Ranking disease control strategies with stochastic outcomes

Preventive Veterinary Medicine 176 (2020) 104906

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Ranking disease control strategies with stochastic outcomes a,

b

a

c

L.J. Verteramo Chiu *, L.W. Tauer , Y.T. Gröhn , R.L. Smith

T

a

Department of Population Medicine and Diagnostic Sciences, Cornell University College of Veterinary Medicine, Ithaca, NY, 14850, USA Charles H. Dyson School of Applied Economics and Management, Cornell SC Johnson College of Business and College of Agriculture and Life Sciences, Cornell University, Ithaca, NY, 14850, USA c Department of Pathobiology, University of Illinois, College of Veterinary Medicine, Urbana, IL 61802, USA b

ARTICLE INFO

ABSTRACT

Keywords: Control ranking Decision making under risk Stochastic outcomes

This paper explains how the methodologies of first and second order stochastic dominance, and expected utility using specific risk preferences, can be applied to epidemiology when choosing among control strategies that have stochastic outcomes. We provide a step-by-step guide on how epidemiologists can rank a number of control strategies based on their distribution of estimated benefits. We also explain how the expected utility model and decision maker’s risk preferences can be used to select between outcomes when none stochastically dominates. To illustrate these techniques, we show the ranking of various control strategies for a dairy herd endemically infected with Mycobacterium avium subs. paratuberculosis (MAP) and mastitis, and explain how decision maker’s risk preferences affect the ranking.

1. Introduction In epidemiologic research we are often faced with the problem of ranking competing disease control strategies. When outcomes are nonrandom (deterministic), the single outcome value is sufficient to rank strategies based upon ordering; however, the assumption of non-randomness does not reflect reality. Factors that contribute to the randomness in epidemiology studies include: imperfect knowledge of the disease, multifactorial etiology, and multiplicity of effects (Dohoo et al., 2003). Parameter values, including those from epidemiological modelling like disease transmission and progression rates, are estimates from an underlying distribution (for a comprehensive explanation see Dohoo et al., 2003). The outcome of a model with random parameters can be thought of as a realization of a random process. When many realizations of the outcome are generated, we end up with a distribution of outcomes, one distribution for each epidemiologic control strategy. Among the methods used to choose strategies with stochastic outcomes, the expected utility framework is widely used in many disciplines. The expected utility framework assumes that the decision maker maximizes their utility. Despite its popularity, individuals’ behavior may not always follow the expected utility model.1 In some

instances, individuals may want to minimize the probability of an outcome falling below a specified value, also known as Safety First criterion (Roy, 1952), or maximize the highest (Maximax) or lowest (Maximin) outcome (or similarly, or maximize the output value for a specified percentile of the distribution instead of the highest and lowest values). These decision criteria do not utilize the expected utility approach. The Safety First criterion may be applicable when the cost of a bad outcome is large, and the only criterion is minimizing the likelihood of a bad outcome. For example, it might be desirable to choose the investment that minimizes the probability of bankruptcy given a minimum return. Maximax criterion can be applied when the decision maker is optimistic and the probabilities of the outcomes do not matter, only the maximum possible outcome value. Maximax behavior is expressed in countries that borrow to invest in large scale infrastructure projects (Haddad, 2005). Maximin criterion also does not consider probabilities of the outcome, but minimizes the worse possible outcome (Rawls, 1971). Vietnam’s move away from centralized institutions is viewed consistent with the maximin principle (Haddad, 2005). Although these various decision criteria may be appropriate in certain circumstances, we are interested in demonstrating how a risk averse individual can rank distributions under general specifications of the utility function properties, consistent with the expected utility

Corresponding author at: VT 701, College of Veterinary Medicine, Cornell University, Ithaca, NY 14853, USA. E-mail address: [email protected] (L.J. Verteramo Chiu). Expected utility theory assumes perfect information in the distribution of the outcomes. However, perfect information, or estimation, on the potential outcomes may not be possible. It is generally defined that the situation where the outcome probabilities cannot be estimated is called ‘uncertainty’, whereas the probabilistic estimate is called ‘risk’ (Knight, 1921). An alternative theory to expected utility explaining behavior under risk and uncertainty is Prospect Theory (Kahneman and Tversky, 1979). ⁎

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https://doi.org/10.1016/j.prevetmed.2020.104906 Received 2 October 2019; Received in revised form 6 December 2019; Accepted 24 January 2020 0167-5877/ © 2020 Elsevier B.V. All rights reserved.

Preventive Veterinary Medicine 176 (2020) 104906

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paradigm. Ranking competing strategies with stochastic outcomes depends on the risk preferences of the decision maker (Varian, 1992). A more risk averse individual may choose a strategy different from the strategy chosen by an individual less risk averse. There are decision techniques which can be used with stochastic outcomes, and the technique selected depends upon the setting of the problem and information on the decision maker. Consequently, the purpose of this paper is to demonstrate how to rank stochastic outcomes using the stochastic dominance criterion of first degree stochastic dominance and second degree stochastic dominance. We use the stochastic outcomes generated by Verteramo Chiu et al. (2019) and compare the distributions of net present values (NPV) simulated from applying stochastic dominance and alternative selection criteria on control strategies in a dairy herd endemically infected with MAP and clinical mastitis. We assume that the decision maker has complete information on the outcome distribution of the control strategies. Strengths and weaknesses of the stochastic decision criteria are examined. We compare stochastic dominance with direct expected utility maximization where specific risk preferences of the decision maker are known.

first and second derivatives U (x ) 0 and U (x ) 0 . These properties state that utility is monotonically increasing (U (x ) 0 ) and concave in (U (x ) 0) : larger x produces greater utility, with decreasing marginal utility. The utility from the last unit of x decreases as total x increases. The increasing and concave properties of the utility function defines the owner of that utility function as risk averse. The degree of aversion to risk depends upon the specific functional form and parameters of the utility function. Some utility functions commonly used are the log function, U (x ) = ln(x ) ; and the exponential function, U (x ) = 1 e x , where λ is the Arrow-Pratt risk aversion coefficient (explained later). Using the log function, the utility of $1,000 of wealth is 6.9, and the utility of $1,100 is 7, which illustrates that utility increases with wealth. The incremental utility of the additional $100 is 0.1, and the marginal utility is 0.001, since each additional dollar of wealth increases utility by 0.001 at an initial wealth of $1,000. Similarly, the incremental utility of additional $100 of wealth when the initial wealth is $2,000 is lower at 0.05, exhibiting decreasing marginal utility. 2.3. Expected utility Because the outcome distribution of each control strategy is unobserved, it is estimated. When outcome is stochastic, as is usually the case, the sum of all utility values derived from each unique outcome realization weighted by its probability of occurring can be used to determine the preferred control. This is called the expected utility of a control strategy, and states that a decision maker behaves as if maximizing the expected value of a utility function over a set of possible outcomes. The limitation of directly calculating expected utility is that the specific utility function for the decision maker must be known, and that is normally not the case. Given control outcomes NPVa and NPVb, option a is preferred over option b if

2. Materials and methods In order to understand how decision makers with different risk preferences may rank control strategies differently, we will first review some preliminary concepts of NPV, utility, expected utility, risk, and risk aversion. We will then define and explain first and second degree stochastic decision criteria. 2.1. Net present value In epidemiology, we are interested in selecting the best control strategy among all strategies available for implementation. The benefit of strategies can be measured in any metric of interest to the decision maker, such as disease prevalence or cost. In this paper we focus on the net present value of control strategies. The net present value is the sum of the discounted net cash flows generated by an investment, or decision, over a predetermined number of years, and thus includes both benefits and costs of a control, with adjustments for temporal occurrence. It is defined as: T

NPV = t=1

(t ) (T ) + (1 + r )t r (1 + r )T + 1

EU (NPVa) =

0

U (x ) PNPVa (dx ) > EU (NPVb) =

0

U (x ) PNPVb (dx ) (2)

Where EU is expected utility, P is the probability of observing outcome x . To illustrate the expected utility function, suppose x has two possible values for NPVa, 5 and 15, with estimated probabilities 0.3 and 0.7, respectively, the expected utility using a log utility function is E [U (x )] = 0.3ln (5) + 0.7ln (15) = 2.38. If NPVb has the same NPV outcomes but with the probabilities reversed, its expected utility is E [U (x )] = 0.7ln (5) + 0.3ln (15) = 1.94 . In this example, NPVb will be chosen over NPVa since it has a larger expected utility.

(1)

where, t is time in years, r is the discount rate, T is the final year of the analysis, and π is the annual net cash flow. The second term represents terminal wealth, measured as the last period’s net cash flow going forward into perpetuity after year T. The discount rate is the sum of the risk-free interest rate (usually government treasury bill rate) plus a risk premium component. The risk premium is the extra return over the riskfree rate that investors are expected to receive to engage into a risky enterprise. If the annual cash flows are measured in real terms (no inflation included), then an inflation rate is not included in the discount rate. The decision maker selects the control strategy that produces the highest NPV, but that value may be stochastic.

2.4. Risk measure and aversion When two control strategies have the same expected outcome but one of them has a larger variance, a risk averse individual would prefer the control strategy with the smaller variance. When distributions have different means and variances other criteria are used to rank them, which are discussed later in this paper. The expected utility of a control strategy with potential outcomes 5 and 15, each with equal probability, will be lower for a risk averse individual than the expected utility of a strategy with a potential certain outcome of 10, even though the expected value is the same in both cases. This is illustrated in Fig. 1 using any increasing and concave utility function U (x ) . The level of utility derived from the control strategy with random outcome, a, is lower than the utility of the same expected outcome but with no risk, a’. The degree of aversion to risk is captured in the shape of the utility function. Higher risk aversion implies a more concave utility function, and hence a larger distance between a and a’. Risk aversion may differ across individuals. Factors that affect risk aversion include gender (females tend to be more risk averse (Croson and Gneezy, 2009)); age (older people tend to be more risk averse (von Gaudecker et al., 2011)); and education (it has been shown

2.2. Utility functions The utility function, U (x ) , quantifies individual’s satisfaction (utility) from wealth or the consumption of goods and services, x , as long as individual’s preferences are well-behaved2 (Varian, 1992). The utility function is typically defined as continuous, twice differentiable with 2 Well behaved preferences are complete, reflexive, transitive, continuous, and strongly monotonic (Varian, 1992).

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Fig. 1. Expected utility of a control strategy with output values 5 and 15 with equal probability, denoted by a = 0.5 U(5) + 0.5 U(15), and utility of the expected value of the outcome, denoted by a’= U(10), using a concave utility function. Because a’ > a, the decision maker is risk averse: a certain outcome is preferred over a stochastic outcome with the same expected outcome.

to be negatively related to risk aversion (von Gaudecker et al., 2011)). Risk preferences are also domain dependent. Soane and Chmiel (2005) state that risk preferences may not be consistent across the domains of work, health and personal finance. Given two competing controls with the same expected value, a risk averse individual would prefer the distribution of the control with the smaller spread. Similarly, if an individual is indifferent between choosing two controls with the same expected outcome, but one control has an uncertain outcome while the other one has a sure outcome, the individual is risk neutral. In this case, the utility function is a straight line. Alternatively, individuals are risk seeking if they prefer the lottery with uncertain outcome to the one with the sure outcome, both with the same expected outcome. The shape of the utility of a risk seeking individual is convex in the outcome. These two situations of risk neutral and risk seeking behaviors are atypical. A way to measure risk aversion is the Arrow-Pratt absolute risk aversion coefficient (Pratt, 1964; Arrow, 1971). This is defined as the negative ratio of the second derivative of the utility function for some payoff x , U (x ), divided by the first derivative, U (x ) . The absolute risk aversion coefficient is denoted by A (x ) =

U (x ) U (x )

x . Under CRRA, risk preferences are not affected by the level of wealth. 2.5. Stochastic dominance Two decision criteria that can be employed and valid for any concave utility function are the first and second order stochastic dominance (Hadar and Russel, 1969; Rothschild and Stiglitz, 1970). These criteria do not require knowing the specific risk preferences of the individuals, only that the individual has increasing utility and is risk averse. Appling the criteria only requires knowing and comparing the cumulative distribution function (CDF) of the outcomes of competing controls. The stochastic dominance criterion has been used in various studies. For instance, Russell et al. (1984) used stochastic dominance criterion in the selection of machinery complements that maximizes farm profits. Benítez et al. (2006) applied stochastic dominance to select conservation policies for forest and agroforest systems in Ecuador. Olynk and Wolf (2009) used second degree stochastic dominance to recommend reproductive programs to dairy farmers. Stacey et al. (2007) used stochastic dominance (first order) to evaluate among various farm interventions to reduce the prevalence of Escherichia coli O157:H7 in cattle and sheep. Smith et al. (2014) compared biosecurity strategies for bovine viral diarrhea virus in cow-calf operations using stochastic dominance rules. If one distribution first order stochastically dominates (FSD) another one, it means that the dominant distribution’s CDF is located to the right of the dominated distribution’s CDF for at least some output values and is never to the left of the dominated distribution. For instance, if we are comparing the net present value of control strategies a and b, then strategy b FSD strategy a if, P (NPVb > x ) P (NPVa > x ) for all x , and P (NPVb > x ) > P (NPVa > x ) for some x . Graphically, the CDF of strategy b is to the right of the CDF of strategy a. Fig. 2 illustrates three competing distributions, a, b, and c, where a is dominated by b and c; and b is dominated by c. A distribution that FSD another one is preferred under any increasing utility function, regardless of its shape. FSD is the consequence of the local non-satiation property, where more (of anything that produces utility) is preferred to less. In Fig. 2, for any probability, outcome is always lower in output a than in outputs b or c. The output value will be higher for the dominant strategy at every percentile. When one distribution second order stochastically dominates (SSD) another distribution, the dominant function has a smaller area under the CDF from 0 (or lowest outcome value) to every other outcome value, than a dominated strategy. Second order dominance of strategy b over a, states that

. Since U (x ) measures

the curvature of the utility function, a higher value of U (x ) relative to U (x ) produces a higher A (x ) , indicating a more concave utility function and a greater difference between the utility of a stochastic outcome and a sure outcome. A (x ) is negative and zero for a risk seeking and risk neutral individual, respectively, and positive for a risk averter. The exponential utility function U (x ) = 1 e x , has a constant absolute risk aversion (CARA) coefficient A (x ) = . Usually risk aversion is stated in relative terms. The measure of relative risk aversion (RRA) is R (x ) =

U '' (x ) U ' (x )

x , where A (x ) is multiplied

by wealth. This is often used to define risk aversion of individuals, since it relates risk relative to wealth level. This measure is used when it is assumed that changes in initial wealth and income from the risky project are changing proportionally, decreasing the willingness to take risks. Wealthier people would hold a larger proportion of risky assets 1 (Bar-Shira et al., 1997). The following utility function, U (x ) = 1 x 1 , if > 0 and 1, is also called the constant relative risk aversion (CRRA) utility function, where is the relative risk aversion coefficient3 . The utility function becomes U (x ) = ln(x ) , if = 1. The exponential utility function, described before, has a linear RRA coefficient, R (x ) = 3 Other risk preferences include DRRA (decreasing relative risk aversion), IRRA (increasing relative risk aversion). DRRA and IRRA imply that as wealth increases the decision maker would hold a higher, and or lower, percentage of her income in risky assets, respectively.

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When two control CDFs cross multiple times but neither control SSD the other, a decision maker might still prefer one control over the other if the specific risk aversion of that decision maker is known. That is because SSD only requires risk aversion without knowledge of the specific degree of aversion to risk. A sufficiently risk averse individual would prefer controls d or e over control f in Fig. 3 because the lowest outcome under d is higher than the lowest outcomes under e or f. 2.6. Data To illustrate the use of stochastic dominance in decision making, we used data generated from Verteramo Chiu et al. (2019). A summary description of the data generation is found in the appendix. Verteramo Chiu et al. (2019) estimated the net present value per cow under 123 control strategies applied to a MAP infected dairy herd. The net present value per cow is estimated for a 5-year period. For each control program, Verteramo Chiu et al. (2019) simulated an NPV distribution consisting of 100 iterations.4 We used the results of the scenario of high MAP prevalence (20 % initial MAP prevalence) with mastitis interaction (Mastitis Association Scenario (MA) in Verteramo Chiu et al., 2019) to compare their control strategies. We compared some top ranked controls by FSD and SSD, as well as one of the lowest ranked controls to provide better clarity of the concepts. These comparisons are done numerically. We also illustrate FSD and SSD by plotting the CDFs of selected controls. Situations where no control SSD another are also provided, and the expected utility, using an exponential utility function, is shown under different values of risk aversion coefficient.

Fig. 2. Empirical cumulative distribution functions of three competing controls outputs. Control a is dominated by b and c. Control b is dominated by c. Dominations are first order stochastic.

AUC (NPVb )[0: x ] < AUC (NPVa )[0: x ]

(3)

for all possible values of x, with strict inequality for some x, where AUC(i)[0:x] is the area under the curve of the cumulative distribution function of i from 0 to x. Because any distribution that FSD another distribution by definition also SSD the other distribution, SSD is applicable only when cumulative distributions cross. For instance, in Fig. 2, distribution c not only FSD distribution a and distribution b, but c also SSD a and b. In Fig. 3, none of the distributions FSD another distribution, but cumulative e does SSD cumulative d because the area under distribution e is always greater than the area under distribution d for all x outcome values. Comparing distributions e and f, which cross each other once, f does not SSD cumulative e. For low values of x the area under cumulative e is less than the area under cumulative f, a requirement for SSD, but at higher outcome values the area under e is greater than the area under cumulative f, which violates the SSD decision rule. In this example, a risk neutral decision maker may prefer distribution f over e, but a sufficiently high risk averse individual may prefer distribution e. The CDFs can cross multiple times, such as distribution d and e, as long as condition (3) holds.

3. Results Table 1 ranks the top 10 controls by FSD and SSD as measured by the proportion of dominated controls out of the total 123 controls evaluated. Table 1 shows that control number 6 FSD 89 % of the rest of the controls, while control number 68 FSD 84 % of the rest of the controls. Fig. 4 shows the CDF of control number 6, control number 68, and control number 2, which are the first, third, and the lowest ranked (tied in 106th with 17 other controls) control by FSD criterion, respectively. We find that control numbers 6 and 68 FSD, and consequently SSD, control number 2, but neither control numbers 6 nor 68 FSD each other; however, control number 68 SSD control number 6. Graphically, the CDF of control number 6 has a higher probability of producing a lower NPV per cow than control number 68. The distribution of control number 6 has thicker tails; it also has a higher probability of getting a high output than control number 68. This means that regardless of the utility function used and risk preferences of the decision maker, control number 68 will produce a larger expected utility value than control number 6. Next, we compare two distributions, control numbers 6 and 67, shown in Fig. 5. Even though control 6 has a smaller area under the curve than control 67 for all NPV values, control 67 has a smaller area under the curve for low NPV values. Thus, neither of these two controls SSD the other. Control 67 ranks 5th and 1st under FSD and SSD criteria, respectively. However, it does not SSD control 6. Their CDFs cross multiple times, one at the lower part of the distribution and a few more on the top part of the distribution. Since neither control SSD the other, the decision to choose one over the other depends on the risk preferences, as measured by the relative risk aversion coefficient, of the decision maker. Examples of relative risk aversion coefficient are 5.4, for Kansas

Fig. 3. Empirical cumulative distribution functions of three competing controls outputs. Control e second order stochastically dominates (SOSD) control d. Control f, although with a smaller area under the curve, does not SOSD the other controls because of control f’s larger probability of realizing smaller outcomes.

4 For a description of the NPV estimation, the control strategies, and the model simulation see Verteramo Chiu et al. (2019). The NPV data are available at https://databank.illinois.edu/datasets/IDB-7539223

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preferences, we used the RRA values of -15, -5, 0, 5, 10, 15, and 20 to represent decision makers that are risk seeking (RRA values of -15 and -5), risk neutral (RRA value of 0), and 4 degrees of risk aversion (RRA values of 5, 10, 15, and 20). Comparing all the 123 control strategies from Verteramo Chiu et al. (2019), we find that control number 68 has the largest expected utility for a very wide range of relative risk aversion coefficients (-10 through 18). These values include risk seekers (negative RRA). For RRA values larger than 18, control number 68 is ranked second and control number 67 first. Control number 67 starts at rank 4 for a risk neutral (RRA = 0), and progressively increases in rank as RRA increases. Table 2 shows the ranking of the top 10 controls by expected utility for different values of relative risk aversion coefficient. Negative RRA values indicate risk seeking behavior. Although risk seeking behavior is atypical, we present the rankings for some negative RRA values for completeness. The outcome distributions that have a probability of a high value are preferred in this case. As the RRA increases, risk aversion increases, and those outcomes with a lower probability of realizing a low value are preferred. Looking at control 7, we see that it started at rank 10 for RAA= -15, eventually reaching rank 3 for RRA = 20. Control 6 shows an inverse pattern than control 7, starting in the top rank for RRA= -15, and decreasing to rank 8 for RRA = 20. Other controls did not change ranking very much (e.g., control 68), and some others switched up and down as RRA changes (e.g., control 1 has its highest ranking at RRA = 5 and 10). The ranking of controls that do not SSD each other depend on the risk preferences of the decision maker, since risk preferences are associated to observable characteristics (e.g., age, gender, education, etc.), likelihood of adoption of certain controls could be potentially estimated based on those characteristics.

Table 1 Top ten strategies and their proportion of dominated strategies by FSD and SSD out of the total 123 controls evaluated. Control definitions are available in Verteramo Chiu et al. (2019). Control ID

FSD

Control ID

SSD

6 37 68 69 67 1 7 97 98 38

0.89 0.86 0.84 0.83 0.82 0.82 0.80 0.80 0.79 0.71

67 68 1 7 69 97 98 37 6 39

0.95 0.95 0.94 0.93 0.93 0.93 0.92 0.91 0.90 0.90

wheat farmers (Saha et al., 1994); 0.615 median value for Israeli farmers (Bar-Shira et al., 1997). Gómez-Limón et al. (2003) found that about 41 % of their sample showed a high relative risk aversion coefficient, greater than 25. Using a CRRA utility function, we estimated the expected utility of controls 6 and 67 for relative risk aversion coefficient (RRA) values ranging from 0.1–10 at 0.1 intervals. These values were used to illustrate a range of risk aversion behaviors, from very small risk aversion to high risk aversion. The results are shown in Fig. 5. We found that the expected utility of control 6 is larger than that of control 67 for RRA coefficients from 0.1–5.9. When the RRA coefficient is 6, the expected value of control 67 becomes larger than that of control 6. This indicates that decision makers would choose control 6 over control 67 unless they have a very high degree of risk aversion, since control 6 has a higher probability of realizing a lower value than control 67. In general, if there is no strategy that SSD the others, we can still find a strategy that is preferred over the others by the expected utility criterion. This ranking is contingent on the specific risk aversion value of the decision maker. To illustrate changes in control ranking on risk

4. Discussion This paper explained step-by-step methodologies for ranking stochastic outcomes using stochastic dominance (first and second degree), and expected utility under various degrees of relative risk aversion coefficients. We focused on the problems commonly faced by

Fig. 4. CDFs of three control strategies. Control number 2 is a low ranked control, control numbers 6 and 68 are the first and third ranked controls, respectively, by FSD among all controls.

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Fig. 5. CDFs of two competing controls that neither one SSD the other. Control 6 has a smaller area under the curve for all NPV values, but control 67 has a smaller area under the curve for small NPV values.

how risk preferences can determine the selection of control options. A general characterization of individuals’ risk preferences, say lowmedium-high risk aversion, may be sufficient to demonstrate the changes in ranking of control strategies, and also to select controls according to that characterization.

Table 2 Ranking of the top 10 controls by expected utility for different relative risk aversion coefficient values. Control definitions are available in Verteramo Chiu et al. (2019). Control ID Rank

RRA= -15

RRA= -5

RRA = 0

RRA = 5

RRA = 10

RRA = 15

RRA= 20

1 2 3 4 5 6 7 8 9 10

6 68 1 99 67 98 69 37 97 7

68 6 1 67 69 98 99 7 37 97

68 6 1 67 69 98 99 7 97 37

68 1 6 67 69 7 97 98 99 37

68 1 67 7 6 69 97 98 37 39

68 67 1 7 97 69 6 39 37 98

67 68 7 1 97 69 39 6 37 98

5. Conclusions Ranking of stochastic outcomes can be performed by using FSD and SSD. FSD implies SSD. FSD is relevant for decision makers with increasing utility functions and SSD is relevant for decision makers with increasing and concave utility functions, which includes normal utility functional forms. When an outcome SSD another one, but does not dominate by FSD, any risk averse decision maker (those with a concave utility function) would prefer the outcome that SSD. However, when no outcome dominates the others by second (and hence by first) order stochastic dominance, ranking of stochastic outcomes can be performed using the expected utility framework. In this case, the ranking would be affected by the specific risk preferences of the decision maker. Under this circumstance, risk preferences of the decision maker need to be considered when selecting a strategy to implement. The expected utility of the competing controls under various relative risk aversion coefficients, characterizing the decision maker, should be estimated and reported.

epidemiologists in which the best control strategy needs to be selected among many, and provide examples from Verteramo Chiu et al. (2019), which ranked 123 potential MAP control strategies in a dairy herd. When no control SSD (and thus FSD) another one, we can find a dominant control by incorporating decision makers’ risk preferences and selecting the control with the highest expected utility. Selecting a dominant control strategy by incorporating decision makers’ risk preferences implies that those risk preferences can be observed or estimated. Arrow-Pratt risk coefficients are commonly estimated through lotteries or other hypothetical choice experiments. However, these estimations may not represent the actual risk preferences of the decision maker when facing with a non-hypothetical decision. Preferences change over time and context (Lönnqvist et al., 2015). Framing of the risk event also affect people’s risk perceptions (Charness et al., 2013). Our intention in this study is to demonstrate

Acknowledgements The authors gratefully acknowledge funding provided by the National Institute of Food and Agriculture of the United States Department of Agriculture [NIFA Award No. 2014-67015-2240], and by the USDA-NIFA AFRI [grant # 2014-67015-2240] as part of the joint USDA-NSF-NIH-BBSRC-BSF Ecology and Evolution of Infectious Diseases program. The funding sources played no role in the research.

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Appendix A A brief description of the model and of the controls used in this study are presented next. The model and results from Verteramo Chiu et al. (2019) were used in this study. This model simulated a closed dairy herd with 1000 cows and an initial infection level of MAP (Mycobacterium avium sub. paratuberculosis) of 20 %. Calves can be born susceptible or infected in utero. Susceptible calves, heifers and adult animals, can get infected by contact with shedding animals, becoming transiently infected. Transiently infected heifers and adults may transition to latent progressing or non-progressing animals. Latent progressing animals may transition to low shedding and then to high shedding animals, or it may become latent non progressing. For the detailed description of the modelling approach and results please refer to Verteramo Chiu et al. (2019). Details of the controls used in this study are shown in Table 3. Table 3 Description of the controls used in this study, their expected net present value and standard deviation (SD). Values are in USD. Control ID

Control Name

Expected NPV

SD NPV

1 6 7 37 39 67 68 69 97 98 99

Do nothing (business as usual) Annual ELISA, cull high Annual ELISA, cull after 2 Biannual ELISA, cull all Biannual ELISA, cull after 2 Annual Continuous ELISA, cull all Annual Continuous ELISA, cull high Annual Continuous ELISA, cull after 2 Biannual Continuous ELISA, cull all Biannual Continuous ELISA, cull high Biannual Continuous ELISA, cull after 2

20,229,145 20,256,645 20,000,275 19,908,484 19,683,773 20,124,452 20,417,547 20,096,789 19,976,623 20,059,620 20,032,668

1,492,467 1,692,623 1,402,866 1,641,560 1,297,032 1,454,216 1,560,492 1,557,348 1,463,786 1,676,497 1,744,335

Control 1 refers to continue operations as usual with no intervention to eliminate MAP. It is important to note that the average productive lifetime of a dairy cow in a commercial dairy operation in the U.S. is less than three lactations; because of this, there are few cases of highly contagious cows at any given time. ELISA refers to testing all cows by means of ELISA test. Annual and Biannual refers to performing a herd test (e.g., ELISA) every year or twice per year, respectively, until five consecutive negative whole herd tests are achieved. Continuous refers to performing the herd test indefinitely. Cull high refers to culling all high shedding cows that tested positive, cull all refers to culling all test positive cows (regarding of their shedding level). Cull after 2 refers to cull all test positive cows after their second test. Appendix B. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j.prevetmed.2020.104906.

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