text:
# CVS difference for arm/source/08.mss

Differences between 1.84 and version 1.85
Log of other versions for file arm/source/08.mss

```--- arm/source/08.mss	2010/05/08 06:31:33	1.84
+++ arm/source/08.mss	2010/06/11 07:27:55	1.85
@@ -1,10 +1,10 @@

-@Comment{\$Date: 2010/05/08 06:31:33 \$}
+@Comment{\$Date: 2010/06/11 07:27:55 \$}
@LabeledSection{Visibility Rules}

@Comment{\$Source: e:\\cvsroot/ARM/Source/08.mss,v \$}
-@Comment{\$Revision: 1.84 \$}
+@Comment{\$Revision: 1.85 \$}

@begin{Intro}
@redundant[The rules defining the scope of declarations and the rules defining
@@ -3271,7 +3271,7 @@
@ChgAdded{Version=[3],Text=[The rule also minimizes cases of implicit
conversions when the tag check or the accessibility check might fail. We
-  word this this way because access discriminants should also be disallowed if
+  word it this way because access discriminants should also be disallowed if
their enclosing object is designated by an access parameter.]}
@end{Reason}
@begin{Ramification}
@@ -3291,7 +3291,6 @@
Note that this and the ambiguity rule have to be @LegalityName@;s.
@end{Ramification}

-@ChgRef{Version=[3],Kind=[Revised],ARef=[AI05-0149-1]}
@Defn2{Term=[preference], Sec=(for root numeric operators and @nt<range>s)}
There is a @i{preference} for the primitive operators (and @nt<range>s)
of the root numeric
@@ -3302,14 +3301,7 @@
@nt<range>) of the
type @i{root_integer} or @i{root_real}, and the other is not,
the interpretation using the primitive operator (or @nt<range>)
-of the root numeric type is @i{preferred}.@Chg{Version=[3],New=[ Similarly,
-there is a preference for the equality operators of the @i{universal_access}
-type (see @RefSecNum{Relational Operators and Membership Tests}). If two
-acceptable interpretations of a constituent of a
-complete context differ only in that one is for an equality operator of the
-@i{universal_access} type, and the other is not, the interpretation using the
-equality operator of the @i{universal_access} type is
-preferred.@Defn2{Term=[preference], Sec=(for universal access equality operators)}],Old=[]}
+of the root numeric type is @i{preferred}.

@begin{Reason}
@leading@;The reason for this preference is so that expressions involving
@@ -3322,12 +3314,23 @@
...
@key[end] @key[if];
@end{Example}
+@end{Reason}

-  @ChgRef{Version=[3],Kind=[Revised],ARef=[AI05-0149-1]}
-  @ChgAdded{Version=[3],Text=[The preference for @i{universal_access} equality
-  operators is necessary because of implicit conversion
-  from an anonymous access type to a named access type, which would
-  allow the equality operator of any named access type to be used
+there is a preference for the equality operators of the @i{universal_access}
+type (see @RefSecNum{Relational Operators and Membership Tests}). If two
+acceptable interpretations of a constituent of a
+complete context differ only in that one is for an equality operator of the
+@i{universal_access} type, and the other is not, the interpretation using the
+equality operator of the @i{universal_access} type is
+preferred.@Defn2{Term=[preference], Sec=(for universal access equality operators)}]}
+
+@begin{Reason}


text:
# Why would the term "dynamically typed" be considered a misnomer?

In the book "Types and Programming Languages", the author writes:

The word "static" is sometimes added explicitly - we speak of a "statically typed programming language", for example - to distinguish the sorts of compile-time analyses we are considering here from the dynamic or latent typing found in languages such as Scheme, where run-time type tags are used to distinguish different kinds of structures in the heap. Terms like "dynamically typed" are arguably misnomers and should probably be replaced by "dynamically checked", but the usage is standard.

I am not sure if I understand this. Does the author mean that, in type theory, all programs are "statically typed"?

The discussion in the section surrounding that paragraph in Pierce's book explains why this is so. In particular, consider the definition of "type system" given on the page before:

A type system is a tractable syntactic method for proving the absence of certain program behaviors by classifying phrases according to the kinds of values they compute.

Note how aspects like "syntactic method" and "classifying phrases" mean that it applies to "source" programs, i.e., inherently is a static notion. Moreover, in actual type theory, programs that do not "type-check" are not even considered programs. The type system defines the set of well-formed programs. Something that's ill-typed has no meaning.

One could argue that this is the wrong definition. But it is shared by most researchers in the field and certainly by the ones working on type theory. It matches the notion of a "type" as established in mathematics and logic (lambda calculus, type theory). On the other hand, it would be difficult to come up with a sufficiently accurate definition of "type system" that would encompass "dynamic typing" without bordering on technically meaningless.

So from that technical point of view, a dynamic language is untyped, in the same way the untyped lambda calculus is (or equivalently, it has only one single type, such that every program is accepted as well-typed). Dynamic checks are not really on types but on the shape of values, which sometimes carry type-like "tags".

• Everything you say is correct -- under the 2-compute-phases assumption. But often you have more than 2 compute stages, e.g. in HPC or in web-programming, and what is 'run-time' to the $n-1$-stage is 'compile-time' to the $n+1$-stage. This is not really reflected in TAPL, and also usually ignored in the part of type-theory that comes from logic. Jun 27 '20 at 12:29
• @MartinBerger, doesn't it carry over to staged programming as well? Each stage effectively is a DSL for generating programs of the next stage. And for each stage individually, this definition still applies. Or am I missing something? Jun 27 '20 at 16:44
• It's a fine quote, but I wish it said "absence or presence of certain program behaviors". The way it's written liveness properties of programs are excluded. Jun 28 '20 at 11:52
• Note that people also talk about "type system" in the context of dynamically typed languages, even tho it means something a bit different in that context since it does not talk about a syntactic criterion on the source code (e.g. lispcookbook.github.io/cl-cookbook/type.html) Jun 28 '20 at 14:17
• @MartinBerger, re "programs that do not "type-check" are not even considered programs": this is a technicality but still true in most type theories, where ill-typed terms (sometimes called pre-terms) simply have no meaning. For some more advanced type systems this actually is a necessity, since you wouldn't even be able to assign meaning, at least not without jumping through hoops (for a practical PL example, take Haskell with type classes). FWIW, standard preservation/progress proofs of type soundness do not depend on the existence of an untyped semantics. Jul 9 '20 at 9:07

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text:
# Tag Info

## Hot answers tagged axiom-of-choice

5

This is not that incredible if you understand what's going on. Given a second-countable space $X$, fix a countable basis $\{U_n\mid n\in\Bbb N\}$. Now you can observe that every open set is uniquely determined by the basis elements it contains, namely the function $U\mapsto\{n\in\Bbb N\mid U_n\subseteq U\}$ is an injective function. And since the real ...

5

$\bf 1.$ The Axiom of Choice Given a set $S$, to say that $S$ is not empty is to say that $\exists x(x\in S)$ (in English: there exists some $x$ such that $x$ is an element of $S$). First-order logic has an inference rule which allows us to move from $\exists x(x\in S)$ to some [new] constant symbol $s$ such that $s\in S$. This process, called existential ...

4

The proof is not trivial to come by. Originally given by Hodges, the proof shows a variant of Zorn's lemma can be proved from the assertion that every commutative ring with a unit has a maximal ideal. Wilfrid Hodges, Krull implies Zorn, J. London Math. Soc. (2) 19 (1979), no. 2, 285--287. Some decades later, Banaschewski gave a somewhat different ...

4

That is correct, if the real numbers are a countable union of countable sets, then measure theory as we know it goes out the window. The same can be said if the real numbers are a countable union of countable union of countable sets, and so on. Essentially for the same reason. To your question, the Lebesgue measure is the completion of the Borel measure. So ...

3

Nope. It does not imply that. This is a result due to Derrick and Drake. The idea, in its essence, is to add $\aleph_n$ Cohen reals, for each $n$, and take only the sets which are definable by a bounded part of the forcing.

3

The discrete topology on a countable set is second countable; it has a countable base consisting of the singleton sets. Thus, the assertion that all second countable spaces are Lindelöf implies that every cover of a countable set $N$ by subsets of $N$ has a countable subcover. I will show that the latter (ostensibly weaker) assertion implies ...

3

Since I am quite unfamiliar with schemes, I'll follow a hunch.If $i$ and $j$ are such that $p\in U_i\cap V_j$ then there is some affine open set $W\subseteq U_i\cap V_j$ such that $p\in W$. Or at least, for every $p$ there exists at least one pair of $i$ and $j$ such that the above is true. In its current form, the proof suggests choosing $i(p)$ and ...

3

Not every set of ideals has a maximal elements. It is true that with the axiom of choice, the set of all ideals has a maximal element, but it is not true that every set of ideals has a maximal element. For example, look at $\Bbb Z[\{x_n\mid n\in\Bbb N\}]$ and consider the set of ideals which are generated by $\{x_i\mid i<n\}$ for each $n\in\Bbb N$. This ...

2

Take the subset $E$ in $A\times \mathcal{P}(A)$ defined by $\{(a,B) : a\in B\}$. Then the second projection is a suejective map onto $\mathcal{P}^*(A)=\mathcal{P}(A) -\emptyset$. A right inverse, composed with the first projection, yields a selecting function as in AC.

2

Well. As with the failure of the axiom of choice in full, I should point out that the failure of any such arbitrary choice principle is not limited to "interesting sets". The axiom of choice could be in full force when considering all the sets relevant for most mathematicians. Then, it might fail badly somewhere up in the clouds of the set theoretic ...

2

Yes. Assuming the axiom of choice the answer is positive. You can find the proof in W.R. Scott's paper: Scott, W. R. "Groups and cardinal numbers." Amer. J. Math. 74, (1952). 187-197. The axiom of choice is used there for all manner of cardinal arithmetics. Without the axiom of choice it is no longer necessary that the proof can go through. Because ...

2

Since you seem to already have considerable information about ordinal numbers, you might use the following proof of Hartogs's theorem, which seems slightly more efficient than Asaf's because you don't need to well-order equivalence classes. Given $X$, there is a set $W$ of all well-orderings of subsets of $X$. Each such well-ordering is isomorphic to a ...

2

I don't know anything about sheaves, but I can give a "no" answer to what Hanno mentions in his comment as a special case. Namely, it is consistent with $\mathsf{ZF}$ that there are families of abelian groups $(A_i : i \in \mathbb{N})$ and $(B_i : i \in \mathbb{N})$ and a family of maps $(\varphi_i : i \in \mathbb{N})$ such that each $\varphi_i$ is a ...

2

Here's the thing about $\Bbb R$. It's linearly ordered. And this means that finite subsets are well-ordered canonically by that linear ordering. In particular, this means that the countable union of finite sets of real numbers is always countable. So the usual proof works, with the minor modification that now we need to appeal to the properties of $\Bbb R$ ...

1

Jyotirmoy writes that: [To prove the Banach-Tarski theorem] requires the Axiom of Choice. This statement needs some qualification. A bit of background: the standard collection of set-theoretic axioms is called ZFC, where "C" stands for the axiom of choice. Deleting the axiom of choice yields a smaller collection of axioms called ZF. Here's the key ...

1

This statement allready follows from the addition theorem, which states that $A \times \{0,1\}$ is equipotent with $A$ for every infinite $A$ (as on $A \times \{0,1\}$, $(a,x) \mapsto (a, 1+x)$ is fixed-point-free). The addition theorem is known to be strictly weaker than the axiom of choice (a result by Sageev).

1

If you look at the fourth section of Hodges paper, Wilfrid Hodges, Six impossible rings, J. Algebra 31 (1974), 218--244. He proves (Section 3, Th. 1) that the implication that you are asking for is not provable without the axiom of choice. Namely, the following chain of implications is irreversible without using Zorn's lemma: \text{Every non-empty ...

1

Yes, it's fine. Note that the use of the axiom of choice is only in proving that a choice function $g$ exists, and we fix this choice function before starting the recursion. Afterwards it's just recursion. Also note that you don't choose from $f(n)$, but rather from $\{y\mid f(n)<y\}$, which is nonempty since $f(n)$ is not maximal.

1

Yes, this is fine. In fact you don't even need the full strength of the axiom of choice: the axiom of dependent choice is all that you're really using.

1

This seems to be a typo (or a small mistake, if you prefer), and the proof goes along just fine if you also require that $\{\Lambda\}$ is open in each $X_a$. The space is still compact. You can find the same proof, minus the typo, on the Wikipedia page of the Tychonoff's theorem.

1

Assuming $\sf ZF$ we can prove Hartogs theorem. Namely, for every set $X$ there exists a set $A$ and a well-ordering of $A$ such that there is no injection from $A$ into $X$. The proof does not use the axiom of choice. It goes by considering all the well-orders whose domain is a subset of $X$, then considering the equivalence classes under the order ...

1

Revised, since there was an oversight in my original argument. I’ve now looked at Tarski’s paper; the very old-fashioned notation is a worse problem than the French. I’ve updated it and filled in some details. The proof in question is on pages $94$ and $95$. In essence he starts by assuming that there is an injection $\varphi:\wp(X)\to\wp(X)$ such that ...

1

This can be obtained with the following theorem of Kuratowski: $\omega\leq^*\mathfrak a\iff\omega\leq2^\mathfrak a$ Namely, there is a surjection from $A$ onto $\omega$ if and only if there is an injection from $\omega$ into $\mathcal P(A)$, which in turn is to say that there is an injection from $\mathcal P(A)$ into itself which is not a surjection. ...

1

Something is not clear to me about your question. Global choice implies the axiom of choice for sets. Of course it does. Every set is a subclass of $V$, and if $V$ can be well-ordered, then every set can be well-ordered. Requiring that there is an injection from the class of ordinals into $V$ is provable in $\sf ZF$. This function is the identity function. ...

1

If you look at the fourth section of Hodges paper, Wilfrid Hodges, Six impossible rings, J. Algebra 31 (1974), 218--244 he proves (Section 3, Th. 3) that the implication that you are asking for is not provable without the axiom of choice. Note that the prove is in $\sf ZF+DC$ in which ACC is equivalent to the statement "Every ideal is finitely ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
SERVING THE QUANTITATIVE FINANCE COMMUNITY

deano1949
Topic Author
Posts: 3
Joined: December 14th, 2013, 8:36 pm

### How to find the best Bayesian posterior distribution given observed data?

Hi Guys,In order to get posterior distribution in Bayesian Inference, I need to assume a prior distribution and a likelihoods of observed data. According to Fisher's information, we can proxy prior distribution from likelihoods distribution. it makes that the assumed distribution of likelihoods is so important. so my question is what method is the best to obtain likelihood distribution? Assumed you are given a bunch of observed financial data and you dont have any prior knowledge of it, how to get the most appropriate distribution?One of the methods is maximum likelihood. Assume the data is normal distributed ~N(x,y), then seek x and y which maximise its probability. Personally, I doubt it is correct method, as it is an approach of frequentist, which might not be suitable for Bayesian. What method do you guys use to find distribution of likelihoods.Any comment is appreciated. Langyu

Alan
Posts: 10331
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### How to find the best Bayesian posterior distribution given observed data?

Your implied premise is problematic. Say your data is a return series $\{r_i\}$ Initially, people explored the notion that the $r_i$ were best described as IID draws from some univariate density $p(r)$.In that context, your question makes sense -- you want to know the best distribution.But I'd say this approach was generally abandoned by the late 60's, as it failed to account for things like volatility clustering.However, if you want to catch up to the 60's try this classic by FamaNowdays, people tend to look at multivariate process models, often in continuous-time, often with hidden states that must be estimated.So they don't start with a distribution, but a parameterized process. Parameters and hidden statesare estimated (ideally and normatively) by maximum likelihood or good approximations. With the voluminous time series data in finance, such inference rarely depends on priors. p.s. I will add that priors play a strong role in the following sense. People tend to fall in love with classes of favorite processes.So, results from a particular researcher or group tend to always be from their favorite class. Given the vagaries of the whole(academic) notion that a stochastic process actually describes financial returns, and the zillions of possibleprocesses, this is inevitable.
Last edited by Alan on January 6th, 2014, 11:00 pm, edited 1 time in total.

deano1949
Topic Author
Posts: 3
Joined: December 14th, 2013, 8:36 pm

### How to find the best Bayesian posterior distribution given observed data?

Thanks for your comment Alan. Time series process is a classical approach, however, in terms of forecast, its power is relatively low, often over-fitting, and it is not flexible of adjusting coefficients (required often review). Therefore, I am thinking a bayesian approach to forecast financial return.I agree with you that with a large number of observed data, inference is dominated by likelihoods. However, now I have very limited data on the instrument that I am analysing. In this case, prior does matter. Surely, when time moving along, we collect more data, then prior is less meaningful.

Alan
Posts: 10331
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### How to find the best Bayesian posterior distribution given observed data?

I see. Well, as a general comment, these forums can be quite helpful, but you need to be specific.As your question is currently posed, I doubt you'll get much useful. But, if you said something like, just to pick a security in the news,"I am trying to estimate the pdf for Twitter's stock price in one year, but don't have much history.Have tried X, Y, and Z." Then, you would probably get some good suggestions.

deano1949
Topic Author
Posts: 3
Joined: December 14th, 2013, 8:36 pm

### How to find the best Bayesian posterior distribution given observed data?

Thanks so much for your advice. It is really helpful. I am a fresher here, learning the rules and culture of this forum. TBH It is pretty cool and useful forum, I should had discovered it lot earlier. Somehow, I think I found the solution. For the bayesian inference, I shouldn't maximise likelihood of observed data, instead, I should maximise posterior. The method is called Maximum Posterior. Using this method, I would be able to calculate backward to obtain variables in the likelihood function, which maximise posterior. It is said such way would resolve over-fitting issue. Another good thing about bayesian is that it not only just give you a point estimate, but also a range of estimates with confidence interval.

Alan
Posts: 10331
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### How to find the best Bayesian posterior distribution given observed data?

outrun, yes, exactly,deano1949, Well, maximum likelihood certainly has confidence intervals. If overfitting is an issue, it suggests you are trying to predict returns ...
Last edited by Alan on January 8th, 2014, 11:00 pm, edited 1 time in total.

Cuchulainn
Posts: 62967
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### How to find the best Bayesian posterior distribution given observed data?

QuoteOriginally posted by: outrunI think Alan himself did something like this for stoch vol models in an upcomming book? Indeed estmate the paramaters *and latent variables (vol)* to maximise the likelihood of observed data.@list posted there?
Step over the gap, not into it. Watch the space between platform and train.
http://www.datasimfinancial.com
http://www.datasim.nl

chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

### How to find the best Bayesian posterior distribution given observed data?

QuoteOriginally posted by: deano1949Thanks so much for your advice. It is really helpful. I am a fresher here, learning the rules and culture of this forum. TBH It is pretty cool and useful forum, I should had discovered it lot earlier. Somehow, I think I found the solution. For the bayesian inference, I shouldn't maximise likelihood of observed data, instead, I should maximise posterior. The method is called Maximum Posterior. Using this method, I would be able to calculate backward to obtain variables in the likelihood function, which maximise posterior. It is said such way would resolve over-fitting issue. Another good thing about bayesian is that it not only just give you a point estimate, but also a range of estimates with confidence interval.Hi, thanks for posting on the topic. I don't understand why MAP would prevent over-fitting (in general and as opposed to ML).I suppose it is just a matter of degrees of freedom. As per the choice between MAP and ML, I am not an expert but I see ML just as a special case of MAP (uniform prior).I'd be great if you could point to some resources on the topic.

chocolatemoney
Posts: 322
Joined: October 8th, 2008, 6:50 am

### How to find the best Bayesian posterior distribution given observed data?

Follow-up thoughts: if your sample is small, some priors may in-fact reduce the degrees of freedom of your model, by constraining the parameters in a margin. I missed the part where the OP mentions that he's dealing with limited sample sizeI'd be interested in reading something more on the topic. Any recommendation?
Last edited by chocolatemoney on January 20th, 2014, 11:00 pm, edited 1 time in total.

AnalyticalVega
Posts: 2260
Joined: January 16th, 2013, 5:03 am

### How to find the best Bayesian posterior distribution given observed data?

QuoteOriginally posted by: chocolatemoneyFollow-up thoughts: if your sample is small, some priors may in-fact reduce the degrees of freedom of your model, by constraining the parameters in a margin. I missed the part where the OP mentions that he's dealing with limited sample sizeI'd be interested in reading something more on the topic. Any recommendation?If the sample test data is small, you are more likely to have bias in your estimator. The amount of overfitting is a probability (PBO) that needs to be calculated.

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text:
Temperature-controlled supply chains (cold chains) require an unbroken chain of refrigeration to maintain product quality and safety. This study investigated cold chains for farmed oysters raised in the Chesapeake Bay, one of the largest shellfish-growing regions in the United States, and sold live to the half-shell market in surrounding states. Temperature sensors were used in boxes of oysters from February to September 2017, which generated 5,250 h of temperature data. Thirty-nine businesses participated in the temperature sensor study, and 26 of those businesses participated in interviews to further understand how cold chains function. Internal oyster temperatures were measured above 50°F (10°C) for over 1 h in 19% (7 of 36) of shipments, which is a temperature that exceeds National Shellfish Sanitation Program criteria. The highest internal oyster temperature recorded in any shipment was 54.5°F (12.5°C). Some parts of the cold chain had difficulty maintaining storage temperatures below 45°F (7.2°C) in warmer months when Vibrio control plans were in effect. We modeled the effects of temperature on Vibrio parahaemolyticus. The model predicted moderate bacterial growth before oysters were under temperature control, but cold chains prevented further bacterial growth and provided a moderate drop-off in V. parahaemolyticus abundance.

Seafood is made available to consumers via a network of producers, processors, freight carriers, wholesalers, food retailers, and restaurants. This network is called a supply chain, which creates economic opportunities for businesses and is a critical component of the food system. Temperature-controlled supply chains (i.e., cold chains) require an unbroken chain of refrigeration to maintain product shelf life and quality, as well as food safety (1). Another feature of supply chains is the level of traceability—whether finished products can be tracked from the farm to the consumer (2). A high level of traceability can support a wide range of functions, including product differentiation, identification of origin and production methods, fraud prevention, waste reduction, facilitation of food recalls, and improvement in consumer trust (2, 3, 13, 16). Some seafood products have considerable traceability issues due to the degree of complexity tied to such factors as international supply chains, species substitution, and other factors. The U.S. government will implement a traceability program in 2018 for 14 priority species of seafood (22).

Seafood supply chains are complex, and very little is known about them outside of the seafood industry. There are 2,500 businesses in the United States that are certified to harvest, process, and distribute molluscan shellfish (33), and to our knowledge, there has never been an effort to map these supply chains. Understanding the size, shape, and performance of supply chains is critical for implementing traceability programs and ensuring food safety.

There are specific regulations that pertain to certain shellfish species that are often eaten raw, owing to the associated foodborne illness risks. The U.S. shellfish regulations include a tag system to track key attributes about shellfish from harvest to consumers (24). These tags are used by government agencies, while performing shellfish-borne disease tracebacks and product recalls, and by consumers to aid in product differentiation.

There are also regulations related to handling of product during harvest, processing, and distribution to reduce some risks associated with shellfish-borne diseases (10, 24, 29). States with confirmed cases of Vibrio vulnificus or Vibrio parahaemolyticus or both must develop Vibrio control plans (VCPs) for warm seasons to further mitigate risks associated with these bacteria (24). V. vulnificus and V. parahaemolyticus are gram-negative bacteria naturally found in estuarine and marine environments, which are becoming more abundant and spreading to new geographic areas due to climate change (34). V. vulnificus is the leading cause of death related to seafood consumption in the United States (5). V. parahaemolyticus causes gastroenteritis that is usually self-limiting but can result in business losses devastating to the oyster industry. From 1996 to 2010, infections associated with Vibrio-contaminated seafood, including oysters, tripled (0.09 to 0.28 cases of vibriosis per 100,000 people calculated by Cholera and Other Vibrio Illness Surveillance and 0.15 to 0.42 cases of vibriosis per 100,000 people calculated by the Foodborne Diseases Active Surveillance Network) (25).

Temperature control of shellfish during harvest, processing, and throughout the distribution is essential to control Vibrio growth. Storing oysters at ambient conditions allows V. vulnificus and V. parahaemolyticus to multiply inside the oyster shell (8, 12). One approach to reduce risk of Vibrio growth involves postharvest processing methods, such as pasteurization, high pressure processing, or rapid freezing, which have been developed to reduce pathogen risks in shellfish (4). These methods, however, can negatively affect the taste of raw oysters (6). Icing is a method to control Vibrio growth in lieu of postharvest processing (15, 20, 27). In comparing cooling methods, ice slurries perform slightly better at rapidly cooling oysters than layered ice, and both methods achieve cooling faster than refrigeration alone (15, 18). Cooling oysters too quickly with ice may lead to a negative side effect of gaping (e.g., open shells), which leads to loss of oyster liquor, reduced shelf life, and possibly oyster death (20). Additionally, storing oysters below freezing could kill the animals, whereas prolonged storage at or above 50°F (10°C) can degrade product appearance, odor, and texture (21). Previous oyster temperature studies have primarily focused on the harvest or production level and have not adequately investigated the role of cold chains in maintaining product quality and safety.

The aim of this study was to gain a better understanding of temperature control in shellfish cold chains within the Chesapeake Bay region to support modeling efforts and inform policy development. Oysters are the most valuable marine aquaculture species in the United States, and the Chesapeake Bay is the largest estuary in the United States and one of the top oyster-producing regions of the country (23, 31). We asked seafood businesses about their protocols and practices for handling oysters and their perceptions of cold chains. Then, we sought to answer two research questions: (i) Are oyster temperatures different across the stages of the supply chain? (ii) What is the modeled abundance of V. parahaemolyticus and the associated risk of gastroenteritis in the supply chain? Finally, we identified areas in which the seafood industry can improve cold-chain performance for oysters.

### Study design

We assessed the performance of regional cold chains for molluscan shellfish by using a mixed methods approach involving temperature sensor measurements and interviews with businesses. We focused on farmed Eastern oysters (Crassostrea virginica) sold live to the half-shell market. Our rationale for this focus was that this product form is a growing segment of the shellfish industry and maintaining proper temperature control for live oysters is challenging. The study area included farmed oysters raised in the Chesapeake Bay and distributed to surrounding states: Virginia, Maryland, Delaware, Washington, DC, and Pennsylvania. We excluded all wild-caught oysters, shucked oysters, and frozen half-shell oysters from the study, as well as any product harvested outside the Chesapeake Bay and sold outside the study region. The study was reviewed by the Johns Hopkins School of Public Health Institutional Review Board.

### Participant recruitment

Participants were recruited into the study on the basis of chain sampling methods, starting with local and regional seafood wholesalers in February 2017. Wholesalers then referred us to other participants in their own supply chains (oyster producers, freight carriers, food retailer, and restaurants). Participants were contacted by phone or e-mail and given a one-page description of the study and a consent form. As an incentive, participants were provided data about the performance of their own cold chain. The inclusion criteria were being an active business in the regional shellfish supply chain, being over 18 years of age, speaking English, and agreeing to participate in the study.

### Survey tool

We developed a survey tool to collect information from businesses in the supply chain, and a subset of these questions pertained to cold-chain performance. The responses to this subset of questions are reported with our temperature data analyses to provide important context. The survey was administered in parallel with the temperature sensor study and was performed as an in-person or phone interview. Survey responses were shared with participants to check accuracy.

### Temperature sensors

Temperature sensors (SmartButton, ACR Systems Inc., Surrey, British Columbia, Canada) were used to track oyster temperature and ambient air temperature at 1-min or 10-min intervals starting at harvest and through the supply chain. The manufacturer reported that the sensors have a working range of −40 to 185°F (−40 to 85°C), which is within the range of temperatures we expected to observe in our cold chains. The reported accuracy of the sensors was ±1.8°F from −22 to 113°F (±1.0°C from −30 to 45°C). We independently tested the interbutton variability in our laboratory by using simulated field conditions and determined it was 1°F (0.6°C). We used a pilot study with wholesalers to determine the variability among three oyster boxes shipped to the same final destination and determined that a single box was adequate to make generalizations about temperatures in warehouses and trucks (see Supporting Information [Supplemental Material] Fig. S1). Handheld analog thermometers, calibrated in an ice water bath, were used for spot readings on farms.

### Harvest and on-farm processing

We visited six oyster aquaculture operations to monitor the harvest and on-farm processing. Each farm was visited once during months in which VCPs were in effect. Virginia producers are required to follow V. vulnificus and V. parahaemolyticus plans from 1 May to 30 September, while Maryland producers are required to follow V. parahaemolyticus plans from 1 June to 30 September. Three of the six farms were visited again in March when VCPs were not in effect, and the ambient temperature in the region ranged from the middle 30s to middle 50s in degrees Fahrenheit. To protect the anonymity of participants, we do not report farm origin or state in our results.

The temperatures of the harvest water, ambient air, wash water, and the walk-in refrigerator were measured. Notes were also taken about sun exposure and processing methods. After oysters were harvested and unloaded at the dock, SmartButton sensors were inserted into oysters to measure temperature at 1-min intervals during washing, grading, boxing, and storage. All oysters containing temperature sensors were wrapped in red duct tape to prevent their introduction into the food supply. Just before the product was shipped, the sensors reading at 1-min intervals were removed, and new sensors reading at 10-min intervals were inserted inside one oyster per 100-count box. (Sensors set at 10-min intervals increased sensor operating life up to 2 weeks.) Farms that sell products in mesh bags were asked to place the bags inside a box. One SmartButton sensor was taped to the outside of each 100-count wax box to measure ambient air temperature in shipments. Between four and eight boxes were tracked from each farm. Stamped envelopes, a study description, and a note card to record arrival times were enclosed in each box to allow the final recipient (food retail or restaurant) to return the sensors to the Virginia Seafood Agricultural Research and Extension Center at Virginia Tech. Producers were offered financial compensation for boxes of oysters used in the study.

### Statistical analyses

Temperature sensor data were downloaded by using manufacturer software (TrendReader, ACR Systems Inc.), analyzed in Excel (Microsoft Corporation, Redmond, WA) and graphed in Prism (Version 6, GraphPad, La Jolla, CA). One-way analyses of variance (ANOVAs) with repeated measures and Greenhouse-Geisser corrections were used to compare the mean temperature among groups. Shipments with incomplete data were removed from the one-way ANOVAs. If significance was observed in an ANOVA, Tukey's multiple comparison test was used with individual variances computed for each comparison. The t tests were used to compare mean temperatures by step of supply chain in VCP months versus non-VCP months.

To better understand temperature outliers, we classified each shipment by the number of times temperature sensor values exceeded certain National Shellfish Sanitation Program (NSSP) criteria. These criteria were oysters with internal temperatures >50°F (10°C) or a shipping environment >45°F (7.22°C). We also added another criterion for cold abuse, oysters or their environment held at <35°F (1.67°C). This is not part of the NSSP; however, oysters held below 35°F are susceptible to gaping. We noted if shipments were above or below the criteria for 1 h or more (on the basis of readings taken at 10-min intervals).

### V. parahaemolyticus modeling

We modeled the expected abundance of V. parahaemolyticus in oysters and the associated risk of gastroenteritis by using internal oyster temperature data. Statistical models were based on the U.S. Food and Drug Administration's risk assessment of V. parahaemolyticus in raw oysters (11). The models first estimated the abundance of bacteria at the point of harvest and determined bacterial growth rate when exposed to temperatures higher than a refrigeration threshold of 41.7°F (5.4°C), as well as bacterial die-off rate when stored at or below the refrigeration threshold. The V. parahaemolyticus abundance estimation was modeled iteratively so that the previous estimation informed future abundance, whereas the calculation of risk was based on the level of bacteria at a given time point. The abundance at the point of harvest was estimated as a function of harvest water temperature by using the following equation:

$$V = - 0.63 \times 0.1 \it W$$

where |$$$V$$$| is the log-transformed number of V. parahaemolyticus bacteria per gram of oyster meat (Vp/g) at the time of harvest, and |$$$W$$$| is surface water temperature (°C) measured at the time of harvest. Vibrio abundance postharvest was calculated by using a growth or die-off model, which can be expressed as follows:

$${A_i} = \left\{ {\matrix{ {{A_{i - 1}} + 0.00372\left( {{T_i} - 5.4} \right)} \hfill&{{\rm{if}}\ {T_i} \ge 5.4\circ {\rm{C}}} \hfill \cr {{A_{i - 1}} - 0.0003} \hfill&{{\rm{if}}\ {T_i} \lt 5.4\circ {\rm{C}}} \hfill \cr } } \right.$$

where |$$${A_i}$$$| is the log-transformed Vp/g at time point |$$$i$$$|, |$$${A_{i - 1}}$$$| is abundance at the previous time point (10 min prior) and its value at the point of harvest is |$$$V$$$|, and |$$$T$$$| is the ambient air temperature (°C) measured at time point |$$$i$$$|. Risk was calculated at each time point by using a beta-Poisson dose-response model, which can be expressed as

$${R_i} = 1 \times {10^{ - 5}} \times \left( {1 - {{\left( {1 + {{D_i} \over {3.54 \times {{10}^7}}}} \right)}^{ - 0.6}}} \right)$$

where |$$${R_i}$$$| is the expected number of gastroenteritis cases per 100,000 servings of one dozen oysters, and |$$${D_i}$$$| is the dose of pathogenic V. parahaemolyticus that is estimated by using the following equation:

$${D_i} = 36 \times {10^{A_i}}$$

Given that the observable doses only occurred on the linear portion of the estimated beta-Poisson slope, a linear version of the model was approximated by using the Taylor series:

$${R_i} = 1 \times {10^{ - 5}}\left( {{{0.6} \over {3.54 \times {{10}^7}}}} \right) \times 36 \times {10^{A_i}} = 0.0061 \times {10^{A_i}}$$

For all modeling efforts, air temperature at the time of harvest was held constant in the model until sensors were placed in oysters; water temperature at harvest was substituted if air temperature was not measured. All modeling was performed in R statistical software, version 3.4.3 (28).

### Overview of study population

We recruited 39 participants for the temperature-tracking portion of the study, and 64% of participants agreed to be interviewed. Participants' roles in the supply chain are reported in Table 1. Most groups (i.e., producers, freight carriers, wholesalers, wholesale delivery companies, and food retailers) had high enrollment for the interview, with the exception of restaurants.

TABLE 1

Study population in the Chesapeake Bay and self-reported cold-chain temperatures

Figure 1 provides a simplified diagram of the supply chain on the basis of interviews and our observations. Most products were transported from producers to wholesalers by a third-party freight carrier. Wholesalers operated their own fleets of refrigerated trucks and delivered products to local food retail and restaurant customers. Four oyster producers in the study used direct sales, which typically used direct to consumer third-party freight carriers.

FIGURE 1

Supply chains for oysters produced in the Chesapeake Bay and sold in surrounding states. Data on supply chains were collected from interviews with participants.

FIGURE 1

Supply chains for oysters produced in the Chesapeake Bay and sold in surrounding states. Data on supply chains were collected from interviews with participants.

Close modal

### Oyster producers: site visits

We visited six farms during VCP months and collected interview and temperature data related to harvest and on-farm processing (Tables 1 and 2 and Fig. 2, and Tables S1 and S2 contain values in degrees Celsius). On average, the harvest water temperature was 73.2 ± 4.0°F (range: 70 to 80°F) (22.9 ± 2.2°C; range: 21.1 to 26.7°C) and the air temperature at harvest was 70.3 ± 1.5°F (range: 68 to 72°F) (21.3 ± 0.8°C; range: 20.0 to 22.2°C). Processing steps were similar among the producers and involved different combinations of washing, grading, hand sorting, boxing, and mechanical refrigeration.

TABLE 2

Temperature control during oyster harvesting and on-farm processing during six farm visits, June to September 2017

FIGURE 2

Internal oyster temperature during harvest and on-farm processing at six farms in the Chesapeake Bay. (A and B) Producer 1 at two time points on the same day (A: n = 2, 7 a.m. harvest; B: n = 2, 12 p.m. harvest), (C) producer 2 (n = 4), (D) producer 3 (n = 4), (E) producer 4 (n = 3), (F) producer 5 (n = 4), and (G) producer 6 (n = 2). The dashed line represents the NSSP temperature criteria. The red line is the mean.

FIGURE 2

Internal oyster temperature during harvest and on-farm processing at six farms in the Chesapeake Bay. (A and B) Producer 1 at two time points on the same day (A: n = 2, 7 a.m. harvest; B: n = 2, 12 p.m. harvest), (C) producer 2 (n = 4), (D) producer 3 (n = 4), (E) producer 4 (n = 3), (F) producer 5 (n = 4), and (G) producer 6 (n = 2). The dashed line represents the NSSP temperature criteria. The red line is the mean.

Close modal

Mechanical refrigeration was the primary means of cooling among producers in this study; even producers who used layered ice or ice slurries did so to prevent the product from warming in ambient conditions and not in place of mechanical refrigeration. Half of the producers used ice during processing: one producer used an ice-filled hopper, one producer used an ice bath to draw out worms, and two producers used ice to chill bagged products before mechanical refrigeration. Two producers did not use same-day shipment and preferred to instead chill products overnight in walk-in refrigerators.

### Oyster producers: time to temperature control

Important benchmarks for Vibrio control are the time to temperature control and the time to reach an internal oyster temperature ≤50°F. The NSSP defines temperature control as maintaining the environment at ≤45°F by using ice, mechanical refrigeration, or other approved means (27). To compare across all farms, we used the time to mechanical refrigeration as the starting point for temperature control. Producers in this study achieved temperature control on average in 2.5 ± 1.0 h (range: 1.2 to 4.0 h; Table 2). Products reached an internal oyster temperature ≤50°F within 5.7 ± 3.0 h (range: 3.1 to 10.2 h) after harvest (Table 2).

### Oyster producers: temperature profiles

Figure 2 presents annotated temperature profiles for each of the six producers during VCP months of June to September. Several details are important to note in Figure 2. Figure 2A and 2B are harvests that occurred on the same day at one farm—the first, a large morning harvest, and the second, a smaller harvest to fulfill last-minute orders. Figure 2C has a notable point of inflection in temperature when box lids were added during refrigeration that slowed product cooling, which produces a similar insulating effect as putting a lid on a cup of hot coffee. The producer in Figure 2C appeared to have an undersized refrigerator chiller, which also increased the time to cool the product. Producers in Figure 2D and 2G processed oysters the day before shipment. In Figure 2E, the producer washed and refrigerated the product immediately after harvesting and then later rewashed the product with 60°F (15.56°C) tap water, which created a temperature spike. In Figure 2F, the producer used liberal amounts of ice at every stage of processing and achieved rapid, staged cooling.

### Supply chain: study population

We measured internal oyster temperature and environment temperature throughout supply chains with temperature sensors. A total of 156 sensors were used from February to September 2017, including 34 sensors implemented as part of a pilot study (see Fig. S1). The overall return rate for sensors with usable data was 81%, which generated 5,250 h of temperature data. The sensors without usable data (19%) included sensors that were knocked off of boxes, tampered with, lost, discarded, or did not contain usable data due to sensor malfunction or human error in programing. (Five boxes of Chesapeake Bay oysters with temperature sensors were shipped to California [n = 1], Illinois [n = 2], Minnesota [n = 1], and North Carolina [n = 1] and were not included in this data set because they shipped to destinations outside of the Chesapeake Bay region.)

### Supply chain: temperature by stage of supply chain

Oysters entered the supply chain warmer in VCP months than a non-VCP month; however, the final oyster temperature at the retail level of the supply chain was the same (Supporting Information text and Table S3). After determining that VCP was a significant factor, we pooled the temperature sensor data by VCP status and by stage of supply chain (Fig. 3).

FIGURE 3

Strip plots of (A and B) internal oyster temperature and (C and D) environmental temperature for shipments in Vibrio control plan (VCP) months (B and D: n = 26) and a non-VCP month (A and C: n = 12). The black bar is the mean value, red or grey circles are individual samples, and the dashed line represents the NSSP temperature criteria. Producer values are point estimates for the temperature just before pickup, while all other values are the average temperature reading for each sample.

FIGURE 3

Strip plots of (A and B) internal oyster temperature and (C and D) environmental temperature for shipments in Vibrio control plan (VCP) months (B and D: n = 26) and a non-VCP month (A and C: n = 12). The black bar is the mean value, red or grey circles are individual samples, and the dashed line represents the NSSP temperature criteria. Producer values are point estimates for the temperature just before pickup, while all other values are the average temperature reading for each sample.

Close modal

We hypothesized that internal oyster temperatures would decrease as the product moved through the cold chain, a concept that agrees with self-reported cold-chain temperatures in Table 1. In a non-VCP month, there was no significant difference in internal oyster temperature comparing all supply chain groups (producers, freight carriers, wholesale, wholesale delivery, food retailers or restaurants; ANOVA: F = 2.4, P = 0.18). There was a significant difference in the box temperature among all groups (ANOVA: F = 6.1, P = 0.03), but these differences were not statistically significant when comparing neighboring groups, such as producers to freight carriers, freight carriers to wholesalers, or wholesalers to food retailers or restaurants.

In VCP months, there were significant differences in both the internal oyster temperature (ANOVA: F = 76.4, P < 0.0001) and the box temperature (ANOVA: F = 38.8, P < 0.0001). Pairwise comparisons suggest that producers, when holding product under temperature control, maintain boxes at cooler temperatures than freight carriers (P < 0.05), but producers and freight carriers had similar internal oyster temperatures. Freight carriers maintained boxes at warmer temperatures than wholesalers (P < 0.0001), which led to warmer internal oyster temperatures (P < 0.0001). Box temperatures were not different between wholesale and wholesale delivery or between wholesale delivery and food retailers or restaurants, suggesting that a relatively uniform environmental temperature was maintained along this portion of the supply chain. Internal oyster temperatures were cooler in wholesale delivery than wholesale (P < 0.01), perhaps due to the use of ice during delivery. There was no difference between internal oyster temperatures during wholesale delivery and at food retailers or restaurants. (See Table S4 for P values from all tests.)

### Supply chain: outlier analysis

In addition to comparing mean values, it is also useful to analyze outliers when the temperature was warmer or colder than expected for sustained periods of time. Overall, 7 (19%) of 36 of shipments had internal oyster temperatures greater than 50°F for more than 1 h (all were in VCP months). The product temperature in these seven shipments exceeded NSSP criteria; the maximum internal oyster temperatures were 50.9 (for five shipments), 52.7, and 54.5°F (10.5 [for five shipments], 11.5, and 12.5°C). Over four-fifths of shipments (21 [81%] of 26) in VCP months were held in storage conditions above 45°F for over 1 h.

Internal oyster temperatures were less than 35°F for more than 1 h in 10 (28%) of 36 shipments, which put products at risk for freezing. Cooler internal oyster temperatures were more common in a non-VCP month than in VCP months.

### Supply chain: exemplar temperature profiles

We plotted temperature profiles of six shipments to show typical examples of temperature control issues (Fig. 4). (Temperature profiles for all shipments are available in Fig. S2.) Figure 4A depicts a 100-count waxed box shipped in March 2017. The box was iced by the producer just before freight carrier pickup, and the product remained at near freezing temperatures for ∼24 h, which could kill oysters or reduce shelf life (we did not visually inspect the box for gaping or mortalities). Figure 4B shows a 50-count box of oysters shipped direct to a consumer in July 2017. The packaging was a polystyrene cooler containing gel packs nested inside a cardboard box. The box was shipped by using a direct to consumer freight carrier with 2-day ground delivery. The product temperature slowly climbed from the middle 30s to the low 40s in degrees Fahrenheit during shipment, but remained well below 50°F, indicating that 2-day shipment under these circumstances was acceptable. Figure 4C and 4D show short and long periods of time when 100-count boxes of oysters were outside of temperature control. The internal oyster temperature slowly rose and then fell once temperature control was recovered by the wholesaler. Figure 4D shows one of five shipments in which internal oyster temperatures exceeded 50°F for more than 1 h (as described previously). We suspect that these issues could be due to the product being stored on a loading dock between trips in a refrigerated truck. Figure 4E depicts a spike in temperature during delivery of a 100-count box of oysters to a food retailer or restaurant in March 2017. The product temperature stabilized after the box was moved to the walk-in refrigerator. Figure 4F was more severe and shows a food retailer or restaurant in June 2017 whose refrigerator was either malfunctioning or set at an unsafe temperature for storage of shellfish.

FIGURE 4

Six examples of temperature-related issues in Chesapeake Bay–farmed oyster cold chains. Temperature profiles for all oyster shipments are provided in the Supporting Information. Temperature profiles of internal oyster temperatures and environment temperatures from harvest to food retailers or restaurants. The sensor sampling interval was 1 min at the oyster producer and 10 min in the supply chain. The dashed line represents the NSSP temperature criteria. P, producer; T, freight carrier (truck); W, wholesale; R, food retailer or restaurant; C, consumer. The number following P, T, W, or R was assigned to each participant to provide anonymity.

FIGURE 4

Six examples of temperature-related issues in Chesapeake Bay–farmed oyster cold chains. Temperature profiles for all oyster shipments are provided in the Supporting Information. Temperature profiles of internal oyster temperatures and environment temperatures from harvest to food retailers or restaurants. The sensor sampling interval was 1 min at the oyster producer and 10 min in the supply chain. The dashed line represents the NSSP temperature criteria. P, producer; T, freight carrier (truck); W, wholesale; R, food retailer or restaurant; C, consumer. The number following P, T, W, or R was assigned to each participant to provide anonymity.

Close modal

### Modeling V. parahaemolyticus abundance and health risks

Models of V. parahaemolyticus abundance and associated risks for all shipments are displayed in Figure 5. Temperature abundance profiles for each shipment are available in the Supporting Information. V. parahaemolyticus models estimated initial abundance at the time of harvest, which ranged from 30 to 108 counts per g. The model predicted internal growth of V. parahaemolyticus on the farm and before temperature control was no more than 25 counts per g overall. After temperature control was achieved, there was a gradual decline in abundance across all shipments, except one (Fig. 4F), which remained at temperatures above the refrigeration threshold throughout the majority of sensor measurements. The mean estimated V. parahaemolyticus abundance at the end of all shipments' cold chains (when the product was delivered) was 50.5 counts per g, with a range of 17.8 to 115.13 counts per g. The expected risk of gastroenteritis was relatively low across all shipments, with a mean risk of 0.03 cases per 100,000 servings of raw oysters and with a range of 0.01 to 0.07 cases per 100,000 servings. (Modeled V. parahaemolyticus abundance and associated risks for each shipment are available in Fig. S3.)

FIGURE 5

Estimated Vibrio parahaemolyticus abundance per gram oyster tissue (left y axis) and the risk of illness as cases per 100,000 servings (right y axis) from oysters produced in the Chesapeake Bay and shipped to surrounding states. Estimations of both can be displayed simultaneously due to the linear approximation of the beta-Poisson dose-response model. Vibrio abundance at harvest was estimated on the basis of water temperature, and growth in supply chains was calculated by using iterative temperature-based models. The black lines represent the abundance or risk of individual oyster shipments. The blue line depicts the mean abundance or risk across all oysters estimated by using a generalized additive model, and the grey band displays the corresponding 95% confidence interval. Vp/g, V. parahaemolyticus bacteria per gram. This figure only presents data from shipments made in Vibrio Control Plan months.

FIGURE 5

Estimated Vibrio parahaemolyticus abundance per gram oyster tissue (left y axis) and the risk of illness as cases per 100,000 servings (right y axis) from oysters produced in the Chesapeake Bay and shipped to surrounding states. Estimations of both can be displayed simultaneously due to the linear approximation of the beta-Poisson dose-response model. Vibrio abundance at harvest was estimated on the basis of water temperature, and growth in supply chains was calculated by using iterative temperature-based models. The black lines represent the abundance or risk of individual oyster shipments. The blue line depicts the mean abundance or risk across all oysters estimated by using a generalized additive model, and the grey band displays the corresponding 95% confidence interval. Vp/g, V. parahaemolyticus bacteria per gram. This figure only presents data from shipments made in Vibrio Control Plan months.

Close modal

### Main findings

To our knowledge, this is the first study in the United States to assess the performance of cold chains for shellfish. During summer months when VCPs were in effect, internal oyster temperatures were higher among producers, freight carriers, and wholesalers than those same groups in March, a month with cooler weather and no VCP. We attribute this difference to warmer harvest waters and higher storage temperatures among some producers and freight carriers in summer months. The NSSP advises that shellfish should be held below 50°F internal temperature and maintained in an environment that is 45°F or below by using ice or mechanical refrigeration (24). When modeling the effects of temperature on V. parahaemolyticus, we found moderate bacterial growth before oysters entered the cold chain. Once oysters were under temperature control, models suggest no further V. parahaemolyticus growth occurred, and there was a moderate drop-off in bacterial abundance. This indicates that the cold chain was effective in mitigating the risk of V. parahaemolyticus, even when oysters were stored slightly above the NSSP limits. Exposure to warm temperatures immediately following harvest and before temperature control is achieved appears to be the period of greatest risk for V. parahaemolyticus growth in Chesapeake Bay oyster supply chains. Our findings are supported by a validation study confirming that the model agrees with V. parahaemolyticus growth rates in postharvest Chesapeake Bay oysters (26).

### Comparison to previous work

Performance of shellfish cold chains is relatively understudied; we identified only one report from Australia that used temperature sensors to assess cold chains (19). In that study, 42 to 50% of shipments were not in compliance with Australian Shellfish Quality Assurance Programs due to a combination of time and temperature issues, including cold abuse (19). Supply chains in the Chesapeake Bay faced similar challenges related to temperature control; however, it is difficult to compare between the studies because different regulations are used in each country. Similar to the Australian study, we found each group within the supply chain had aspects of temperature control that could be improved. In fact, several participants noted that continual improvement is what they seek in their own business and for businesses they deal with. One participant in our study noted, “selling oysters is a team effort up and down the supply chain.” What follows is a discussion of temperature control at each step of the cold chain and identified areas for improvement.

### Oyster production

Achieving rapid temperature control of harvested oysters is critical for food safety. Our findings were comparable to mechanical refrigeration used by Jones et al. (15) but slower than using ice for cooling. Differences between studies may be due to the experimental design, because our study tracked the temperature of the entire harvest with multiple sensors. Producers who wish to improve product cooling may consider using temperature sensors for internal validation studies with the help of local seafood extension agents or state regulators. Several state regulators in the northeast have implemented temperature sensors at farms to learn more about shellfish harvest and processing. If cooling the product for same-day shipping is challenging, oysters can be harvested and stored overnight in refrigeration to ensure adequate cooling has been achieved.

### Transportation

Freight carriers in our study transported products from producers to wholesalers by using refrigerated trucks maintained at a median self-reported temperature of 34°F (1.4°C). As one freight carrier described it, “my job is to maintain temperature only, not to bring the temperature down.” A certain amount of time outside refrigeration is expected, as products are loaded and unloaded at facilities. In some cases, we measured environmental temperatures above 45°F, and when prolonged, they affected internal oyster temperatures. Two areas that could be improved are reducing the time products spend unrefrigerated at loading docks or transfer stations and adding supplemental ice if products are stored at loading docks or transfer stations for extended periods of time.

### Wholesale

Wholesalers purchased oysters from producers and sold them to food retailers and restaurants. Wholesalers had the most advanced mechanical refrigeration systems of any group in the supply chain and provided a stabilizing force on the cold chain with self-reported temperatures between 38 and 42°F (3.3 and 5.6°C). Our sensor data suggest environmental temperatures in storage and delivery were not an issue for wholesalers. Holding oysters at near freezing temperatures in cooler months was an issue that could be improved upon and could have been caused by excessive icing during delivery to food retailers and restaurants. Wholesalers could contribute to improving overall performance of their cold chains by conducting their own or third-party quality control audits, which some participants indicated as an area for future work.

### Retailers and restaurants

For food retailers and restaurants, oysters are just one of many products they purchase. Oysters can be ordered most days of the week, and product turnover is high. Large food retailers, such as supermarket chains, specify product temperatures in their hazard analysis and critical control point plans and can influence the practices of upstream businesses. Oysters in this study were maintained in refrigerators set to a self-reported temperature of 38°F, and our sensor data indicate that temperature control was not an issue for restaurants or food retailers. There was no consistent method for processing and storing oysters; this is an area in which education, training, and cross talk between the restaurant industry and seafood industry could be useful. Participants were aware of food safety practices specific to shellfish, such as maintaining tags for 90 days, and few had been involved in product recalls. We did not track oyster mortality; however, some participants noted that on occasion boxes had to be returned or discarded due to gaping. Addressing the underlying reasons behind gaping could be a topic for future studies.

### Vibrio modeling

The cold-chain temperature data supported models of Vibrio growth and illness risks, and these models indicate a very low risk of gastroenteritis from consuming the oysters in this study. The models, derived from the U.S. Food and Drug Administration's risk assessment of V. parahaemolyticus in raw oysters (32), have been widely incorporated into cold-chain supply regulations in shellfish (24), and its framework continues to be used when estimating the risk of V. parahaemolyticus (30). In this study, abundance and risk estimates were restricted to values of central tendency. Further incorporating the known variability (e.g., abundance across oyster samples) and parameter uncertainty into these models would likely increase the estimated range for the risk of gastroenteritis from consuming oysters in the sampled supply chains. Future work will incorporate such variability and uncertainty to provide a probabilistic risk assessment of these cold chains. Future work could also improve upon the existing risk models. For example, the current model framework currently relies solely on water temperature to determine bacterium abundance at the point of harvest (32). However, many studies have indicated that additional environmental parameters, such as salinity and turbidity, can improve risk estimation (14), particularly in the Chesapeake Bay (9). Furthermore, the risk assessment was completed in 2005, and since then a number of studies have further investigated the population dynamics of V. parahaemolyticus during a variety of harvest and postharvesting practices (7, 9, 15, 17, 21). Incorporating these studies into future modeling efforts of the supply chain will likely further improve our understanding of V. parahaemolyticus populations and the risk of foodborne illness from consuming raw oysters. Others have developed predictive models of V. parahaemolyticus growth in Pacific oyster supply chains under different scenarios (11).

### Strengths and limitations

There are several strengths and limitations of the study. The temperature sensors used in the study had significant benefits in their small size and ability to fit inside an oyster and take readings over several weeks. The sensors, however, required manual inspection, did not measure humidity, and were relatively expensive (US\$50 each). The sensors did not have global positioning system or radio frequency identification capabilities, so we contacted participants to ascertain the time that boxes arrived or departed facilities or both. Our study included a limited number of visits to producers; repeated visits were performed at half of the study farms, and visiting more farms would improve our ability to generalize about temperature control at the farm level. Our farm visits were observational, and future research could manipulate variables to assess the effectiveness of different practices related to harvesting and handling. Many freight carriers, wholesalers, and retailers and restaurants were repeatedly sampled during the study period, which increased our ability to generalize about this part of the supply chain. The oyster supply chains in the study did not involve international shipments, and future cold-chain studies should include international seafood supply chains.

Farmed oysters are an environmentally sustainable product requiring no feed inputs, and opportunities to refine food safety in production, processing, and supply chains can support growth within the industry. For many, oysters are a luxury food; however, lessons learned from oyster cold chains can be applied to other refrigerated seafood products in the United States and elsewhere. We hope the findings of this study provide a better understanding of cold chains for shellfish, provide baseline data for modeling Vibrio growth, and support industry and government regulations and policy development.

This research was primarily supported by the National Oceanic and Atmospheric Administration Saltonstall-Kennedy Grant Program (16GAR008). Some materials were purchased by the Johns Hopkins Center for a Livable Future with a gift from the Greater Kansas City Community Foundation. Modeling efforts performed by B.J.K.D were supported by the National Institutes of Allergy and Infectious Diseases (grant 1R01AI123931-01A1).

Supplemental material associated with this article can be found online at: https://doi.org/10.4315/0362-028X.JFP-18-044.s1.

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