# Incentivizing hospital infection control

^{a}Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544;^{b}Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544;^{c}Princeton Environmental Institute, Princeton University, Princeton, NJ 08544;^{d}Fogarty International Center, National Institutes of Health, Bethesda, MD 20892;^{e}Center for Disease Dynamics, Economics & Policy, Washington, DC 20036

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Contributed by Simon A. Levin, January 25, 2019 (sent for review July 18, 2018; reviewed by Lauren Ancel Meyers and Timothy C. Reluga)

## Significance

As populations age and hospitalization becomes more common, healthcare-associated infections, including many caused by drug-resistant pathogens, are increasing in importance. Because healthcare facilities share patients, healthcare-associated pathogens spread easily between facilities. Each institution, therefore, has an incentive to free ride on the infection control efforts of other institutions. A feasible policy is to compensate hospitals for infection control. Here we explore the effect of different forms of subsidies on hospitals’ infection control. Under a limited budget, we find that the most effective subsidy is a dollar-for-dollar matching subsidy. Additionally, we show that a subsidy should be preferentially given to hospitals with lower transmission rates. Economic incentives strongly influence hospital behavior and understanding these is a useful complement to epidemiological analysis alone.

## Abstract

Healthcare-associated infections (HAIs) pose a significant burden to patient safety. Institutions can implement hospital infection control (HIC) measures to reduce the impact of HAIs. Since patients can carry pathogens between institutions, there is an economic incentive for hospitals to free ride on the HIC investments of other facilities. Subsidies for infection control by public health authorities could encourage regional spending on HIC. We develop coupled mathematical models of epidemiology and hospital behavior in a game-theoretic framework to investigate how hospitals may change spending behavior in response to subsidies. We demonstrate that under a limited budget, a dollar-for-dollar matching grant outperforms both a fixed-amount subsidy and a subsidy on uninfected patients in reducing the number of HAIs in a single institution. Additionally, when multiple hospitals serve a community, funding priority should go to the hospital with a lower transmission rate. Overall, subsidies incentivize HIC spending and reduce the overall prevalence of HAIs.

Healthcare-associated infections (HAIs) are a serious danger to patient safety (1⇓–3). In the United States roughly 1.7 million patients admitted to hospitals each year contract an HAI (4). HAIs cause increased mortality, longer hospital stays, and a large financial burden for patients and healthcare systems (5). Furthermore, HAIs are linked to the development of antimicrobial resistance in pathogenic bacteria (6).

Patients and healthcare workers act as vectors that transmit infections between healthcare institutions. Since the pathogens responsible for HAIs can persist on body surfaces for long periods of time (1), when patients discharged to the community are rehospitalized, they can spread infection and colonization between hospitals and long-term care facilities (7, 8). Controlling HAIs is especially challenging since many individuals admitted are already colonized from other institutions (9).

A hospital that invests in hospital infection control (HIC) lowers transmission in its own wards, which decreases the basic reproduction number

Policymakers can offer subsidies as an incentive to encourage spending and control the spread of HAIs (14, 15). We develop coupled mathematical models of epidemiology and hospital behavior in a game-theoretic framework to investigate how hospitals may change spending behavior in response to various implementations of subsidies. Our focus is to understand how to use subsidies to incentivize optimal infection control.

## Mathematical Model and Analysis

### Single-Hospital Game.

Consider a single-hospital population admitted from a large catchment population. Assuming HAI transmission occurs only within hospitals, we can approximate the proportion of patients admitted from the community already colonized as a constant value κ. The dynamics of the proportion of patients colonized within the hospital (represented by the variable X) are governed by Eq. **1**:

Infection control lowers the transmission rate of the pathogen, but with diminishing marginal returns. Mathematically, the function

#### Equilibrium analysis and optimization.

A hospital will choose an optimal value h such that the total costs are minimized. If the average hospital stay is short [in the United States it is

If the proportion of community admissions is nonzero, then Eq. **1** has two equilibria: One is negative and the other is strictly positive and therefore the only relevant equilibrium. If

#### Subsidies.

In general, when a subsidy is offered by a policymaker, the total investment in infection control is a function of both the hospital spending h and the subsidy s. The policymaker is concerned solely with reducing the number of people infected, usually given some limited budget. We consider three specific formulations of a subsidy for HIC: a fixed-amount subsidy, a subsidy for each uninfected patient at equilibrium, and a dollar-for-dollar matching subsidy.

Under a fixed subsidy a predetermined amount of money s is given to the hospital to spend on HIC. The total investment in this scenario is then *SI Appendix*, a fixed subsidy will simply replace a hospital’s own optimum spending, even for more general forms of

A subsidy for each uninfected patient at equilibrium has total investment *SI Appendix*) and may result in a negative cost. As formulated here, we assume that the subsidy value is predetermined and reduces a hospital’s total cost after equilibrium has been reached. Redirecting these funds into HIC without causing complicated recursion is better suited for a repeated game, which we do not consider here.

Finally, a dollar-for-dollar matching subsidy implies that for every dollar a hospital spends on HIC, the policymaker will provide s dollars to also be spent on HIC. The total investment is

As can be seen in Fig. 1, a dollar-for-dollar matching subsidy appears to outperform the other subsidy types. The number of patients infected at equilibrium given a limited budget from the policymaker is lower for the matching grant for the parameter range and functional forms considered. For this reason, we focus on this type of subsidy.

We consider a subsidy to be effective if it encourages a hospital to spend more than it would without a subsidy

*Theorem 1* describes the existence criteria for when a matching subsidy will be effective in the extreme case when

### Theorem 1.

*A region of effective subsidies will exist for* *when*

### Proof:

The minimum of the cost function occurs when **2**. □

This existence condition can be rearranged to

A closed-form analytical solution is no longer possible when

In general, a hospital may be “free riding” if a subsidy is not effective since it lowers its own spending as more subsidy is offered. Fig. 2, *Top* and *Bottom* shows that behavior may differ between a “free rider”—a hospital which does not increase spending when others do—and a “cooperator”—a hospital that increases spending when others do. In the language of game theory and economics, for a free rider a subsidy is a strategic substitute for the hospital’s own HIC resources and the institution is displaying policy resistance. For a cooperator a subsidy is a strategic complement and the institution is displaying policy reinforcement.

The strongest characteristic influencing into which category a hospital falls is the intrinsic transmissibility. This attribute may be associated with hospital size, ward size, or hospital type. However, the proportion colonized in the community, κ, also plays a role in determining hospital behavior. Hospitals with low intrinsic transmissibility may act as cooperators if the proportion admitted colonized is high.

### Multiinstitutional Games.

The single-hospital case offers a stylized model to understand how identical hospitals all receiving subsidies may interact. We then explore a two-player game as a case study in subsidy allocation.

Let X and Y be the proportions colonized in a focal hospital and all other hospitals, respectively, such that the total number of hospitals is n. Z replaces the constant κ as a variable describing the proportion colonized in the community. Patients are still discharged into a common catchment population, but are admitted to each institution at a rate r. The dynamics for hospitals are the same as in the single-hospital case; however, each has its own transmission function and choice of amount to invest. Differences between hospitals could be defined in multiple different ways, i.e., different sizes (different contributions to the community) or different propensity to admit from a recently hospitalized population (primary vs. tertiary care). Having nonidentical transmission functions provides an easy way to describe these differences and is analogous to having different values of

The system of differential equations describing the situation is

The dynamics are now more complicated and thus we use a cumulative cost function of the form below,**9**, given the strategy

When two heterogeneous hospitals, i.e., one with low intrinsic transmissibility and one with high intrinsic transmissibility, play against each other, their responses to the other hospital’s actions will be very different. Fig. 3 shows the response curves for three different subsidy values laid over a heat map of the resulting equilibrium of infection. Nash equilibria occur at the intersection of the response curves. By comparing the level of infection at the Nash equilibria, we note that if a subsidy can be offered to only one hospital, the optimal solution for reducing infection is to offer it to the hospital with the lower transmission rate. We assume that any combination of matching grant values can be offered to this pair of hospitals.

When the hospitals have identical transmission functions, the response curves are entirely symmetrical. When hospitals have the same transmission rate, the best outcome is obtained by offering the entire subsidy to a single hospital rather than splitting it equally between the two.

The number of infected patients at Nash equilibria spending for a range of subsidy values is shown in Fig. 4. The axes are the total amount of money a public health authority would provide to either hospital in the same heterogeneous case as in Fig. 3. The asymmetry of the heat map shows that the lowest prevalence appears to be obtained when more money is given to the hospital with lower transmissibility (predisposed to free riding). While some benefit is gained no matter to whom the subsidy is given, the worst outcome seems to occur when each one receives similar amounts of incentive. In the case of a limited budget, sloping downward diagonal lines in Fig. 4 denote equal total amounts of money spent by the public health authority (the sum of money given to each hospital). The point at which infection is lowest is along the axis corresponding to most of the budget being devoted to the hospital with low intrinsic transmission.

## Discussion

In this paper, we explored how public subsidies for HIC could alter the behavior of hospitals and how best to deploy limited subsidy resources.

We considered three types of subsidies: a subsidy for each uninfected patient, a fixed subsidy, and a matching subsidy. Assuming a fixed level of subsidy resources, the least effective at reducing infection was a subsidy tied to the number of uninfected patients at equilibrium, which is equivalent to a tax on infected patients. A fixed subsidy was moderately effective; however, it simply replaces a hospital’s own spending. The fixed subsidy has no effect on the transmission of HAIs until the subsidy contributes all of the HIC spending and the hospital invests zero. The matching subsidy was the best at reducing infection for the same cost to the policymaker. Furthermore, we derived a theoretical rule for when a matching subsidy will be effective. For hospitals with high transmission and/or high levels of patients being admitted already infected, a matching subsidy will result in a hospital increasing its own spending.

In our study, we observed two kinds of behaviors on the part of hospitals: Free riders spend less when offered funding and cooperators spend more when offered funding. An intuitive approach to deploying scarce subsidy resources might be to offer encouragement to a hospital that increases spending when others do. However, the results of our mathematical model suggest that in a two-hospital system incentivizing the institution with a lower transmission rate is better for the overall outcome of the system. Since this two-player game acts as a case study in subsidy allocation, future work should explore the nuances of more complex network combinations of n hospitals with various transmission functions. Furthermore, an optimal control model may be able to tease apart the feedbacks between local and global infection control efforts as well as the differences between optimal and suboptimal spending conditions.

Previous work has explored similar epidemiological optimal allocation problems in metapopulations with varying conclusions (20, 21). In a two-patch susceptible-infected-susceptible system with an annual budget for treatment it may be optimal under certain conditions to devote the entire budget to the least-infected group (22⇓–24). A similar theoretical question is addressed by Klepac, Laxminarayan, and Grenfell (25) with respect to vaccine allocation between two patches. In their model a limited budget should initially be allocated to the patch closest to the herd immunity threshold.

In the context of this paper, the group that has the lower transmission rate should receive the incentive. This group is closest to eradicating the infection and significantly reduces the proportion of patients infected in the community. In effect, this is focusing on reducing the local infection in a single patch to reduce the total global prevalence. Individuals colonized in the community play a large role in the transmission dynamics and so reducing the community prevalence is most effective at reducing the global prevalence. If the money from a subsidy was split, then there would not be as strong an impact on patients admitted already colonized. Thus, the marginal return on an additional subsidy dollar is much higher for a lower transmission rate.

We have made two strong assumptions to support analytic tractability. First, hospitals are identical in all ways other than their transmission rate as a function of investment. Moreover, within a hospital we assume the patient population is well mixed. The differences in transmission rate between institutions may be a reflection of hospital size or function (such as tertiary care facilities); however, they cannot reflect nonrandom admittance from the catchment population. If patients are more likely to be admitted to one hospital over another, that hospital may contribute disproportionately to colonization. Implementing any of these heterogeneities is important for future work. We predict that such an extension will not qualitatively change our results and that the subsidy would still be allocated to a single institution. Our second major assumption is that hospitals must choose a level of investment which cannot be changed once implemented. Future work could extend this model to an optimal control problem that may determine the most effective investment as a function of time.

Unfortunately, we lacked data to confirm how hospital spending decisions are made. Such data could reveal whether there are changes in hospital budgeting when comparing those with different financial incentives or community structure. Our model predicts that rational actors would respond optimally when presented with incentives; however, there may be other factors at play in decision making.

Our results are applicable not only to individual hospitals. Large multifacility systems managed by healthcare corporations could also benefit from considering the behavior change resulting from infection control incentives. Additionally, this type of framework could be applied to consider vaccination or treatment in a metapopulation.

Within the overlap of epidemiology and economics, this research provides another example of the way in which incentives can alter behavior (sometimes in unexpected ways) and fundamentally change the outcome of an epidemic. More broadly we offer a model that can guide policymakers in enacting subsidy programs which result in encouraging investment in hospital infection control.

## Acknowledgments

The authors thank B. Morin and V. Vasconcelos for their helpful feedback during the analysis. The reviewers greatly improved the quality of the manuscript. R.L. was supported by Grant IPA 16IPA1609427 from the US Centers for Disease Control and Prevention and a US Health Policy grant from the Center for Health and Wellbeing at Princeton University.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: sedrohan{at}princeton.edu, slevin{at}princeton.edu, or ramanan{at}cddep.org.

Author contributions: S.E.D., S.A.L., B.T.G., and R.L. designed research; S.E.D. performed research; and S.E.D. and R.L. wrote the paper.

Reviewers: L.A.M., University of Texas at Austin; and T.C.R., The Pennsylvania State University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1812231116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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