Corona 4: infection rates and cool graphs

Covid-19: the fun never ends. So you've heard about the lockdowns beginning to "ease" in Europe, the redneck / right-wing nut protestors in the US, and definitely -- for a while now -- about flattening the curve. But how, other than going stir crazy inside and politicians and business owners getting worried about re-elections and profits, respectively, how can we know when it might be safe(-ish) to start lifting restrictions, going back to "normal" (whatever that will be), and -- most importantly -- knowing how and when the virus might be starting to go away? The answer is (or more accurately, part of the answer is): the infection rate (aka "R_t: the effective reproduction number", even better aka "The Metric We Need to Manage COVID-19").

R_t is, as the name of this article suggests, the "effective reproduction number" or "infection rate" and in this case we of course mean for Covid-19 (what else would anyone be talking about these days?). Similar in some ways to the replacement fertility rate, which is the average number of children that would be born to a woman over her lifetime required to sustain a given population, R_t can tell us what we can expect to happen in a population -- but in this case, how the virus might keep spreading or start to slow down. For replacement fertility, the important number is right around 2.0 (it's a little higher due to some other factors like child mortality rates but for our purposes and in general for the developed world you can think of it as 2.0). If the number is < 2.0 in a country, eventually (without immigration), the population will disappear. If it's > 2.0 then the population will grow. Easy, right?

It's similar with the effective reproduction rate. But in this case, it tells us the average number of people that an infected person will infect. If it's > 1.0 then the virus will keep spreading. If it's < 1.0, then the virus will eventually disappear. As Tomas Pueyo put it in this excellent article The Hammer and the Dance: "If R is above 1, infections grow exponentially into an epidemic. If it’s below 1, they die down." Let's look at a simple example:

1. Imagine 10 people have the virus and we are experiencing R_t = 2.0. Then those 10 people will each infect (on average) 2 other people. So then we have 20 new cases (eventually the original 10 people will get better, we hope, or, sadly, will stop being counted as "active cases" in another way). Those 20 people, with the same infection rate, will each infect another 2 people each, and now we have 40 new cases. Et cetera...

2. Now imagine that 10 other people have the virus, but because of social distancing, self-isolation, and other "restricting" measures, the R_t has been pushed down to, say, 0.85. Those 10 people will then infect a further ~8.5 people (we can round up to 9 to be conservative), who will then infect 9*0.85 = 7.65 (or around 8), who will then infect 8*0.85 = 6.8 (or around 7), until, eventually, there's no one being infected.

So, obviously, the point of the lockdown measures (among other things) is to push R_t down towards 0. And we can only consider opening things up if we are confident that we can keep R_t below 1. From The Hammer and the Dance again:

In Wuhan, it is calculated that R_t was initially 3.9, and after the lockdown and centralized quarantine, it went down to 0.32.

But once you move into the Dance [easing lockdown measures], you don’t need to do that anymore. You just need your R_t to stay below 1: a lot of the social distancing measures have true, hard costs on people. They might lose their job, their business, their healthy habits…

As long as you can keep the infection rate under 1 until there's a vaccine, then we can prevent another outbreak. With the background in place, we can now finally get to the big question and (hopefully) some answers: what is R_t currently in different places, what has it been in the past, and where is it going?

What is R_t right now?
Unfortunately, as you can probably imagine, we can't compute an exact value for R_t. It is a latent (hidden) variable. We can observe the effects of the infection rate, though, which is (of course) the number of people being infected. The way one does this is (duck for cover): statistics! Probabilistic inference! And most exciting of all, Bayesian inference! What a coincidence that the author of this blog's PhD thesis was on this very topic! Now, there are very few things in the world that tend to be as boring and/or annoying as someone explaining their PhD thesis work, so I will try to keep the explaining part to a minimum, and the cool results part to a maximum. Sadly, there is some background that I just can't help myself but feel that I have to lay out here. It will be super basic though.

Bayesian statistics uses Bayes theorem to understand degrees of belief in events happening. The main important thing that distinguishes it from frequentist statistics is that we infer probabilities of events based not only on running many trials and building up statistics, but on our prior belief (encoded as a probability) that something will (or will not) happen. We're mainly interested in computing posterior probabilities which (thanks to Bayes theorem) are understood to be proportional to the likelihood of our data given some proposition being true, times our prior belief that the proposition in question really is true. Tiny obligatory simplified example of coin tossing time. In pure frequentist statistics, if we flip a coin 5 times and each time it comes up heads, our model will predict that this coin will be heads 100% of the time. In Bayesian statistics, if we encode a prior belief that, like most coins, it will come up heads 50% of the time, then our model will have a more sensible prediction that (depending on the likelihood and prior functions that we chose) will be closer to predicting that heads will come up maybe a tiny tiny bit more often than tails. If we keep running experiments and after 1000 trials (for example) we still see heads coming up like 950 times, then the evidence (aka likelihood) will overwhelm our prior belief and we'll again come to the conclusion that this coin is not very fair and is in fact weighted to come up heads a lot more often.

It's well known that we are not able to count every single person who gets infected (for example, recent research suggests that Covid-19 may be between 50 and 85 times (!!) as prevalent as official figures suggest). Thankfully, however, as long as we use the same method to count the number of people infected, it doesn't really matter for our purposes if it over- or under-estimates the raw numbers -- as long as it's more or less consistent.

Our method
This is great because I love Instagram so much: the guys who created Instagram have created a website, rt.live, which not only calculates values of R_t for all of the US states (and graphs them really nicely), but also explains their method and provides some code that we will use to extend their analysis to places that readers of this particular blog may care more about; namely, Germany and Canada.

As Kevin Systrom does at rt.live, we will use a variation of the Bettencourt and Ribeiro method (probably no relation to the Milwaukee Admirals sniper Mike Ribeiro). This method uses Bayes' Theorem with the following setup:

P ( R_t | k ) = P ( k | R_t )  P ( R_t ) / P ( k )

This gives an expression to calculate the probability of R_t given k new cases, or P ( R_t | k ). This is the posterior probability (see above) that we're interested in. This is equal to:

- The likelihood of seeing k new cases given an infection rate of R_t, P ( k | R_t ), multiplied by
- The prior probability (or prior belief) of what R_t might be before we've seen any data, P ( R_t ), divided by
- The probability of seeing this many new cases in general, P ( k ).

Oh dear, this is already getting a bit complex. But please bear with me for just a little bit longer and then we'll get into the nice graphs, I promise. I will spare you the actual inference calculation, but we need to at least understand what we will use to model each of the expressions mentioned above.

First, the likelihood function. Since we're modeling counts of something, we will use the Poisson Distribution. From our friend wikipedia, this models the "probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event". It has a single parameter λ (lambda) which is the expected number of occurrences. After some derivation, we can connect λ with k (number of new cases) and our famous rate R_t. It's like this:

which can then be plugged into the Poisson Distribution and we get a nice formula for our likelihood:

Remember that we know k_t-1 (it's the number of new cases on the previous day). Those who have studied biology will probably recognize our expression for λ as a standard exponential growth model. Note that the γ is the reciprocal of the serial interval (γ ≈ 1/7 for Covid-19).

So what does this likelihood function actually look like? Consider an example where we have 4 days of data: on day 1 there were 20 new cases, day 2 had 40, day 3 had 55, and day 4 had 90. Then, the likelihood of R_t for each of these days looks like this:

Next let's look at our prior. Remember this is our prior belief how likely values for R_t without having observed any data. Because the value of R_t for one day will probably be very close to its value on the previous day, we use a normal (Gaussian) distribution with a mean equal to the value of R_t on the previous day. Therefore our prior looks like this:

where σ is a hyperparameter that we can estimate (or just use some "sensible" value) and is the Gaussian's variance. An example prior distribution P ( R_t ) with R_t-1 = 3.9 (an estimate of what the initial value was in Wuhan) and σ = 0.3 (we can choose a good value by maximizing the likelihood of our observed data) looks like this:

What this means is there's a very high probability of the new value being around 3.9 again (in fact it is the most likely value) but there's quite a spread of possible values (and getting exactly 3.9 is actually extremely unlikely). For example, if we integrate the probability distribution function above from 3.0 to 3.5, this tells us the probability of R_t falling between 3.0 and 3.5, and that probability is 18.24 %.

Finally, we're ready to combine the prior probability with the likelihood function and perform inference with our observed data. All we have to do for that is multiply the prior by the likelihood and normalize.

Applying the model to real-world data

The Worldometer is providing up-to-date numbers for new confirmed cases of Covid-19 across the world. I gathered the data for Germany and Canada on Thursday, April 23rd from https://www.worldometers.info/coronavirus/country/germany/ and https://www.worldometers.info/coronavirus/country/canada/ respectively. Here are the total number of cases for Germany and for Canada starting on March 2nd:

Just looking at the graphs above, while Germany has a lot more total cases than Canada (almost 4x as many -- Germany has ~150,000 while Canada only has ~40,000), it looks like right now Germany's cases are growing more slowly, and therefore likely has a lower R_t. Let's find out!

We can see more clearly in the following graphs that indeed Canada's new daily case numbers are currently going up, while Germany's are going down (we also show a "smoothed" curve -- and use that in our model calculations -- because it helps control other unseen errors in the observed numbers like people who didn't get counted, etc.).

Again, the raw numbers are bigger in Germany, but the fact that those numbers are going down is what's important. Let's see those infection rates!!!

The data points show the maximum likelihood estimates of the infection rate for each day and the grey shaded area shows 90% of the probability density interval of our posterior. You can see that we get more confident of the estimate range as we gather more data. In both cases, the good news is that the trend for the infection rate is clearly on the way down. For Germany, as the initial data suggested, the infection rate is slightly lower than it seems to be in Canada and perhaps it is enough for a very cautious gradual re-opening of the economy (but I personally doubt it).

The graph looks nice, but here are some more specific numbers for the last few days, first in Germany (left), and then in Canada (right):

The tables above show the maximum likelihood (ML) estimate -- i.e. the most likely value for R_t followed by the low end of the 90% probability density interval and then the high end. For Germany, our best guess for the current infection rate is 0.77 and is almost for sure not lower than 0.49 and almost for sure not higher than 0.98. For Canada our best guess is 1.08 and it's almost for sure somewhere between 0.81 and 1.30.

So does this mean that things can go back to normal any time soon? I'd say... definitely not.

Corona Check-in #3

I wrote the first Corona Check-in (aka Life in the Trenches) on here exactly two weeks ago but it feels like at least 3 years [note: this ended up being posted a few days after the previous sentence was written]. At the time it felt weird that there were less people in the office than normal and that the streets were a little more empty than usual. Later it was strange that restaurants were allowed to be open but only for limited hours. Now, as anyone who hasn't been living under a rock for the last month knows (or those living in the "Big Brother" house who allegedly continue to be cut off from the outside world and therefore have no idea what's going on), things have, to put it lightly, escalated. We're down to essential services in basically all of the modern world and many of us are in various states of lock-down. In France, breaking the lock-down rules results in a fine of a few hundred Euros for the first offence, and then quickly climbs up to six months in prison for a second (and if you think carefully about what prisons are like especially right now, you really don't want to be going to one). India's 1.3 billion people have been put into lock-down as of the other day, with the governors of some states giving orders to "shoot on sight" any offenders. Here in Germany, things are less draconian for now, but I continue to see people not taking it very seriously (e.g. walking to the grocery store -- which is of course allowed -- I saw a big group of runners out together... come on people...) and I think that will lead to more forceful restrictions in due course.

I continue to put my very best effort in for working from home, but to be honest it's all but impossible. There are some positive things coming out of this, uhh... "incident", however. For one, there is a lot of funny Twitter and Instagram content coming out. I'm especially impressed by the sarcastic and self-deprecating parents trying to make light of surviving this thing with their kids at home. Some really funny people out there on the Internet.


What a weird existence.

Life at the Covid-19 Epicentre: Report #2

A lot has changed in just a few days. In Germany we're not yet on full lock-down like they are in France and Italy, but things feel very strange indeed. In theory all "non-essential" shops are closed but I went for a run yesterday (I checked and it's allowed -- "even encouraged") and saw even an ice cream shop open. I believe officially restaurants are allowed to be open but only from 11:00 to 15:00 and everyone has to be at least 1.5 metres apart. One thing that seemed very normal was going to the grocery store. It wasn't overly crowded or eerily empty -- just normal. In general, all of the shelves were stacked how I'm used to seeing them (even the toilet paper aisle!) except there was a lot less selection in the pasta aisle. Normally it's basically an entire aisle of different brands and types and yesterday there was basically only a few boxes of whole wheat pasta left sitting in the corner (yuck).

Our office is officially completely closed and -- like most office workers -- we are in work-from-home mode. This presents interesting and difficult challenges especially since, as we all know, schools are closed and will be for the foreseeable future. Not only adults are getting cabin fever; kids seem to perhaps even go crazier more quickly. But of all the things that are closed vs. things that are operating as usual, a lot of it feels very arbitrary. We keep getting mail, for example, which seems to me a perfect way for a virus to continue traveling across the globe. Delivery places are all open but must follow a "no touch" delivery procedure where you pay online and the delivery guy rings your bell and leaves the food at the door. All well and good, I suppose, but I'm kind of more concerned with the person who actually made the food than the guy who delivered it, to be honest.

Reports coming out of England and (of all places) Alberta have officials suggesting that schools will stay closed until at least September. If the virus and the inability to travel or be social or buy the things you need doesn't scare you, the idea of your kids being at home with out for the entire spring, summer, and potentially the fall definitely will.

Signing off from the trenches.

Life in the trenches (aka Europe)

As I entered the final year of my 30's, the world entered the uncharted territories of the long-awaited Great Global Pandemic. I would say it's only in the last day or so that things have started to really kind of feel weird -- even unsettling. The office attendance has been dwindling slowly for the last couple of weeks but hit the precipice today where it really feels like a Ghost Town. Walking the streets of the city centre is kind of neat but also rather eery because there really are noticeably a lot less people out and about. So far (as of last night) the grocery store was business-as-usual but some small groups of products were conspicuously lacking in stock (e.g. pasta and some canned foods). I think this could quickly change perhaps over today and tomorrow especially as things are closed -- as always -- on Sunday. All schools, kindergartens, and daycares were cancelled today until mid-April (the 17th to be exact -- so far). Fun times ahead for sure. How serious are things getting? Sure, schools are cancelled, no one's at work, and it's hard to find your favourite pasta. BUT, the Starkbierfest was just cancelled. Bavarians cancelling a rich-in-history festival celebrating the creation of a beer for the sole purpose of getting through religious fasting? Something really profound is clearly happening. What's next? I suppose only time will tell.

Spanien

In February 2020 Pemulis and family -- Joelle, Helga, and Heinrich -- went to Spain. Just as the Covid-19 (aka Coronavirus) was ramping up its path of destruction, Pemulis and Co. attempted to escape said path by traveling to beautiful Andalusia and try to if at all possible reduce some of their daily stresses.

TL;DR: sort of worked, sort of didn't.

One nice thing about living in the general environs of the geographical centre of Europe is that it's fairly easy and relatively inexpensive to get to all sorts of nice places. That includes, of course, the region of Andalusia, and following a 2.5 hour uneventful-ish flight originating from Franz Josef Strauss we touched down in the wonderful city of Málaga (or, as is the case with nearly all airports, just outside of the city). It was here that we picked up our trusty Peugeot, hoped that I installed the child seats at least relatively correctly, and asked Siri to guide us up into the mountains around Ernest Hemingway's favourite Spanish city, Ronda.

We arrived at our temporary home in the hills north of Ronda just before nightfall. As is always the case, we were in a rush to get our children fed so that they might become a little less whiney and difficult, and then get them to bed. While generally much appreciated, the 4-course meal probably wasn't the best thing to be placed on the agenda for this particular evening, but the food was good and no one left the hotel due to our children ruining their expensive romantic dinner (that we know of).

Despite being exhausted, it took the usual 8 hours to get Heinrich to sleep which -- if you're doing the math at home -- means that we didn't sleep at all. But that's nothing out of the ordinary so the following morning our somehow-used-to-not-getting-any-sleep-at-all bodies managed to get everyone dressed and to breakfast before everything was dismantled for the day. Three aspects of Spanish culture that really align with my worldview / life philosophy / whatever you want to call it, are: (1) late meals; (2) taking forever to do anything; and (3) sleeping all afternoon (aka the famous "siesta"). Unfortunately, each one of those aspects -- which I can't stress enough attune with my vibe, so to say, just perfectly -- are 100% diametrically opposed to how a child lives his or her life (especially our children). Perhaps "taking forever to do anything" fits quite nicely with some of Helga's predispositions such as her affinity to requiring 45 minutes to take even the first bite of a single piece of toast before leaving for Kindergarten in a mad rush. But things like waiting for food to come and staying in your seat or remaining somewhat quiet are so completely foreign to her that one would think me and my daughter consist of entirely different species. Let's not even think about the possibility of having a siesta...

One very nice thing about our hotel and that matched point (1) above (late meals) is that we had a nice setup whereby the chef would whip up some pasta around 6:30, Helga and Heinrich would (mostly) eat it, we'd put them to bed, and then head back to the dining room around 9:00 for the "real" dinner. We actually pulled this off on 4 out of 7 nights which is pretty impressive. You've already read about the first night where we arrived to the hotel at the actual dinner time anyways so we all ate together. The two other nights where a romantic adult dinner did not work out were on non-consecutive evenings where each of us suffered some kind of stomach bug. But every single other night the kids went to sleep, we walked down to the dining room, ate a long drawn-out dinner with very good wine from the region, and returned to the room without any major disasters having occurred. Not bad!

Because we stayed in a somewhat mountainous region, we were excited to get in some training runs for our quickly approaching 64 km ultra trail race in the Alps. That didn't exactly work out as we hoped it might, but we did manage to do a little bit of running/hiking and rather than actually getting any training in, the main thing that these few outings accomplished was showing us that we are completely and woefully unprepared for this upcoming event and there's basically no way that we will be because who has time / possibility to go to the mountains and run for 6 hours when you have two small children? Kind of a dumb mistake on our part but I've made worse mistakes in my life, so...

Above you'll see my longest run of the trip which was still only 10 km. I managed to get some "climbing" in but if you can read the contour lines in the picture it was only about 400 m elevation change at most. But that tiny amount destroyed my legs for the entirety of the vacation and it was only 10 km and at most 400 m up. The race (in about 100 days from now) is 64 km through the Alps with 5000 m of climbing. Shiiiiit as they say.

Anyways, other parts of our trip were also nice. The food in Spain is very good and not very expensive. It was sunny every day and around 20 degrees. One day we met up with a friend and his daughter who live in Málaga in a nearby town -- Setenil de Las Bodegas -- which is famous for many of the buildings being built right into the side of the rocky mountain. It's pretty cool...

Another day we drove into Ronda proper, walked on the famous bridge, visited the famous palace (at least the outside of it), and of course ate some pizza. Throughout all this I'm glossing over the fact that it was never all that relaxing because Helga was always doing something like trying to run away or have a screaming fit in the street or jump into oncoming traffic or refusing to eat what she ordered or spilling food all over herself and into her hair or Heinrich was crying because he was tired or hungry or needed to be changed or whatever. But throughout it all we never had to actually make any food or clean any dishes so in that sense things were somehow maybe somewhat easier than at home. It didn't necessarily feel like it at the time though... Strange...

All in all: 9/10. Nice place, good weather, great food, not enough to keep kids occupied.

Small Fry

Editor's Note: this is an entry from our "submitted blogs by readers" series. The author has requested to remain anonymous.

Many weeks ago I read an excerpt from a book called Small Fry in a magazine called Vanity Fair (symbolism of worldly ostentation and frivolity?). The book is a memoir by Steve Jobs's (Apple co-founder, eventual CEO then outcast then CEO again) daughter, Lisa Brennan-Jobs. The excerpt is short -- around 3000 words -- and sad. Like uber-sad. One of the saddest things I ever read. At first I thought "I won't read that book, it seems too sad" but I was drawn to it. Like a moth to a flame (I should trademark that). I pre-ordered the book on a website called Amazon (heard of it?) and even though it was not set to be released for another 10 days or so, it arrived early, in late August. Every page of the book is sad (so at least the excerpt wasn't false advertising). But I just couldn't stop reading it. I read the whole book -- 381 pages -- in one day while I was supposed to be working (something to do with trains). I was addicted to the sadness and though reading it made my heart feel heavy, I enjoyed the feeling, like although it was unpleasant, it was something strong and it meant something. Somehow it was important. 

I felt intense sympathy for the author (not empathy though because come on). I wanted to hug her. In some strange fantasy I imagined that I could meet her and we could be friends and I could comfort her and say things like "you are a good person and no one should have ever done those mean things to you". Of course it's ridiculous but I thought I could tell her something like you deserve to be loved too.

I'd never thought so much about the actual author behind the words, even for other autobiographies (in the highly originally titled "Clapton: The Autobiography" which believe it or not is an autobiography by Eric Clapton, I just kind of felt bored once he stopped being a crazy drug and alcohol addicted rockstar). I wondered how her, or anybody really, remembers such specific memories from their lives. ... His voice was so sharp that I wondered why they bought them at all. "We need them to make ours," he said. So computers make computers, I thought. After that we said goodbye to the men and walked back up stairs. How does she remember these scenes and who said specifically what and at what time? Is she inventing it? Do other people have memories like this? I can't remember detailed conversations from earlier this morning, let alone many years ago. Is she special in that way and that's why she can write such an affecting book? Or am I deficient in some way that I can't imagine being able to remember details like this?

I read the book so fast that I was sad when it was over (obviously one would feel sad after reading all that sadness, but I meant I was sad that there wasn't more sadness to read about!). And whenever I would pick up the book after just a few sentences it would change who I was. I would start making plans in my head that at that time there was a strange mixed feeling of both believing that I would do these things but also knowing that once I stopped reading and a little bit of time went by I would go back to normal and wouldn't feel as strongly about doing those things anymore and most definitely wouldn't actually do them.

Once, I held a brief thought that was: I'm glad I bought her book because she deserves to have success. I want her to know that she's talented both because I think she is but also because she's suffered. Then I think about how she's allegedly inherited millions of dollars from her father and then money doesn't seem as important. And then I think that money can't do anything because of the emotional abuse that she seems to have encountered as a child. And that causes me to realize that I believe that once someone has been emotionally abused that they can never be whole, no matter what happens much later in life -- even being a successful writer. And extending that further maybe any kind of trauma means you can never be whole or fully happy, whatever that might be. But everyone eventually goes through something sad in life and so if that thought is true then from the moment we're born we're just on an eventual course to never being whole again. That's pretty depressing. Good thing it was just a brief thought. And maybe I should stop reading such sad books.

Imagine being able to do things

First a message to the future: Helga and Heinrich, if you read this, please don't feel bad. It's just that you took up a lot of time and were really quite exhausting. But I'm pretty sure all kids back then were like that. Of course now that it's the future modern science has probably figured out how to keep babies / children happy and the advances in child psychology, philosophical understanding of the requirements of the soul to live a good life, and discoveries in biochemistry, neurology, and nanotechnology have rendered misunderstandings and difficulties in family life to the dustbin of history. But back when I wrote this we didn't have all that and so I just needed to vent a little bit.

Now the content.

Imagine being able to do things! To seek inspiration for new (or even old but re-tooled) subject matter to fill in the spaces on this here blog, I took a trip down memory lane and visited some of our earlier entries back in the pre-November-2015 days, and it really seems like things were pretty nice back then! For example, there is a veritable boatload of pictures of Pemulis and Joelle together from soon after we arrived in Grenoble. I defy you to find more than one from the last year or more. We don't have time for that sort of thing! And even if we did, the ravages that time and sleep deprivation have inflicted on my body including most definitely its appearance prevent me from wanting to be anywhere within hundreds of metres of Joelle when the possibility of being framed by a camera lens is extant because having those contrasts of my aged hideousness beside her continuous beauty fills the Instagram filters with dread, foreboding, and any other word meaning roughly the same thing. But yeah, having those pictures of us together looking young is pretty nice. Glad I took em and posted them.

Next come the photographs of the food we both made and purchased so that we could consume. But let's focus primarily on the former. Imagine having time to make food! I refer of course to actual food rather than the umpteenth rendition of cheesy pasta or grilled cheese (thank goodness for cheese and its ability to melt on to and around food). One notices further in these gastronomically themed photographs that we were often (or perhaps essentially always) able to enjoy such delights of the culinary senses paired with some very fine French wines. Imagine drinking an entire bottle of wine in this political climate! One surely needs more than our standard 3 hours of heavily interrupted sleep per night to even dream of partaking in such an activity.

One might move next to the trips. The weekend getaways. The long-loved European capital city trip. Amsterdam. Paris. Rome. Instead if one were able to actually find the time to go somewhere our hard-earned money would more than likely rather be spent on hauling everybody to one of two places: (1) Canada which of course isn't bad at all but it's a little far for a long weekend; or (2) some "kid-friendly" hotel which requires both a swimming pool and a Spielplatz and which then implies there will highly unfortunately exist other guests who are classified as children. Too, too bad.

But I'm sure that one day, not far from now, Pemulis and Joelle will wake up after a long night's uninterrupted sleep and will hear no pitter-patter of small feet running across the hardwood or (more realistically) screaming and yelling from a fight over a toy. They will realize that the golden years of their children's childhoods has passed them by and they are now alone and they will long for the days of yore much as this author is doing right now for another bygone time. But then again, maybe not...