The Basic Principles
To overcome the antibiotics dilemma and associated complications, we propose a refunding scheme to incentivize the development and appropriate use of antibiotics. The main properties of the refunding scheme are as follows:
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1.
An antibiotics fund should be started with initial contributions from industry and public institutions, similar to the recently established AMR Action fund. In addition, all antibiotic use is charged with a small fee which is channeled continuously into the antibiotics fund.
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2.
Firms that develop new antibiotics obtain a refund from the fund.
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3.
The refund for a particular antibiotic is calculated with a formula that satisfies the following three properties:
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There is a fixed payment for the successful development of an antibiotic, i.e., an antibiotic that is approved by 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 Kremer (1998), as it is equivalent to an advanced market commitment. Pharmaceutical companies know that once a patent for a new antibiotic is awarded, they will be reimbursed part of their development costs.
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The refund is strongly increasing with the use of the new antibiotic for currently resistant bacteria, compared to other newly developed antibiotics for this purpose. This part is the resistance premium.
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The refund is declining in the use of the antibiotics for non-resistant bacteria, compared to other antibiotics used for this purpose. This part is the non-resistance penalty.
The objective of our refunding scheme is to financially incentivize pharmaceutical companies to reorient their R &D efforts and sales strategy toward narrow-spectrum antibiotics, using a minimum-size antibiotics fund. As we will demonstrate below, all above elements (1–3) are necessary to achieve this purpose.
Several remarks are in order to summarize the application areas of refunding schemes and the challenges arising in the context of antibiotic resistance. First, refunding schemes are widely discussed in the environmental literature. These schemes are meant to provide incentives for firms to reduce pollution (Gersbach and Winkler 2012). Second, simple forms of refunding schemes could also be used in other contexts where pharmaceutical companies have only little financial interest in investing in drug research, due to potentially low sales volumes. This is, for instance, the case for orphan drug development and vaccine research for viral infections, including SARS and Ebola, or enduring epidemic diseases as described by Bell and Gersbach (2009). However, for such cases, refunding schemes are much easier to construct, since they can solely rely on the usage, e.g. the number of vaccinated individuals. For antibiotics—because of the antibiotics dilemma—one has to construct new types of refunding schemes with “sticks and carrots”: The carrots for using the antibiotic against bacterial strains resistant against other antibiotics and the sticks for using the antibiotics against wild-type strains. This stick-and-carrot complication does not arise in the context of the aforementioned (simple) refunding schemes.
Refunding Schemes for Two Antibiotics
We illustrate the working of the refunding scheme with a simple model. It includes two elements:
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There is a fixed amount, denoted by \(\alpha \), which a pharmaceutical company obtains if it successfully develops a new antibiotic \(\mathrm {B}_i\), i.e. an antibiotic approved by a public health authority.
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There is a variable refund that is determined by the following refunding function:
$$\begin{aligned} g(f_{1 \mathrm {B}_i}y_1, f_{3\mathrm {B}_i}y_3) = \beta \frac{f_{3\mathrm {B}_i}y_3}{\gamma f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3}, \end{aligned}$$
(6)
where \(i\in \{1,2\}\) (to represent antibiotics \(\mathrm {B}_1\) and \(\mathrm {B}_2\)) and \(\beta \) and \(\gamma \) are scaling parameters, with \(\beta \) being a large number and \(\gamma \) satisfying \(\gamma \ge 0\). The parameter \(\beta \) determines the refund per drug unit and \(\gamma \) controls the non-resistance penalty. The refunding function \(g(f_{1\mathrm {B}_i}y_1, f_{3\mathrm {B}_i}y_3)\) measures the relative use of the new antibiotic in compartment 3 (A-resistant strains) compared to the total use of the antibiotic. The use of the antibiotic in the wild-type compartment is weighted by the parameter \(\gamma \). Note that the refunding function g satisfies the following properties.
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It is bounded according to \(0 \le g(f_{1\mathrm {B}_i}y_1, f_{3\mathrm {B}_i}y_3) \le \beta \).
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It is increasing in the use for \(\mathrm A\)-resistant bacteria in comparison with other newly developed antibiotics used for this purpose: \(f_{3\mathrm {B}_i}y_3\).
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It is declining in the use of antibiotics for non-resistant bacteria in comparison with other antibiotics used for this purpose: \(f_{1\mathrm {B}_i}y_1\).
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It reaches a maximum if the antibiotic is only used to treat A-resistant strains and 0 if it is only used for non-resistant strain treatment.
Note that our refunding scheme uses three free parameters \(\alpha \), \(\beta \), and \(\gamma \). We will show in the subsequent section that all three parameters are necessary to achieve the objective of the refunding scheme.
The total refund that a successful pharmaceutical company receives in the time interval [0, T] for developing an antibiotic \(\mathrm {B}_i\) is given by
$$\begin{aligned} R_i(T) :=\alpha +\int _0^{T} \beta \frac{f_{3\mathrm {B}_i}y_3 (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3)}{\gamma f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3}\, \mathrm {d}t. \end{aligned}$$
(7)
For \(\gamma =1\), the refund is solely determined by \(f_{3 \mathrm {B}_i}y_3\) and the use of antibiotics in compartment 1 is irrelevant for the refund. For \(\gamma >1\), the use of antibiotics in compartment 1 decreases the refund, and thus the use of the antibiotics for non-resistant bacteria is penalized. As we will see in our numerical examples, for small values of \(\epsilon \) (see Eq. (1)), such penalties may not always be needed, but we certainly need them for higher values of \(\epsilon \).
Incentivizing Development
Table 1 Overview of the main refunding scheme parameters. The values that are listed in the last column are used to perform a calibration of the refunding scheme in Sect. 3.5. We use the same parameters for antibiotics \(\mathrm {B}_1\) and \(\mathrm {B}_2\) (i.e., for \(i=1,2\))
We next focus on how our refunding scheme can incentivize a pharmaceutical company to invest in R &D for new antibiotics and in particular for new narrow-spectrum antibiotics. We assume that the pharmaceutical company makes a risk-neutral evaluation of such R &D investments.Footnote 6 For this purpose, we first consider the situation without refunding. For simplicity, we neglect discounting. Then, without refunding (i.e., without \(R_i(T)\)), the net profit of the company under consideration that invests into the development of an antibiotic \(\mathrm {B}_i\) is
$$\begin{aligned} \pi _i = q_i (p_i-v_i) \int _0^T \frac{\mathrm {d} C_{\mathrm {B}_i}}{\mathrm {d}t}\, \mathrm {d}t - K_i = q_i (p_i-v_i) \int _0^{T} (f_{1 \mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3)\, \mathrm {d}t - K_i,\nonumber \\ \end{aligned}$$
(8)
where \(K_i\) denotes the total development costs of \(\mathrm {B}_i\), and \(q_i\) is the probability of success when the development is undertaken. Moreover, \(p_i\) is the revenue per unit of the antibiotic used in medical treatments for the pharmaceutical company under consideration, and \(v_i\) are the production costs per unit. An overview of the main parameters used in our refunding scheme is provided in Table 1.
Note that in our example with two antibiotics, \(f_{3 \mathrm {B}_i}=1\), since only drug \(\mathrm {B}_i\) can be used against A-resistant strains. We assume that without refunding, \(\pi _i\) is (strongly) negative, because of high development costs \(K_i\) and low success probabilities \(q_i\). The task of a refunding scheme is three-fold: First, it has to render the development of new antibiotics commercially viable. Second, it has to render the development of narrow-spectrum antibiotics against resistant bacteria strains more attractive than the development of broad-spectrum antibiotics. Third, if a narrow-spectrum antibiotic is developed that is also effective against wild-type strains, but less so than others, the refunding scheme should make its 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 \(\pi _i\) becomes zero and investing into antibiotics development just becomes commercially viable. We assume that the pharmaceutical company continues to receive \(p_i\) per unit of the antibiotic sold.Footnote 7
The general break-even condition for a newly developed antibiotic \(\mathrm {B}_i\) is
$$\begin{aligned} \begin{aligned} K_i&= q_i (p_i-v_i) \int _0^{T} (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3) \, \mathrm {d}t + q_i R_i(T)\\&= \alpha q_i + q_i\int _0^{T} \left[ \beta \frac{f_{3\mathrm {B}_i}y_3}{\gamma f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3} + (p_i-v_i)\right] (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3) \, \mathrm {d}t. \end{aligned} \end{aligned}$$
(9)
Clearly, refunding increases the profits from developing new antibiotics since \(R_i(T)>0\). There are many combinations of the refunding parameters \(\alpha \), \(\beta \), and \(\gamma \) that can achieve this break-even condition. However, and more subtly, the refunding has to increase the incentives for the development of narrow-spectrum antibiotics more than those for broad-spectrum antibiotics. This can be achieved by an appropriate choice of the scaling parameter, as we will illustrate next.
In Eq. (9), we have (implicitly) assumed that there is a life-time T for the drug and that the company wants to achieve break-even over that period. There are two caveats to this assumption.
First, some (smaller) biotech companies cannot raise enough capital in the market to finance the initial development, as financiers prefer immediate over future rewards. Hence, such companies would need to achieve profits above break-even levels in order to be attractive for investors, as the investment is long term.
Second, we have neglected many sources of uncertainty about the future revenues the new antibiotic will generate, such as uncertainties about prices, volume, life time (including new antibiotics produced by competitors), and production costs. Such uncertainties will typically call for additional risk premia that have to be added to the break-even condition. Or, in other words, the break-even condition in expected terms has to be achieved in a shorter time period. Typically, such time periods can be in the range of five to ten years or a bit more, but not much longer.
Critical Conditions for Refunding Parameters
To derive the critical refunding parameters, we assume that the parameter \(\alpha \), which satisfies \(0< \alpha < K_i\), is given, and thus, a fixed share of the R &D costs is covered by the antibiotics fund. We also assume that \(\alpha +\pi _i<K_i\), where \(\pi _i\) is the profit without refunding. Based on the break-even condition (see Eq. (9)), we obtain the following general condition that the parameters \(\beta \) and \(\gamma \) have to satisfy:
$$\begin{aligned} \beta = \frac{K_i-\alpha q_i-q_i(p_i-v_i)\int _0^T(f_{1 \mathrm {B}_i}y_1 + f_{3 \mathrm {B}_i}y_3)\mathrm {d}t}{q_i\int _0^T \frac{f_{3 \mathrm {B}_i}y_3}{\gamma f_{1 \mathrm {B}_i}y_1 + f_{3 \mathrm {B}_i}y_3}(f_{1 \mathrm {B}_i}y_1+f_{3 \mathrm {B}_i}y_3)\mathrm {d}t}. \end{aligned}$$
(10)
The goal of our refunding scheme is to incentivize pharmaceutical companies to produce narrow-spectrum antibiotics \(\mathrm {B}_2\) that are only used against currently resistant strains (see treatment IV in Sect. 2). Thus, the refunding scheme has to satisfy two conditions: first, with the development of antibiotic \(\mathrm {B}_2\), the company achieves break-even. Second, developing antibiotic \(\mathrm {B}_1\) is not attractive, i.e. the profit is negative. To satisfy the first condition, we use Eq. (10) and obtain the optimal refund per unit
$$\begin{aligned} \beta ^*= \frac{K_2-\alpha q_2-q_2(p_2-v_2)\int _0^Tf_{3 \mathrm {B}_2}y_3 \, \mathrm {d}t}{q_2\int _0^T f_{3 \mathrm {B}_2}y_3 \, \mathrm {d}t}, \end{aligned}$$
(11)
where we used that \(f_{1 \mathrm {B}_2}=0\) (see Sect. 2). To achieve negative profit for using \(\mathrm {B}_1\), we need to choose the parameter \(\gamma \) such that developing a broad-spectrum antibiotic \(\mathrm {B}_1\) and applying it in compartments \(Y_1\) and \(Y_3\) (see treatment I in Sect. 2) is not more attractive than developing a narrow-spectrum antibiotic \(\mathrm {B}_2\) according to treatment IV (see Sect. 2). Thus, the refunding scheme needs to satisfy
$$\begin{aligned} \alpha q_1 + q_1\int _0^{T} \left[ \beta ^*\frac{f_{3\mathrm {B}_1}y_3}{\gamma f_{1\mathrm {B}_1}y_1+f_{3\mathrm {B}_1}y_3} + (p_1-v_1)\right] (f_{1\mathrm {B}_1}y_1+f_{3\mathrm {B}_1}y_3) \, \mathrm {d}t - K_1 < 0.\nonumber \\ \end{aligned}$$
(12)
If we evaluate Inequality (12) as an equality, we obtain a critical value for \(\gamma \) (i.e., a critical non-resistance penalty), denoted by \(\gamma ^*\), for certain values of \(\beta ^*\), \(p_1\), and \(v_1\). For \(\gamma >\gamma ^*\), Inequality (12) holds. Inequality (12) together with Eq. (11) imply that it is more profitable to produce a narrow-spectrum antibiotic \(\mathrm {B}_2\) and obtain a higher refund than to develop a broad-spectrum antibiotic \(\mathrm {B}_1\) and sell more units. We observe that this critical value is uniquely determined, since the left side is strictly decreasing in \(\gamma \). We discuss conditions for the existence of \(\gamma ^*\) in the next section.
Third, we need to make sure that a narrow-spectrum antibiotic \(\mathrm {B}_2\) is not used for wild-type bacterial strains (see the 50/50 treatment III in Sect. 2). Since a narrow-spectrum antibiotic may also be effective against wild-type strains, the refunding scheme should exclude any incentives to use \(\mathrm {B}_2\) in compartment \(Y_1\). In terms of our refunding scheme, this could be achieved by replacing the 50/50 treatment involving antibiotic \(\mathrm {B}_1\) on the left-hand side of Eq. (12) with the 50/50 treatment involving antibiotic \(\mathrm {B}_2\). Note that the resulting critical value for \(\gamma \), which we denote by \(\gamma ^{**}\), is different from \(\gamma ^*\). An alternative to imposing this additional constraint on the refunding scheme is to implement strict medical guidelines which demand that less-effective antibiotics should not be used in compartment \(Y_1\).
Together, Eqs. (11) and (12) determine the refunding scheme that ensures that a pharmaceutical company breaks even at time T after developing and effectively using a narrow-spectrum antibiotic, without (primarily) focusing on the development of broad-spectrum antibiotics.
Numerical Example
We now focus on an example to illustrate how our refunding scheme can incentivize the development of narrow-spectrum antibiotics. For this purpose, we use the parameters listed in the last column of Table 1. To work with reasonable population sizes, we apply our refunding scheme to populations with 50, 100, and 150 million people and rescale the corresponding compartments that we used to determine the antibiotic consumption in Fig. 6.
We first determine the critical refund per unit \(\beta ^*\) according to Eq. (11) and show the results in Fig. 7 (a). Since the consumption \(C_{\mathrm {B}_2}\) decreases with \(\epsilon \) (see Fig. 6), the critical refunding parameter \(\beta ^*\) has to increase with \(\epsilon \). Before discussing the corresponding critical broad-spectrum penalties \(\gamma ^*\) and \(\gamma ^{**}\), we briefly summarize the conditions for their existence and distinguish three cases.
- Case I:
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If \(q_i a+ q_i \int _{0}^T \left[ \beta ^*+(p_i-v_i) \right] (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3) \mathrm {d}t - K_i< 0\) (\(i=1,2\) and \(f_{(\cdot )}\) is chosen according to some treatment protocol), we find that Eq. (12) is satisfied for any \(\gamma > 0\), independent of the underlying refunding scheme since, for finite \(\gamma \),
$$\begin{aligned} \beta ^*\frac{f_{3\mathrm {B}_i}y_3}{\gamma f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3}=\beta ^*\frac{1}{1 + \gamma \frac{f_{1\mathrm {B}_i}y_1}{f_{3\mathrm {B}_i}y_3}} < \beta ^*. \end{aligned}$$
(13)
- Case II:
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If \(q_i a+ q_i \int _{0}^T \left[ \beta ^*+(p_i-v_i) \right] (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3) \mathrm {d}t - K_i > 0\) and \(q_i a+ q_i \int _{0}^T (p_i-v_i) (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3) \mathrm {d}t - K_i < 0\), there exists a \(\gamma > 0\) such that the left-hand side of Eq. (12) (for \(\mathrm {B}_i\) and corresponding refunding parameters) is equal to zero.
- Case III:
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If \(q_i a+ q_i \int _{0}^T (p_i-v_i) (f_{1\mathrm {B}_i}y_1+f_{3\mathrm {B}_i}y_3) \mathrm {d}t - K_i > 0\), it is not possible to satisfy Eq. (12) (for \(\mathrm {B}_i\) and corresponding refunding parameters), since \(p_i-v_i\) is too large.
For the parameters of Table 1, we show the resulting values of \(\gamma ^*\) and \(\gamma ^{**}\) as a function of \(\epsilon \) in Fig. 7 (b–d). We observe that \(\gamma ^*\) and \(\gamma ^{* *}\) always exist for the chosen parameters (case II). Case I does not exist in the outlined example, since \(\beta ^*\) (Eq. (11)) is large enough. For the chosen values of \(p_i\) and \(v_i\), we do not observe case III in Fig. 7 (b–d) either. In real-world applications of our refunding scheme, one can always avoid case III by, for instance, reducing the refunding offset \(\alpha \).
To summarize:
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For intermediate consumption of \(\mathrm {B}_2\) in 50/50 treatment (see treatment III in Sect. 2) and corresponding returns, a finite \(\gamma ^{**}\) exists (see Fig. 7 (b–d)). Within the green-shaded regions of Fig. 7 (b–d), Eq. (12) is satisfied for \(\mathrm {B}_1\) and \(\mathrm {B}_2\) (\(\gamma > \gamma ^*\) and \(\gamma > \gamma ^{**}\)), whereas the left-hand side of Eq. (12) is positive for \(\mathrm {B}_1\) and \(\mathrm {B}_2\) within the red-shaded regions (\(\gamma < \gamma ^*\) and \(\gamma < \gamma ^{**}\)).
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Within the orange-shaded regions of Fig. 7 (b–d), either \(\gamma > \gamma ^*\) or \(\gamma > \gamma ^*\).
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If the expected return associated with the \(\mathrm {B}_2\) treatment III of Sect. 2 is too large, there is no \(\gamma >0\) that discourages pharmaceutical companies from developing such drugs.
Generalizations
We now generalize the refunding scheme of the previous sections to account for possible treatment options with more than two antibiotics. We assume that \(N_1\) antibiotics are used currently and that \(N_2\) new antibiotics are developed, such that the total number of (potential) antibiotics is \(N=N_1+N_2\). For the outlined scenario, the underlying resistance dynamics are described by the general antibiotic resistance model in Appendix A. Note that before new antibiotics are introduced, there is at least 1 non-resistant strain and up to \(2^{N_1}-1\) strains that are resistant to some antibiotic. Furthermore, there is one class of bacterial strains that is resistant to all antibiotics currently on the market. The class of microbes that is resistant to all N antibiotics has the index \({\hat{k}}=2^N\). The generalized refunding scheme still consists of a fixed refund \(\alpha \) and a variable refund that depends on the use of the antibiotic in different compartments. The scaling parameter \(\gamma _1\in [0,\infty )\) “punishes” the use of the antibiotic for wild-type strains by decreasing the refund. In addition, \(\gamma _2\) scales the reward of the use of the antibiotic for strains that are resistant to some, but not all, antibiotics currently on the market. Note that \(\gamma _2\) could be negative, such that the refund still increases in the use for partially resistant strains. Lastly, the refund strongly increases in the use for fully resistant strains in the class \({\hat{k}}=2^N\).
The generalized refunding scheme is given by
$$\begin{aligned} g(\tilde{\mathbf {f}_i}) = \beta \frac{\sum _{j=2}^{2^N}f_{j \mathrm {B}_i}y_j}{\gamma _1f_{1 \mathrm {B}_i}y_1+\gamma _2\sum _{j=2}^{2^N-1}f_{j \mathrm {B}_i}y_j + f_{{\hat{k}} \mathrm {B}_i}y_{{\hat{k}}}}, \end{aligned}$$
(14)
where \(\tilde{\mathbf {f}}_i\) denotes the vector of the usage of a newly developed drug \(\mathrm {B}_i\) in all infected compartments \(Y_j\) with \(j \in \{1,\dots ,N\}\) (notation as defined in Appendix A). The use of antibiotics in infected compartment j is \(f_{j \mathrm {B}_i}y_j\).
The break-even conditions can be established as for the model with two antibiotics, but now with adjusted total consumption per antibiotic and with the generalized refunding scheme.
Similar to the extension to more than two antibiotics, the refunding scheme can be generalized when more than one pharmaceutical company should be given incentives to pursue R &D on narrow-spectrum antibiotics. In such cases, the refunding parameters have to be adjusted, such that with lower sales volumes for each company, it is still profitable to undertake R &D investments.