable refund that is determined by the following refunding function: g ( f i y , f i y ) = β f i y γf i y + f i y , (9)where i ∈ { , } (to represents antibiotics B and B ) and β and γ are scaling parameters, with β being a largenumber and γ satisfying γ ≥
1. The refunding function g ( f i y , f i y ) determines the relative use of the newantibiotic in compartment 3 (A-resistant strains) compared to the total use of the antibiotic weighted by theparameter γ . We require the refunding function to be – bounded according to 0 ≤ g ( f i y , f i y ) ≤ β , – increasing in the use for currently resistant bacteria in comparison with other newly developed antibioticsused for this purpose: f i y , – declining in the use of antibiotics for non-resistant bacteria in comparison with other antibiotics used forthis purpose: f i y , – maximal if the antibiotic is only used to treat A-resistant strains and 0 if it is only used for nonresistantstrain treatment.Note that refunding scheme uses three free parameters α , β , and γ . While in the simplest cases, only one or twoparameters would be needed, we will see in the subsequent section that all three parameters are necessary to achievethe objective of the refunding scheme.The total refund that a successful pharmaceutical company receives in the time interval [0 , T ] for developing anantibiotic B i is given by R i ( T ) := α + (cid:90) T β f i y ( f i y + f i y ) γf i y + f i y d t . (10)We note that for γ = 1, the refund is solely determined by f y and the use for compartment 1 is irrelevant forthe refund. For γ >
1, the use of antibiotics for compartment 1 decreases the refund.
C. Incentivizing development
We next focus on the research and development efforts of pharmaceutical companies and on how antibiotics are usedonce they have been developed. For this purpose, we first consider the situation without refunding. Companies areassumed to make a risk-neutral evaluation of the profit opportunities and loss risks from investing in such developments.2For simplicity, we neglect discounting. Then, without refunding (i.e., without R i ( T )), the net profit of a companythat develops an antibiotic B i is given by: π i = q i ( p i − v i ) (cid:90) T d C B i d t d t − K i , = q i ( p i − v i ) (cid:90) T ( f i y + f i y ) d t − K i , (11)where K i denotes the total development costs of B i , and q i the probability of success when the development isundertaken. Moreover, p i is the revenue per unit of the antibiotic used in medical treatments and v i are the productioncosts per unit.Note that in our example with two antibiotics, f i = 1, since only drug B i can be used for A-resistant strains.We assume that without refunding, π i is (strongly) negative, because of high development costs K i and low successprobabilities q i . The task of a refunding scheme is now three-fold: First, it has to render developing new antibioticscommercially viable. Second, it has to render developing narrow-spectrum antibiotics more attractive than developingbroad-spectrum antibiotics. Third, if a narrow-spectrum antibiotic is developed that is also effective against wild-typestrains, but less so than others, the refunding scheme should make the use against wild-type strains unattractive.With a refunding scheme in place, we directly look at the conditions for such a scheme to achieve the break-even condition, i.e. a situation at which π becomes zero and investing into antibiotics development becomes justcommercially viable. The general break-even condition for a newly developed antibiotic B i is given by K i = q i ( p i − v i ) (cid:90) T ( f i y + f i y ) d t + q i R i ( T )= αq i + q i (cid:90) T (cid:20) β f i y γf i y + f i y + ( p i − v i ) (cid:21) ( f i y + f i y ) d t . (12)Clearly, refunding increases the profits from developing new antibiotics. There are many combinations of the refundingparameters α , β , and γ that can achieve this break-even condition. However and more subtly, the refunding hasto increase the incentives for the development of narrow-spectrum antibiotics more than those for broad-spectrumantibiotics. This can be achieved by an appropriate choice of the scaling parameter, as we will illustrate next. D. Critical conditions for refunding parameters
To derive the critical refunding parameters, we assume that the parameter α , with 0 < α < K i , is given and thusa fixed share of the R&D costs is covered by the antibiotics fund. Based on the break-even condition (see Eq. (12)),we obtain the following general condition that the parameters β and γ have to satisfy: β = K i − αq i − q i ( p i − v i ) (cid:82) T ( f y + f y )d tq i (cid:82) T f y γf y + f y ( f y + f y )d t . (13)The goal of our refunding scheme is to incentivize pharmaceutical companies to produce narrow-spectrum antibioticsB that are only used against currently resistant strains (see treatment strategy IV in Sec. IV). Thus, the refundingscheme has to satisfy two conditions: first, with the development of the antibiotic B , the company achieves break-even. Second, developing antibiotic B is not attractive, i.e. the profit is negative. To satisfy the first condition, weuse Eq. (13) and obtain the optimal value β ∗ = K − αq − q ( p − v ) (cid:82) T f y d tq (cid:82) T f y d t , (14)where we used that f = 0 (see Sec. IV). To achieve negative profit for using B , we need to choose the parameter γ such that developing a broad-spectrum antibiotic B and applying it in y and y (see treatment strategy I inSec. IV) is not more attractive than developing a narrow-spectrum antibiotic B according to treatment strategy IV(see Sec. IV). Thus, the refunding scheme needs to satisfy αq + q (cid:90) T (cid:20) β ∗ f y γf y + f y + ( p − v ) (cid:21) ( f y + f y ) d t − K < . (15)If we evaluate inequality 15 as an equality, we obtain a critical value for γ , denoted by γ ∗ , for certain values of β ∗ , p , and v . For γ > γ ∗ , it is more profitable to produce a narrow-spectrum antibiotic B and get a higher refund3 Figure 8.
Critical refunding parameters.
We show the critical refunding parameters β ∗ (a) and γ ∗ (b–d) for different (cid:15) and population sizes 50 × , 100 × , and 150 × . The critical refunding parameter β ∗ increases with population size(a). A finite γ ∗ and γ ∗ is indicated by the black solid line, whereas the left hand side of Eq. (15) is always positive in thegrey-shaded region (b–d). We numerically solve Eqs. (7) with a classical Runge-Kutta scheme in the time interval [0 , T ] with T = 100 setting λ = 100, d = 1, c = 1 . b = 0 . r i = (2 − k )0 . k is the number of effective antibiotics in the respectivelayer), h = 1, s = 0 . α B = γ B = (cid:15) ≥
0. The initial conditions are x (0) = 50, y (0) = 33 . y (0) = y (0) = y (0) = 0. Allcompartments were rescaled according to the shown population sizes. than to develop a broad-spectrum antibiotic B and sell more units. We observe that this critical value is uniquelydetermined, since the left side is strictly decreasing in γ . We discuss conditions for the existence of γ ∗ in the nextsection.Third, we need to make sure that a narrow-spectrum antibiotic B is not used for wild-type bacterial strains (seethe 50/50 treatment strategy III in Sec. IV). Since a narrow-spectrum antibiotic may be also effective for wild-typestrains, the refunding scheme should exclude any incentives to use B in compartment Y . In terms of our refundingscheme, this could be achieved by replacing the 50/50 treatment strategy involving antibiotic B on the left-hand sideof Eq. (15) with the 50/50 treatment strategy involving antibiotic B . Note that the resulting critical value for γ ,which we denote by γ ∗ , is different from γ ∗ . An alternative to imposing this additional constraint on the refundingscheme is to implement strict medical guidelines which demand that less-effective antibiotics should not be used incompartment Y .Together, Eqs. (14) and (15) determine the refunding scheme that ensures that a pharmaceutical company breakseven at time T by developing a narrow-spectrum antibiotic, does not focus on broad-spectrum antibiotics and thenarrow-spectrum antibiotics is not misused once it is developed. E. Numerical example
We now focus on a simple example to illustrate how our refunding scheme can incentivize the development ofnarrow-spectrum antibiotics. For this purpose, we use the parameters listed in Tab. I. To work with reasonablepopulation sizes, we apply our refunding scheme to populations with 50, 100, and 150 million people and rescale the4 quantity symbol value success probability q i K i α p i
100 USDproduction costs per unit v i
70 USDTable I. Refunding calibration parameters. corresponding compartments that we used to determine the antibiotic consumption in Fig. 7.We first determine β ∗ according to Eq. (14) and show the results in Fig. 8 (a). Since the consumption C B decreaseswith (cid:15) (see Fig. 7), the critical refunding parameter β ∗ has to increase with (cid:15) . Before discussing the correspondingvalues of γ ∗ and γ ∗ , we briefly summarize the conditions for the existence of a critical value γ ∗ and distinguish threecases.Case I: If q i a + q i (cid:82) T [ β ∗ + ( p i − v i )] ( f i y + f i y )d t − K i < i = 1 , f ( · ) is chosen according to some treatmentstrategy), we find that Eq. (15) is satisfied for any γ > γ , β ∗ f i y γf i y + f i y = β ∗
11 + γ f i y f i y ≤ β ∗ . (16)Case II: If q i a + q i (cid:82) T [ β ∗ + ( p i − v i )] ( f i y + f i y )d t − K i > q i a + q i (cid:82) T ( p i − v i )( f i y + f i y )d t − K i < γ ∗ > i and corresponding refunding parameters)is equal to zero.Case III: If q i a + q i (cid:82) T ( p i − v i )( f i y + f i y )d t − K i >
0, it is not possible to satisfy Eq. (15) (for B i and correspondingrefunding parameters), since p i − v i is too large.For the parameters of Tab. I, we show the resulting γ ∗ and γ ∗ in Fig. 8 (b–d). We observe that γ ∗ always exists forthe chosen parameters, whereas γ ∗ only exists for certain values of (cid:15) (case II). Case I does not exist in the outlinedexample, since the chosen β ∗ (Eq. (14)) is too large to satisfy Eq. (15) (evaluated for B and corresponding refundingparameters) for any γ >
0. At the boundary separating cases II and III, we find that γ ∗ diverges. Note that the sizeof region associated with case III increases with population size. To avoid such scenarios in real-world applications ofour refunding scheme, we could, for instance, reduce the refunding offset α .To summarize: • For intermediate consumption of B in 50/50 treatment (see treatment strategy III in Sec. IV) and correspondingreturns, a finite γ ∗ exists (see Fig. 8 (b–d)). Within the green-shaded regions of Fig. 8 (b–d), Eq. (15) is satisfiedfor B and B ( γ > γ ∗ and γ > γ ∗ ), whereas the left-hand side of Eq. (15) is positive for B and B within thered-shaded regions ( γ < γ ∗ and γ < γ ∗ ). • Within the orange-shaded regions of Fig. 8 (b–d), either γ > γ ∗ or γ > γ ∗ . • If the expected return associated with the B treatment strategy III of Sec. IV is too large, there is no γ > VI. REFUNDING SCHEMES: GENERAL CONSIDERATIONS
Using a refunding scheme as constructed above in practice requires a series of additional considerations whichwe address in this section. In particular, it must be possible to apply a refunding scheme for any constellation ofantibiotics use, it must work under a variety of sources of R&D uncertainty, and it must still be effective whendiagnostic and treatment uncertainties are taken into account.
A. Generalizations
We first generalize the refunding scheme for possible treatment with more than 2 antibiotics. We assume thatcurrently, N antibiotics are used. Now, N new antibiotics are potentially developed, such that the total number5of antibiotics is N = N + N . Note that before new antibiotics are introduced, there is 1 non-resistant strain anda total of 2 N − N antibioticshas the index ˆ k = 2 N . The generalized refunding scheme still consists of a fixed refund α and a variable refund thatdepends on the use of the antibiotic in the different compartments. The scaling parameter γ ”punishes” the useof the antibiotic for wild-type strains by decreasing the refund. In addition, γ scales the reward of the use of theantibiotic for strains that are resistant to some, but not all, antibiotics currently on the market. Note that γ couldbe negative, such that the refund still increases in the use for partially resistant strains. Lastly, the refund stronglyincreases in the use for fully-resistant strains in the class ˆ k = 2 N .The refunding scheme is given by: g ( ˜f i ) = β (cid:80) N j =2 f j B i y j γ f i y + γ (cid:80) N − j =2 f j B i y j + f ˆ k B i y ˆ k , (17)where ˜f i denotes the vector of the usage of a newly developed drug B i in all compartments Y j with j ∈ { , . . . , N } .The use in each compartment is given by f j B i y j .The break-even conditions can be established as for the 2-antibiotics case, but now with adjusted total consumptionper antibiotic and with the generalized refunding scheme.Similarly to the extension to more than 2 antibiotics, the refunding scheme can be generalized when more than onepharmaceutical company should be given incentives to do R&D on narrow-spectrum antibiotics. In such cases, therefunding parameters have to be adjusted, such that with lower sales volumes for each company, it is still profitableto undertake the R&D investments. B. Multi-dimensional R&D uncertainties
The development and usage of antibiotics is subject to a variety of uncertainties. In particular, companies may notknow at the beginning of a development process against which type of bacterial strains the drug that might emergewill be effective. Such uncertainties can be taken into account as follows: Suppose a pharmaceutical company starts anR&D process for an antibiotic, but does not know initially, whether it will turn out to be broad or narrow-spectrum,as this will only become known during or, in the worst case, at the end of the development process.A possible solution to this issue is setting the value of β equal to β ∗ + δ for some small δ >
0. Moreover, we set γ at the critical value γ ∗ for the value of β ∗ + δ . Then, starting the R&D investment is profitable and the incentivesfor a narrow-spectrum antibiotic are maximal. If during the R&D process, a narrow-spectrum opportunity emerges,it will be chosen, since profits will be higher than for a broad-spectrum antibiotic. However, the company also breakseven for a broad-spectrum antibiotic. Hence, the company faces no additional risk if it is impossible at the beginningto evaluate whether a broad or narrow-spectrum antibiotics will result from the R&D investment. C. Diagnostics and treatment uncertainties
The refunding scheme depends on the ability of doctors to rapidly identify the strain of bacteria that caused acertain infection. For a fraction of such treatments, this may be impossible - in particular in emergency situationsor when rapid, high-throughput diagnostic devices are unavailable. Note that certain bacterial strains can alreadybe identified in a few hours by using peptide nucleic acid (PNA) fluorescent in-situ hybridization (FISH) tests, massspectroscopy, and polymerase chain reaction (PCR)-based methods [14].The refunding scheme can be readily adapted to allow for diagnostic and treatment uncertainties. For instance,one could base refunding only on diagnosed strains of bacteria against which the antibiotics is used. The refundingparameters have to be adapted accordingly. Basing refunding only on cases in which the bacterial strain has beendiagnosed and reported would provide further incentives for pharmaceuticals to develop fast diagnostic tests to identifythe sources of infections.A further refinement would be to provide a refund in case a newly developed antibiotic is used and turns out tobe effective. Such success targeting would be desirable, but may not be easily implementable in practice. As long asthe success rates of an antibiotic that is effective against particular bacterial strains are known or can be estimatedwith sufficient precision, taking the usage/bacterial strain data would be sufficient to provide desirable incentives toengage in R&D for narrow-spectrum antibiotics.6
D. Small firms and the R&D ecosystem
Both small biotech companies and large pharmaceutical companies play a significant role in developing new an-tibiotics. The flexibility, nimbleness, and flat organizational structure of smaller biotech companies that specializein innovative antibacterial treatments can be very effective for the development of new antibiotics. Therefore, whilerefunding will mostly benefit large pharmaceutical companies, the anticipation of such refunds is expected to alsomotivate smaller biotech companies to step up with R&D efforts. These smaller companies can expect significantrewards when they sell or license their patents to larger companies. Moreover, small biotech companies may receivemuch more start-up funding both from venture capitalists and larger pharmaceutical companies, and one might evenconsider using the antibiotics fund for this purpose as well. Hence, it is expected that the refunding scheme will benourishing for the entire ecosystem that develops new antibiotics.
E. The antibiotics fund and participating countries
A necessary condition for the functioning of our refunding scheme is the existence of an antibiotics fund withsufficient equity to cover R&D incentives. Similar to the recently established AMR Action Fund, which aims atbridging the gap between the pipeline for innovative antibiotics and patients, an antibiotics fund should be startedby initial contributions from industry and public institutions. Since it is in the collective self-interest of the pharma-ceutical industry to solve the antibiotics dilemma—as otherwise many other business lines and their reputation willbe harmed—a significant contribution from the industry to set up the antibiotics fund can be expected, as was thecase for the AMR Action Fund.In addition, a continuous refilling of the fund can be achieved by levying a fee on every use of existing antibiotics.These fees have to be set in such a way that the antibiotics fund will never be empty. Since the (sometimes excessive)use of existing antibiotics (e.g., in agricultural settings [42–44]) creates the resistance problem, levying this fee notonly helps to continuously refill the fund, but it may also help to use existing antibiotics cautiously. Ultimately, theantibiotics fund and the refunding scheme are a mechanism to internalize the externality in antibiotics use, namelythe creation of resistant bacteria.As in the context of slowing down climate change, the ideal implementation would involve a global refunding schemeadministered by an international agency, because reducing resistance is a global public good. However, also similarto implementing climate-change policies, worldwide adoption should be extremely difficult and might be impossibleto achieve. As a starting point, a set of industrialized countries should agree to a treaty that fails if any of them doesnot participate. Once an antibiotic fund has been initiated, a treaty should establish the continuing financing of theantibiotics fund and the refunding scheme. The gains would be large and may lead to long-standing self-enforcingincentives to substantially and continuously increase the chances to develop antibiotics against resistant bacteria. Ifattempts to build a larger coalition fail, the European Union or the US could take the lead and become the firstcountry or coalition of countries that implements a refunding scheme for antibiotics.
VII. CONCLUSIONS
The rapid rise of antibiotic resistance poses a serious threat to global public health. No new antibiotic has beenregistered or patented for more than three decades (see Fig. 1). To counteract this situation, we introduced a novelframework to mathematically describe the emergence of antibiotic resistance in a population that is treated with n antibiotics. We then used this framework to develop a market-based refunding scheme that can solve the antibioticsdilemma, i.e. which can incentivize pharmaceutical companies to reallocate resources to antimicrobial drug discoveryand, in particular, to the development of narrow-spectrum antibiotics that are effective against multiresistant bacterialstrains. We outlined how such a refunding scheme can cope with various sources of uncertainty inherent to R&D forantibiotics as well as with diagnostic and treatment uncertainties. ACKNOWLEDGMENTS
LB acknowledges financial support from the SNF Early Postdoc.Mobility fellowship on “Multispecies interactingstochastic systems in biology” and the US Army Research Office (W911NF-18-1-0345). We thank Margrit Buser,Emma Schepers, and Maria R. D’Orsogna for helpful comments and corrections. LB also acknowledges helpful7discussions with Paul Richter. [1] Antimicrobial resistance factsheet, WHO, available at: (2019). Last accessed: July 10, 2020.[2] Laxminarayan, R. et al.
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Treatment with two antibiotics.
We numerically solve Eqs. (A1) and (A3) with a classical Runge-Kuttascheme setting λ = 100, d = 1, c = 1 . b = 0 . r = 0, r = r = 0 . r = 0 .
2. In panel (a), we set h = 0 (i.e., no treatment)and h = 1 in the remaining panels. Panel (b) shows a solution of Eq. (A1) (“single-antibiotic therapy”). The bottom panelsshow solutions of Eq. (A3) (“combination therapy”) with q = 10 − > s = 10 − (c) and q = 10 − < s = 10 − (d). If q > s ,the gain is smaller for multiple treatment. We use P ∗ and G to indicate the total stationary population size (see Eqs. (4) and(A2)) and gain of uninfected in the considered time interval (see Eq. (5)), respectively. The gain G corresponds to the greyshaded region and the characteristic resistance time scale T / is the time when the proportion of completely resistant strainsis 50% (see Eq. (6)). G / is the gain in the time interval [0 , T / ]. The initial conditions are x (0) = 50, y (0) = 33 .
33, and y (0) = y (0) = y (0) = 0. Appendix A: Combination therapy versus targeted use of antibiotics
To illustrate the difference between combination- and single-antibiotic therapy, we first derive the correspondingmathematical results for n = 2 antibiotics { A , B } [25] and discuss the case where n > n = 2 antibiotics, the corresponding sets of antibiotics for the N = 4 infected compartments are A = { A , B } , A = { A } , A = { B } , and A = ∅ . Based on Eq. (1), the treatment of patients with single broad-spectrum antibioticscan be described by: d x d t = − bx ( y + y + y + y ) + r y + r y + r y + r y + h (1 − s ) ( y + y + y ) + λ − dx , d y d t = ( bx − r − h − c ) y , d y d t = ( bx − r − h − c ) y + 12 hsy , d y d t = ( bx − r − h − c ) y + 12 hsy , d y d t = ( bx − r − c ) y + hs ( y + y ) , (A1)0where we set b ij = b , s ij = s , h ij = h , and use the proportions f = f = 1 / f = 0, f = 1, f = f = 0, f = 1, and f = f = 0.In the absence of treatment, the total stationary population is P ∗ = x ∗ + y ∗ = r + cb + λc − db − dr cb . (A2)Analytical expressions for G and T / for some specific parameter constellations are summarized in Ref. [25]. Wecompare the single-antibiotic therapy and targeted use of antibiotics (see Eq. (A1)) with a broad-spectrum treatmentthat uses combinations of antibiotics A and B:d x d t = − bx ( y + y + y + y ) + r y + r y + r y + r y + h (1 − q ) y + h (1 − s ) ( y + y ) + λ − dx , d y d t = ( bx − r − h − c ) y , d y d t = ( bx − r − h − c ) y , d y d t = ( bx − r − h − c ) y , d y d t = ( bx − r − c ) y + hqy + hs ( y + y ) , (A3)where q is the fraction of double resistances that develop from the combined treatment of the wild-type strain ( Y )with antibiotics A and B. In Eq. (A3), we use the proportions f = 1, f = f = 0, f = 1, f = f = 0, f = 1, and f = f = 0.We show a comparison between the outlined single-antibiotic and combination therapy treatment in Fig. A.1. If q > s , we find that, in agreement with earlier results [25], the single-antibiotic treatment outperforms combinationtherapy. For q < s (i.e., for very small probabilities of double resistance resulting from combination treatment ofwild-type strains), single-antibiotic treatment is not as efficient as broad-spectrum therapy anymore. Appendix B: Properties of the general model
In this Appendix, we establish several properties of the general antibiotic-treatment model (see Eq. (1)). Inparticular, in the absence of antibiotic treatment and for sufficiently strong treatment, we show that the mathematicalstructure of the equation describing the stationary population P ∗ is unaffected by the number of antibiotics anddifferences in treatment protocols. The term “sufficiently strong treatment” [25] means that growth factor b/ ( r N + c N )in the completely resistant compartment is larger than the growth factor b/ ( r i + c i + (cid:80) j ∈ A i f ij h ij ) in any othercompartment y i ( i < N ).In the absence of antibiotic therapy (i.e., h ij = 0 for all i , j ), we find that the stationary population of susceptibleindividuals is x ∗ = ( r + c ) /b and y ∗ = λ/c − d/b − ( dr ) / ( b c ).The stationary solution under sufficiently strong treatment with y ∗ N (cid:54) = 0 implies that y ∗ = y ∗ = . . . = y ∗ N − = 0.Independent of treatment protocol details, the stationary solution is x ∗ = ( r N + c N ) /b N and y ∗ N = λ/c N − d/b N − ( dr N ) / ( b N c N ). Even if the stationary behavior has a similar form for general numbers of antibiotics n , the dynamicalfeatures and corresponding characteristics such as T / exhibit a more complex dependence on n , which we analyzenumerically in Fig. B.2.We observe that for the considered parameters in the “ q > s n regime” (see Appendix A), single-antibiotic therapytreatment still outperforms combination treatment. The larger the number of antibiotics n , the smaller relativedifferences between various performance metrics (see insets in Fig. B.2).1 Figure B.2.
Treatment with multiple antibiotics.
We numerically solve the n -antibiotic generalizations of Eqs. (A1) and(A3) with a classical Runge-Kutta scheme in the time interval [0 , T ] with T = 600 and λ = 100, d = 1, c = 1 . b = 0 . r i = ( n − k )0 . k is the number of effective antibiotics in the respective layer), h = 1, s = 10 − , and q = 10 − . n . In panels (a)and (b), we show the total stationary population size P ∗ (see Eqs. (4) and (A2)) and the gain of uninfected G (see Eq. (5)); inpanels (c) and (d), we show the time T / when the proportion of completely resistant strains is 50% (see Eq. (6)) and the gain G / in the time interval [0 , T / ]. The insets in each panel represent the ratio of the single and combination therapy values.The initial conditions are x (0) = 50, y (0) = 33 .
33, and y (0) = y (0) = y4