Jeff Kaufman's Writing
https://www.jefftk.com/p
Jeff Kaufman's Writing from post 2264 backen-us/p/honor-system-for-vaccinationHonor System for Vaccination?
https://www.jefftk.com/p/honor-system-for-vaccination
contracovid-1924 Sep 2023 08:00:00 EST<p><span>
In June the social dance I help organize decided to </span>
<a href="https://blog.bidadance.org/2023/06/some-mask-optional-dances.html">drop
its requirement</a> that people be vaccinated and boosted:
<p>
</p>
<blockquote>
Checking vaccination cards at the door requires an additional
volunteer, is a hassle for attendees, and at this point the
communities we draw from are overwhelmingly vaccinated. While we
still recommend staying up to date on vaccinations (including your flu
shot and covid booster in the fall!), we're dropping this requirement
starting with our July 2nd dance.
</blockquote>
<p>
One question we got, both in person and electronically, was why not
keep requiring vaccination on the honor system? Speaking for myself
and not BIDA, I think this is rarely a good policy for events.
</p>
<p>
(I'm setting aside the question of whether vaccination is helpful in
reducing risk to others, and writing all of this under the assumption
that it is. I do think this is unclear, especially if a policy does
not require a recent booster.)
</p>
<p>
The issue is, an honor system is a bad fit when getting caught is very
unlikely and the harms of violating the rule are diffuse and unclear.
If an unvaccinated dancer ignores the policy and does happen to bring
an infection to the dance that vaccination would have prevented, it's
very unlikely that anyone will ever know. There's no feedback
mechanism promoting honesty here. Instead, you're filtering
unvaccinated people for ones who are ok violating the policy, and
nudging more unvaccinated people toward being dishonest in this way.
</p>
<p>
Compare this to a dance in a small community that does payment on the
honor system: an unattended fiddle case by the door for you to drop
your money into. If someone doesn't contribute it's much more likely
that people will notice, and it's also much clearer that when you
defect you're hurting people. There's still some pressure toward
dishonesty, but there's enough pushing the other way that it still
works well.
</p>
<p>
The honor system can work well in some cases, but I don't think this
is one of them.
</p>
<p><i>Comment via: <a href="https://www.facebook.com/jefftk/posts/pfbid0gwa7i884k2nmFPbrBcJbzqu1xFQ2FN8rRfTaAkDmCJwJNdrqsSduNjUC3QykbDvtl">facebook</a>, <a href="https://lesswrong.com/posts/8dDzqC4H57zeNqHC2">lesswrong</a>, <a href="https://mastodon.mit.edu/@jefftk/111119946932019103">mastodon</a></i></p>
/p/hand-writing-mathmlHand-writing MathML
https://www.jefftk.com/p/hand-writing-mathml
mathtech23 Sep 2023 08:00:00 EST<p><span>
When I write posts I use raw HTML. Yes, the modern thing to do is
probably </span>
<a href="https://en.wikipedia.org/wiki/Markdown">Markdown</a>, but HTML
was designed for hand-coding and still works well for that if you
don't want anything especially fancy. But what if you want math?
<p>
Previously when I've wanted to do math I've written it out as
fixed-width ASCII:
</p>
<p>
</p>
<pre>
e^(-7t)
</pre>
<p>
In my editor this looks like:
</p>
<pre>
<pre>
e^(-7t)
</pre>
</pre>
<p>
This is reasonably readable, works anywhere, and I like the aesthetic.
I probably should have stuck with it, but after helping publish <a href="https://naobservatory.org/reports/predicting-virus-relative-abundance-in-wastewater/">a
report</a> that included some traditionally-formatted equations and
learning that <a href="https://developer.mozilla.org/en-US/docs/Web/MathML">MathML</a>
has been supported cross-browser since the beginning of the year (<a href="https://mathml.igalia.com/">thanks Igalia!</a>), I decided to
try it out. I wrote the equations in two <a href="https://www.jefftk.com/p/weekly-incidence-vs-cumulative-infections">recent</a>
<a href="https://www.jefftk.com/p/weekly-incidence-including-delay">posts</a>
in it, and am mixed on the experience.
</p>
<p>
It definitely does look nicer:
</p>
<p>
<math display="block">
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mn>7</mn>
<mi>t</mi>
</mrow>
</msup>
</math>
</p>
<p>
On the other hand, here's how it looks in my editor:
</p>
<p>
</p>
<pre>
<math display=block>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mn>7</mn>
<mi>t</mi>
</mrow>
</msup>
</math>
</pre>
<p>
There's a small learning curve on when to use the different tags, but
mostly it's just very verbose. And I think, needlessly so? That
"<code>-</code>" is an operator, "<code>7</code>" is a number, and
"<code>t</code>" is an identifier could all be the default. Then I
could just write:
</p>
<p>
</p>
<pre>
<math display=block>
<msup>
e
<mrow>
-7t
</mrow>
</msup>
</math>
</pre>
<p>
And we could remove many uses of <code><mrow></code> too: a
series of characters without whitespace separating them could
be already treated as a group:
</p>
<p>
</p>
<pre>
<math display=block>
<msup>
e
-7t
</msup>
</math>
</pre>
<p>
Of course if you wanted to use a character for a non-traditional
purpose you could still mark it up as one, but a good set of defaults
would make MathML much more pleasant. I'd hate to have
to read and write blog posts as:
</p>
<p>
</p>
<pre>
<word><lt>h</lt><lt>e</lt><lt>l</lt><lt>l</lt><lt>o</lt></word>
<word><lt>w</lt><lt>o</lt><lt>r</lt><lt>l</lt><lt>d</lt></word>
<pnct>.</pnct>
</pre>
<p>
I know I'm about <a href="https://web.archive.org/web/19990428145738/http://www.w3.org/Math/mathml-faq.html">25
years too late</a> on this, and I'm happy that a pure-HTML solution is
now cross-browser, but it's still sad we ended up so close to a
comfortable hand-editable solution.
</p>
<p>
(Just use <a href="https://www.mathjax.org/">MathJax</a>? Nope—I
don't want a runtime dependency on JS. Though I could see including a
LaTeX-to-MathML or a MathML-verbosifier step at build time.)
</p>
<p><i>Comment via: <a href="https://www.facebook.com/jefftk/posts/pfbid02EBNF19Y2Z7bf1EzSes2eiL3fdX8RFXBXWUpCzb1Koj4VcFRyCCoxms3KFfWirDwMl">facebook</a>, <a href="https://lesswrong.com/posts/aQDNfCrhqYg95sMDN">lesswrong</a>, <a href="https://mastodon.mit.edu/@jefftk/111114175926556334">mastodon</a></i></p>
/p/fund-transit-with-developmentFund Transit With Development
https://www.jefftk.com/p/fund-transit-with-development
housingtransit22 Sep 2023 08:00:00 EST<p><span>
When transit gets better the land around it becomes more valuable:
many people would like to live next to a subway station. This means
that there are a lot of public transit expansions that would make us
better off, building space for people to live and work. And yet, at
least in the US, we don't do very much of this. Part of it is that
the benefits mostly go to whoever happens to own the land around the
stations.
</span>
<p>
A different model, which you see with historical subway construction
or Hong Kong's <a href="https://en.wikipedia.org/wiki/MTR_Corporation">MTR</a>, uses the
increase in land value to fund transit construction. The idea is, the
public transit company buys property, makes it much more valuable by
building service to it, and then sells it.
</p>
<p>
While I would be pretty positive on US public transit systems adopting
this model, I have trouble imagining them taking it on. Instead,
consider something simpler and more distributed: private developers
paying to expand public transit.
</p>
<p>
Consider the proposed <a href="https://www.somernova.community/master-plan">Somernova
Redevelopment</a>, in Somerville MA:
</p>
<p>
<a href="https://www.jefftk.com/somernova-with-existing-rail-big.png"><img src="https://www.jefftk.com/somernova-with-existing-rail.png" width="550" height="473" class="mobile-fullwidth" style="max-width:100.0vw; max-height:86.0vw;" srcset="https://www.jefftk.com/somernova-with-existing-rail.png 550w,https://www.jefftk.com/somernova-with-existing-rail-2x.png 1100w"><div style="height:min(86.0vw, 473px)" class="image-vertical-spacer"></div></a>
</p>
<p>
This is a proposed $3.3B 1.9M-sqft development, adjacent to the <a href="https://en.wikipedia.org/wiki/Fitchburg_Line">Fitchburg
Line</a>. A train station right next to it would make a ton of sense,
and could be done within the existing right of way without any
tunneling. Somernova briefly mentions this idea on p283, where they
say:
</p>
<p>
</p>
<blockquote>
Introducing a new train station on campus could dramatically reduce
commute times, making all of Somernova within a five minute walk from
the station. We look forward to ongoing dialog about these transit
possibilities with the community and advocates, ensuring we continue
to explore all options for enhanced connectivity longterm.
</blockquote>
<p>
This is pretty vague compared to the rest of the plan, which has a ton
of estimates, but we can make our own. The MBTA recently completed a
long and expensive project to <a href="https://en.wikipedia.org/wiki/Green_Line_Extension">extend the
Green Line</a> along this right of way, which stops at Union Square.
Extending it to Dane Street would require another 0.9km of track and
another station. The overall Green Line extension cost $2.2B for
7.6km, or $290M/km, though this included a bunch of over-designed work
that needed to be thrown away and it <a href="https://transitcosts.com/transit-costs-study-final-report/#case_boston">should
have been far less</a>. This portion is relatively simple compared to
the other work, with no maintenance facility or elevated sections,
though it does include three bridges and moving a
substation. Accepting the $290M/km figure, though, we could estimate
$260M.
</p>
<p>
A $260M extension would raise Somernova's construction costs by under
8%, less if you include the costs of the land, and I expect would
raise the value of the completed project by well more than
that—rents right next to subway stations are generally a lot
higher than farther away. So even though Somernova would not capture
all of the benefits of the new station they would capture enough to
come out ahead.
</p>
<p>
This isn't a new idea: <a href="https://ir.federalrealty.com/news-releases/news-release-details/mbta-board-approves-orange-line-t-agreement">in
2011</a> the <a href="https://en.wikipedia.org/wiki/Assembly_Square">Assembly Row
developers</a> made a deal with the MBTA to fund an <a href="https://en.wikipedia.org/wiki/Assembly_station">infill
station</a> for their development. Because this was just a station it
was <a href="https://web.archive.org/web/20120527221939/http://articles.boston.com/2011-10-06/news/30253264_1_blue-line-orange-line-red-line/2">cheaper</a>:
$15M from the developer and $16M from the federal government.
</p>
<p>
Another place where something like this could make sense is building
housing at Route 16. The other branch of the Green Line Extension,
along the <a href="https://en.wikipedia.org/wiki/Lowell_Line">Lowell
Line</a>, could be extended 1.4km to <a href="https://en.wikipedia.org/wiki/Green_Line_Extension#Mystic_Valley_Parkway_extension">Route
16</a>. Figuring the same $290M/km this would be $400M, though as a
straight-forward project in an existing right of way it <a href="https://transitcosts.com/new-data/">should be possble</a> to do
it for about half that. Next to the site is a liquor store and
supermarket, about 150k sqft:
</p>
<p>
<a href="https://www.jefftk.com/whole-foods-plot-big.png"><img src="https://www.jefftk.com/whole-foods-plot.png" width="550" height="443" class="mobile-fullwidth" style="max-width:100.0vw; max-height:80.5vw;" srcset="https://www.jefftk.com/whole-foods-plot.png 550w,https://www.jefftk.com/whole-foods-plot-2x.png 1100w"><div style="height:min(80.5vw, 443px)" class="image-vertical-spacer"></div></a>
</p>
<p>
Let's say you build ground-floor retail (with more than enough room
for the current tenants) and many stories of housing above it. It's
not currently zoned for this, but zoning is often dependent on transit
access and this is something the city could fix (ex: <a href="https://en.wikipedia.org/wiki/Assembly_Square">Assembly
Square</a> got special zoning). A hard limit on height is <a href="https://www.massport.com/logan-airport/about-logan/logan-airspace-map/">Logan
Airport airspace</a> but that's high enough you wouldn't come close:
</p>
<p>
<a href="https://www.jefftk.com/route-16-logan-airspace-restrictions-big.png"><img src="https://www.jefftk.com/route-16-logan-airspace-restrictions.png" width="550" height="408" class="mobile-fullwidth" style="max-width:100.0vw; max-height:74.2vw;" srcset="https://www.jefftk.com/route-16-logan-airspace-restrictions.png 550w,https://www.jefftk.com/route-16-logan-airspace-restrictions-2x.png 1100w"><div style="height:min(74.2vw, 408px)" class="image-vertical-spacer"></div></a>
</p>
<p>
The airspace rules would allow a ~65 story building, but lets
say you pattern this off Assembly and go for 23 stories: 22
housing, one retail. With 75% average lot coverage (courtyards) this
would be ~2.6M sqft. Building this might cost $600/sqft (but <a href="https://www.jefftk.com/p/better-construction-cost-estimates">I'd love better
numbers</a>), giving ~$1.6B.
</p>
<p>
There's also a UHaul lot next door, at around 110k sqft which could be included:
</p>
<p>
<a href="https://www.jefftk.com/home-depot-plot-big.png"><img src="https://www.jefftk.com/home-depot-plot.png" width="550" height="462" class="mobile-fullwidth" style="max-width:100.0vw; max-height:84.0vw;" srcset="https://www.jefftk.com/home-depot-plot.png 550w,https://www.jefftk.com/home-depot-plot-2x.png 1100w"><div style="height:min(84.0vw, 462px)" class="image-vertical-spacer"></div></a>
</p>
<p>
Building it out the same way would give you another ~1.9M sqft at a
cost of ~$1.1M, for a total of 4.5M sqft for $2.7B. Add in the subway
extension and it's $3.1B.
</p>
<p>
Even with the cost of the extension this would be quite profitable:
new construction right next to a subway station. Factoring in 20% for
affordable housing, you break even at around $850/sqft, and
extrapolating from the listings I see for Assembly you should be able
to get about $1,000/sqft.
</p>
<p>
I'm not sure why we don't see more of this. Are transit agencies not
very open to it? Are the costs too unpredictable? Are private
developers too car-focused? Am I wrong about how much more valuable
transit access makes places? Are cities unwilling to agree to high
density for newly transit-served locations?
</p>
<p><i>Comment via: <a href="https://www.facebook.com/jefftk/posts/pfbid0bq4kXQjkYGMgG64iJMZLGnn476L4kjw1UNiNkzKqp2S6wWE3Ls1udk85r2ZGrv6ul">facebook</a>, <a href="https://lesswrong.com/posts/zCyWKQfioW2ZKudTm">lesswrong</a>, <a href="https://mastodon.mit.edu/@jefftk/111108481472946860">mastodon</a></i></p>
/p/ward-5-jack-perenick-and-naima-saitWard 5: Jack Perenick and Naima Sait
https://www.jefftk.com/p/ward-5-jack-perenick-and-naima-sait
housingpolicysomerville21 Sep 2023 08:00:00 EST<p><span>
I live in Ward 5 in Somerville, where there are two candidates running
for city council this fall: </span>
<a href="https://www.jackforsomerville.com/">Jack Perenick</a> and
<a href="https://www.naimaforsomerville.org/">Naima Sait</a>. On Sunday
both of them stopped by our neighborhood block party, and we got a
chance to talk. As in
<a href="https://www.jefftk.com/p/somerville-election-thoughts">2021</a>, the
<a href="https://www.jefftk.com/apartment_prices/#2023-09-18&2">high
cost</a> of housing continues to be my primary concern for Somerville,
and I asked them both what their views were.
<p>
It was clear that they both care deeply about resolving the housing
crisis, and seemed like they would prioritize trying to improve the
situation, but when we got into the details there were clear differences.
</p>
<p>
I talked to Jack first, and he told me he was strongly in favor of
upzoning, especially along the major streets, in the squares, and
around transit. His view was that allowing developers to build
taller and maximizing the number of units built was our best tool
for bringing prices down. He didn't think rent stabilization would
work well: rents are already much too high. As we talked it became
clear that the specific details of housing policy were something he
had thought a lot about and where he cared about fine distinctions.
</p>
<p>
I talked to Naima later, after getting something to eat, and she was
in favor of affordable housing, especially city-built, and wary of
developers exploiting the community. She was very interested in
tenant protections, and gave the example of rent stabilization. When
I asked if what kind of stabilization she'd like to see, and she told
me that there are lots of different kinds people propose before moving
on to another voter.
</p>
<p>
I think which candidate is best on housing depends primarily on what
policies you think would help the most. Jack's views are close
<a href="https://www.jefftk.com/p/market-rate-food-is-luxury-food">to my own</a>, and cities
that have <a href="https://www.pewtrusts.org/en/research-and-analysis/articles/2023/04/17/more-flexible-zoning-helps-contain-rising-rents">gone
with upzoning</a> have seen decreases in real rents from 2017 to 2023:
</p>
<p>
</p>
<table cellpadding="5">
<tr>
<th>Location</th>
<th>Nominal</th>
<th>Real
</th>
</tr>
<tr>
<td>Minneapolis MN</td>
<td>+1% </td>
<td>-24%
</td>
</tr>
<tr>
<td>Portland OR</td>
<td>+2% </td>
<td> -23%
</td>
</tr>
<tr>
<td>Tysons VA </td>
<td>+4%</td>
<td>-21%
</td>
</tr>
<tr>
<td>New Rochelle NY</td>
<td>+7%</td>
<td>-18%
</td>
</tr>
<tr>
<td>National</td>
<td>+31%</td>
<td>+6%
</td>
</tr>
<tr>
<td>Ward 5, Somerville MA</td>
<td>+31%</td>
<td>+6%
</td>
</tr>
</table>
<p>
<i>Estimates for Ward 5 from comparing the estimated average cost of a
two-bedroom apartment in Ward 5 between <a href="https://www.jefftk.com/apartment_prices/#2017-09-19&2">2017-09</a>
and <a href="https://www.jefftk.com/apartment_prices/#2023-09-18&2">2023-09</a>;
all others are from <a href="https://www.pewtrusts.org/en/research-and-analysis/articles/2023/04/17/more-flexible-zoning-helps-contain-rising-rents">Pew's
analysis</a> of <a href="https://www.apartmentlist.com/">Apartment
List</a>'s rent estimates.</i>
</p>
<p>
While I like both of them and would be happy being represented by
either, I'd much rather see Jack become one of the <a href="https://www.somervillema.gov/departments/city-council">thirteen people</a>
most responsible for directing Somerville housing policy.
</p>
<p><i>Comment via: <a href="https://www.facebook.com/jefftk/posts/pfbid025dszMB2Ls65RM7CsPcSuP4sPL3MBG5KnqaGWDVDFghduNqWpiPw5PX4B1xMsjcdal">facebook</a>, <a href="https://lesswrong.com/posts/hxn6FjzjmM5boKSdB">lesswrong</a>, <a href="https://mastodon.mit.edu/@jefftk/111103252307093392">mastodon</a></i></p>
/p/weekly-incidence-including-delayWeekly Incidence Including Delay
https://www.jefftk.com/p/weekly-incidence-including-delay
bionao20 Sep 2023 08:00:00 EST<p><span>
</span>
<i>Cross-posted from my <a href="https://data.securebio.org/jefftk-notebook/weekly-incidence-including-delay">NAO Notebook</a></i>
<p>
A few days ago I <a href="https://www.jefftk.com/p/weekly-incidence-vs-cumulative-infections">wrote about</a>
some math behind a scenario where you're trying to identify a new
epidemic based on signals proportional to incidence, and ended up
deriving:
</p>
<p>
<math display="block">
<mfrac>
<mrow><mi>i</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow>
<mrow><mi>c</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>ln</mi>
<mo>(</mo><mn>2</mn><mo>)</mo>
</mrow>
<msub><mi>T</mi><mi>d</mi></msub>
</mfrac></math>
</p>
<p>
Where:
</p>
<ul>
<li>
<math display="inline"><mi>i</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, is
incidence ("how many people are getting sick now")
</li>
<li>
<math display="inline"><mi>c</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, is
cumulative infections ("how many people have gotten sick so far")
</li>
<li>
<math display="inline"><mi>k</mi></math>, is
the exponential growth rate.
</li>
<li>
<math display="inline"><msub><mi>T</mi><mi>d</mi></msub></math>, is
the doubling time (redundant with <math display="inline"><mi>k</mi></math>).
</li>
</ul>
<p>
One big problem with this model, however, is that any conclusions you
make today aren't driven by current incidence, but instead some kind
of delayed incidence. There is, unavoidably, time from infection until
you're making your decision, during which the disease is spreading
further:
</p>
<p>
<a href="https://www.jefftk.com/infection-delay-timeframe-illustration-big.png"><img src="https://www.jefftk.com/infection-delay-timeframe-illustration.png" width="550" height="275" class="mobile-fullwidth" style="max-width:100.0vw; max-height:50.0vw;" srcset="https://www.jefftk.com/infection-delay-timeframe-illustration.png 550w,https://www.jefftk.com/infection-delay-timeframe-illustration-2x.png 1100w"><div style="height:min(50.0vw, 275px)" class="image-vertical-spacer"></div></a>
</p>
<p>
If your signal is "people arrive at the hospital and the doctors
notice a weird cluster", then you need to wait for each infection to
progress far enough to result in hospitalization. How do the
conclusions of the previous post change if we extend our model to
account for a delay, but keep the goal of flagging an epidemic before
1% of people have been infected?
</p>
<p>
Recall that last time we estimated that, for something doubling
weekly, when 1% of people have ever been infected then 0.69% of people
became infected in the last seven days. With a delay of a week,
however, by the time we learn that incidence has hit 0.69% many more
people will have been infected and we'd have missed our 1% goal by a
lot. The effect of delay is that during this time the epidemic will
make further progress, which will depend on the growth rate: with a
shorter doubling period there will be more progress. Can we get an
equation relating cumulative infections to delayed incidence?
</p>
<p>
Let's call delay <math display="inline"><mi>d</mi></math>.
Instead of
<math display="inline"><mfrac>
<mrow><mi>i</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow>
<mrow><mi>c</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow>
</mfrac></math>
we now want
<math display="inline"><mfrac>
<mrow><mi>i</mi><mo>(</mo><mi>t</mi><mo>-</mo><mi>d</mi><mo>)</mo></mrow>
<mrow><mi>c</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow>
</mfrac></math>.
That is, what's the relationship between cumulative infections and
what incidence was when the information we're now getting was derived?
</p>
<p>
We can do a bit of math:
</p>
<p>
<math display="block">
<mfrac>
<mrow><mi>i</mi><mo>(</mo><mi>t</mi><mo>-</mo><mi>d</mi><mo>)</mo></mrow>
<mrow><mi>c</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<msup><mi>e</mi>
<mrow><mi>k</mi><mo>(</mo><mi>t</mi><mo>-
</mo><mi>d</mi><mo>)</mo></mrow></msup>
</mrow>
<msup><mi>e</mi>
<mrow><mi>k</mi><mi>t</mi></mrow></msup>
</mfrac>
<!--
<mo>=</mo>
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<mrow>
<msup><mi>e</mi>
<mrow><mi>k</mi><mi>t</mi></mrow></msup>
<msup><mi>e</mi>
<mrow><mo>-</mo><mi>k</mi><mi>d</mi></mrow></msup>
</mrow>
<msup><mi>e</mi>
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<mo>=</mo>
<mi>k</mi>
<msup><mi>e</mi>
<mrow><mo>-</mo><mi>k</mi><mi>d</mi></mrow></msup>
<mo>=</mo>
<!--
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<mo>(</mo><mn>2</mn><mo>)</mo>
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<msub><mi>T</mi><mi>d</mi></msub>
</mfrac>
<msup><mi>e</mi>
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<msub><mi>T</mi><mi>d</mi></msub></mfrac>
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<mfrac>
<mrow>
<mn>2</mn>
<mi>ln</mi>
<mo>(</mo><mn>2</mn><mo>)</mo>
</mrow>
<msub><mi>T</mi><mi>d</mi></msub>
</mfrac>
<msup><mi>e</mi>
<mfrac>
<mrow><mo>-</mo><mi>d</mi></mrow>
<msub><mi>T</mi><mi>d</mi></msub>
</mfrac>
</msup>
</math>
</p>
<p>
What does this look like, for a few different potential delay values?
</p>
<p>
<a href="https://www.jefftk.com/former-weekly-incidence-by-delay-big.png"><img src="https://www.jefftk.com/former-weekly-incidence-by-delay.png" width="550" height="365" class="mobile-fullwidth" style="max-width:100.0vw; max-height:66.4vw;" srcset="https://www.jefftk.com/former-weekly-incidence-by-delay.png 550w,https://www.jefftk.com/former-weekly-incidence-by-delay-2x.png 1100w"><div style="height:min(66.4vw, 365px)" class="image-vertical-spacer"></div></a>
</p>
<p>
My main takeaway is that, under these assumptions, delay matters less
than I would initially have guessed: the sensitivity you need to
design for is driven by the need to catch slow-growing epidemics. For
example, even with up to eleven days of delay a system sensitive
enough to flag an epidemic that doubles every four weeks is sensitive
enough to detect one that doubles every three days.
</p>
<p>
This isn't the whole story, though, because many of the actions that
you would want to do post-discovery are more urgent with higher growth
rates. This means you need to count the delay of your core response
(ex: implementing <a href="https://www.cdc.gov/nonpharmaceutical-interventions/index.html">NPIs</a>)
in the total delay: you can't allocate the entire delay budget to the
detection system.
</p>
<p>
Another thing to note is that this is all under the assumption that
detection depends on incidence above a threshold, and whether this is
actually the case is unclear for several potential systems. For
example, with wastewater sequencing detection my current best guess is
this primarily would rely on the total number of observations of the
pathogen, which would be proportional to (delayed) cumulative
infections and not (delayed) incidence. With detection based on
cumulative infections minimizing delay matters a lot more.
</p>
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