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a soccer win probability model

Last year, I tweeted about a win probability model I created for soccer (or football, depending on where you are from) and the 2019 FIFA Women’s World Cup case study. I promised to blog about this case study I developed for my probability models course. This is a long overdue blog post on this topic.

Here is a portion of the soccer analytics case study.

Soccer win probability model

Soccer (or football, which it is called outside of the the United States) is based on a 90 minute match. The data from FIFA women’s soccer indicate:

  • Home teams score 2.34 goals/match (in regulation)
  • Visiting away teams score 1.71 goals/match (in regulation)

We assume the goals are scored according to a Poisson process with exponentially distributed arrival times. Assume each team scores independently of one another and of the score. We can use the home team data for the team that is favored in the match.

Consider the situation when the home team is down by 1 goal with 4 minutes in regulation. Find the probability that the home team wins. This is a win probability. We do not consider pulling the goalie.

In one possible solution, we divide the match into small increments of, say, a half minute in length. We recursively solve for the home team’s win probability for any score differential and any time. This way, we want to be able to answer questions like this repeatedly for different score differentials and lengths of remaining time.

The derivation is next. You can skip the math if you want and jump to the figures below.

Let the random variable W_i(d) capture the event that the home team wins with a score differential of d with i increments to go.

We want to find P(W_i(d)), the probability that the home team wins if there is a score differential of d with i increments to go.  In our problem, the home team is down by 1 goal with 4 minutes in regulation, yielding P(W_8(-1)) for a win probability with a score differential of -1 with eight 30 second increments to go.

We use a recursive expression to compute the probability of scoring in small intervals of time and put the solution in a spreadsheet. The spreadsheet approach computes our solution, and it allows us to assess a variety of situations, including different score differentials with different amounts of time to go. The boundary conditions with 0 time increments to go are P(W_0(d))=1 if d > 0 (the home team is winning when time expires), P(W_0(d))=0 if d < 0 (the home team is losing when time expires), or P(W_0(d))=1/2 if d = 0 (the match ends in a tie).

There are two ways to compute the probability that a team scores in a small amount of time \delta , which is the length of an increment: (1) we can use exponential interarrival times, or (2) we can use the Binomial approximation. I’ll illustrate the latter approach below. We have to make the time increments small enough such that having at most one goal scored during the time interval is a reasonable assumption.

We compute P(W_i(d)) by conditioning on 𝑌, where 𝑌 captures what happened in increment 𝑖 with a home goal (+1), an away goal (-1), or no goals (0):

P(W_i(d)) = \sum_y P(W_i(d) | Y=y) P(Y=y)

After taking advantage of independent increments, we can simplify this to P(W_i(d)) = \sum_y P(W_{i-1}(d + y) ) P(Y=y) . Here, we recursively solve for P(W_i(d)) by conditioning on what happened last and formulating a new expression based on the win probability with i-1 time increments to go.

Here, P(Y=1) = \lambda_H \delta / 90 , where \lambda_H =2.34 home goals per match. Likewise, P(Y=0) = \lambda_A \delta / 90 , where \lambda_A =1.71 visiting team goals per match.

Let’s look at the answer

We can put this into the spreadsheet and estimate the probability of 0.048 that the home team wins when down by 1 goal with 4 minutes to go. We can see answers to other scenarios. A home team wins with a probability of 0.517 if the match is tied with 5 minutes to go.

What is more interesting is that we can move across these spreadsheet to estimate real-time win probabilities. Every time there is a goal, we jump to another row in the spreadsheet. We jump up a row if the visiting team scores and down a row if the home team scores.

I made two win probability charts for the USA v ENG and USA v FRA games in the 2019 World Cup. I set the USA Women’s National Team as the “home” team since they were favored to win in each of the matches, even though the matches were played in France. You can also see that the home team has a probability of 0.61 of winning when the game begins.

2019 FIFA World Cup 2019 FIFA World Cup

Earlier we assumed that goals are scored according to a Poisson Process. Is that a good assumption? Not exactly (see this post using Premier League data) but it’s not a bad approximation except for the end of game situation when a team pulls their goalie. The model we built above can be easily changed to have time-specific scoring rates. Pulling a goalie is trickier but doable. When pulling a goalie, we have to consider new Poisson scoring rates that depend on time and the score differential.

On a side note, the Poisson process assumption holds up better with National Hockey League data.

A Conversation about COVID-19 with Economists, Sociologists, Statisticians, and Operations Researchers

David Banks organized a conversation with a few researchers about the COVID-19 pandemic that turned into a publication in the Harvard Data Science Review [Link]. I was one of the experts included in this conversation. A description of the article and a link to the full article is below.

A Conversation about COVID-19 with Economists, Sociologists, Statisticians, and Operations Researchers by David Banks, Laura Albert, Jonathan Caulkins, Sylvia Frühwirth-Schnatter, Fiona Greig, Adrian Raftery, and Duncan Thomas

Summary. The participants in this discussion are leaders in areas of economics, sociology, statistics, and operations management that are relevant to the COVID-19 pandemic. They felt that the economic toll of the pandemic would be large, adversely affecting the U.S. and the world for years to come. Even if an effective vaccine is found quickly, some changes (e.g., telemedicine, increased working from home) are likely to be permanent. Some portions of commerce can be safely restarted now, with appropriate social distancing protocols, but other portions (concerts, theaters) cannot. Social distancing is an effective tool for reducing the reproduction number of the disease, and small changes in that figure have nonlinear impacts on the scale of the problem.  It is important to get better estimates for epidemiological modeling, and for this we need disease and antibody testing on representative samples of the population. We also need to re-engineer work and other activities in ways that allow the economy to begin to slide back towards normal operation without increasing the reproduction number.  As restrictions loosen, there will patchwork outbreaks throughout 2020.  Universities will face challenges to their traditional ways of doing things, but will also have opportunities to improve what they do.

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Planning a sabbatical from the beginning

Several professors asked me for sabbatical planning advice and how to get started. I’m not sure what the best way to plan for a sabbatical is, but I am more than happy to share my experience along with lessons learned from my sabbatical at RWTH Aachen in Germany. Your mileage may vary.

Two years ahead of time, I started thinking about where I wanted to go. In the back of my mind, I always wanted to have an extended stay in Germany. I speak German and even earned a minor in German in college, when I studied abroad in Darmstadt, Germany for a summer. Additionally, my children could take German as a foreign language in school before and after the trip, and that was the deciding factor. I also considered the Netherlands after giving a couple of talks there and meeting many excellent faculty members at Dutch universities. Germany borders the Netherlands, so collaborations across the border would be doable.

My colleague Jim Luedtke was the source of inspiration. He took an academic year sabbatical to Chile with his partner and three children, two of whom attended Chilean schools. He was positive and adventurous, and he gave me the confidence to take my family with me on a sabbatical. Others I talked to were overwhelmingly positive about their experiences overseas with a family.

I looked into applying for a Fulbright scholar award early on. The application requirements are country specific. I used that the Fulbright guidelines as a template for overall sabbatical planning. Jim Luedtke lent me his sabbatical application for Chile, which was helpful in understanding what to include and what level to write about my proposed Fulbright project. Fulbright scholar awards in general have a maximum length of four months, which could be wrapped into a longer stay overseas. This would allow me to do a semester long or academic year sabbatical.

Fulbright applications are due in August a year before the trip starts. The spring before the Fulbright was due, I contacted Marco Luebbecke at RWTH Aachen about the possibility of a sabbatical. He said yes and asked me if I would be willing to teach a course in the summer semester (April – July). At this point, it became clear that an academic year sabbatical would be too long, because I would stay for the academic year and summer. This helped me narrow my scope for the sabbatical timeline, and I committed to a Spring semester sabbatical.

The Fulbright application was due in August 2018, and my university sabbatical application for the 2019-20 academic year was due a month later in September 2018. I could work on both applications at once. I proposed a sabbatical to start in January 2020, almost a year and a half before leaving for Germany. A few months later after I submitted my Fulbright application, I was delighted to learn that I was selected for a Fulbright award and that my university sabbatical was approved.

Living in Germany for more than three months requires becoming a resident and embracing German bureaucracy. U.S. citizens who visit Germany can stay on a tourist visa for 90 days. A longer stay requires applying for a residence permit, registering with immigration, and other requirements. This meant that I had to have a lease (not have an extended stay with Air BnB), arrange for health insurance that is required by law, open a bank, and have all sorts of documents for my children.

The Department of State and the Fulbright Commission were fantastic. The Fulbright Commission helped me with health insurance and gave me checklists and verification letters for all the requirements. I cannot say enough about the support I was given. They helped me acquire necessary international health insurance (required for residence in Germany) and helped me understand the residency requirements in Germany. They also provided a letter that I needed to register as a resident, register with immigration, and open a bank account. And importantly, they were extremely helpful during my sudden return to the United States in March 2020 during the COVID-19 pandemic.


Read more blog posts about my 2020 sabbatical here and see my tweets using the “PunkRockORinGermany” hashtag on twitter. I have posts about my sabbatical plan and how I prepared for a semester overseas. I had to end my trip to Germany early due to the COVID-19 outbreak.

articles about re-engineering society post COVID-19

Here is what I’ve been reading about how COVID-19 will forever alter the world in which we live.

To start, the Centers for Disease Control has mitigation resources for re-engineering communities, schools, workplaces, and events (“Guidance for Where You Live, Work, Learn, Pray, and Play”). It is worth visiting. They have guidance for many institutions, including:

Related: Ed Yong explains how COVID-19 works in the Atlantic, which underlies many of the re-engineering requirements as we move forward


Re-engineering our facilities

  1. How ten cities throughout the world adapted public transit in the wake of COVID-19
  2. Possible improvements to airplanes to reduce virus transmission 
  3. How one theater company is bucking the trend by not canceling plays and implementing a slew of precautions, including removing 70% of seats and having no intermissions. I hope the arts can adapt, because I would love to see a play and the symphony in the future.
  4. Citylab: how the United States public transit can survive the coronavirus has
  5. Why are meat processing facilities so vulnerable to COVID-19 outbreaks? “At least 9,400 positive cases have been tied to meatpacking plants across 28 states.” The work is labor intensive in crowded, poorly-ventilated environments where workers may encounter 1000 other workers per day. From the Post Crescent.

Re-engineering our processes

  1. NPR: 9 ways K-12 schools will be different when the reopen, including staggered schedules, smaller classes, and new calendars.
  2. In an op-ed, Drs. Sheldon Jacobson and Janet Jokela suggest ways big land-grant research universities could slowly and intentionally reopen while mitigating risks to faculty, staff and students/.
  3. It will not be business as usual for universities in the coming academic year. The Chronicle has a list of college plans for the fall.
  4. German barbers and hairdressers can reopen with new, strict requirements, including styling chairs at least 1.5 meters from each other; hair washing before a cut is required to help kill off possible viruses; no face to face services; and no magazines or free coffee in the waiting areas.
  5. Runners World covers running during a pandemic.

Re-engineering our social groups

  1. New York City requires wearing face masks in public. NYC is distributing face masks in public parks. And Deutsche Welle explains the face mask laws in Germany.
  2. Slate’s parenting podcast “Mom and Dad are Fighting” discussed “quarantine pacts” where families can group together to manage care and work responsibilities as a method to manage risk in a time of quarantine living.
  3. The Nation discusses many of the challenges for non-nuclear families introduced by social distancing and quarantines that also hints at what solutions should encompass.
  4. Child care allows every other industry to work, but it’s not clear what child care during the pandemic will look like. Citylab has a roundup of coronavirus child care related issues
  5. Workplaces need to adjust to new leave laws and family leave obligations and homeschooling realities. The New York Times reports on how mothers are managing homeschooling with work, and the Washington Post note that there are many lawsuits over family leave and discrimination

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articles about forecasting and the Covid-19 pandemic

Here is what I’ve been reading about forecasting and COVID-19:

  1. Nature has a nice article on forecasting: The simulations driving the world’s response to COVID-19
  2. The Kaiser Family Foundation has a nice introduction to forecasting models for epidemiology: COVID-19 Models: Can They Tell Us What We Want to Know?
  3. Fotios Petropoulos and Spyros Makridakis published an article in PLOS ONE entitled Forecasting the novel coronavirus COVID-19
  4. The mathematics of predicting the course of the coronavirus from Wired
  5. Why it’s so hard to see into the future of Covid-19: human behavior from Vox
  6. FiveThirtyEight on Why it’s so freaking hard to make a good COVID-19 model
  7. Finally, science writer Ed Yong has written a few articles about COVID-19 for The Atlantic that focus on the future, including How the pandemic will end based on what we know from other related viruses, Our Pandemic Summer, and Why the Coronavirus is so Confusing 

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articles about supply chains and Covid-19

Here is what I’ve been reading about supply chains in the COVID19 era:

  1. Vox on food and meat supply chains
  2. Why there are toilet paper shortages
  3. The Society for Risk Analysis recorded a webinar called “COVID Conversations on Risk” (Ep 3) about food security
  4. The Washington Post on why meat supply chains are vulnerable to Covid-19 outbreaks: they are massive facilities with thousands of workers in close proximity 
  5. Why you can expect shortages of pharmeceuticals but not toilet paper (nice job, Anna Nagurney)
  6. On personal protective equipment (PPE) supply chains
    1. Read about the origins of the Strategic National Stockpile
    2. CDC guidance for optimizing supply chains
    3. The Washington Post reports on a factory that produces a chemical used in PPE
  7. The modern supply chain is snapping from The Atlantic. Note this came out a month ago.
  8. We need a stress test for critical supply chains (nice job David and Edith Simchi-Levi)
  9. And finally, Belgians are urged to eat French fries at least twice a week as changes in consumption during the pandemic lead to a massive potato surplus


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how to organize your Tupperware cabinet

Rumor has it that many people are taking advantage of the extra time at home during the COVID-19 pandemic stay at home orders to organize their food container/Tupperware cabinet. It’s such a popular pandemic activity that The Onion published a satirical article about it.

My food container cabinet is always organized and optimized. I took a picture below. The secret to organizing my cabinet is to formulate it as an integer programming model and then solving it to optimality. I have my kids resolve the integer programming model every time they unload the dishwasher and put the dishes away.

I’m kidding.

I organize my cabinet with a heuristic. I nest my food containers in a few stacks that fit in the cabinets. Once I organize the cabinet, there are a few food containers that do not nest with the nicely organized stacks. The most important part of this process is to remove the few food containers that do not nest with the other containers (see the bottom picture). A few problematic food containers can lead to chaos and disorder. I do not throw these out. Instead, I use them to store items throughout my house (e.g., in my toiletries cabinet and tool cabinet). 

I have not purchased food containers in awhile. When I do, I purchase them based on their nestedness properties.

How do you organize your food container cabinet?

An organized and optimized food container cabinet

The key is to remove the pieces that don’t nest.

Related posts:

on vampires, exponential population growth, and scientific literacy

Eleven years I wrote a tongue-in-cheek blog post about vampires and stochastic processes. I was inspired by my course material about Markov chains and branching processes, which has application to the spread of infectious disease, to the vampire population dynamics in the Twilight series and other teenage vampire stories that were very popular at the time.

I have a great deal of skepticism about vampires.

Here’s my problem with vampires. I have a hard time believing that there would be just a few vampires out there and that the existence of vampires would be such a well-kept secret. After all, they reproduce rather easily (a single vampire could create thousands of offspring, whereas there are limits to human reproduction) and vampires don’t die easily. If there were vampires, they would almost certainly outnumber humans (but then vampires would run out of food).

This argument becomes even more overwhelming if you model a vampire population as a branching process or birth-death process and assume that each vampire in the population has probability Pj of producing j offspring (with j=0,1,2,… ). The vampire population would either explode or die out, depending on the expected number of offspring per vampire. But if you take into account the fact that vampires live many, many generations (they’re virtually immortal) and may create thousands of offspring, the population explodes (if you assume that each vampire creates at least one vampire, on average, before it dies). With those numbers, vampires would not be living under the radar–they would be everywhere!

I have yet to see a vampire movie that implicitly assumes that there is a reasonable model for vampire population dynamics (using a stochastic process framework or something else). And frankly, I’m pretty disappointed. Until I am offered a reasonable explanation for why there aren’t more vampires, I won’t be able to jump on the vampire bandwagon.

This issue had been bothering me since I first saw The Lost Boys, long before I knew about Markov chains. I enjoyed The Lost Boys, but I did not enjoy it’s inability to acknowledge exponential vampire population growth. Markov chains later helped me understand why my skepticism was valid.

The post went viral. Life was interesting for awhile. Twilight fans hated me.

My blog post was never intended to be taken seriously. It was not a serious critique of vampires, because vampires aren’t real.

Once in awhile, I google myself to see what turns up. Over the years, I have found that several vampire fan news sites and blogs that existed at the time (teenage vampire stories were very popular at the time) picked up my blog post and wrote serious articles about it.

Some of the concerning coverage of my post was on vampire fan websites that no longer exist online. A decade ago I remember discovering a vampire fan website for teenage girls with a domain that may have been that made a serious two minute “vampire news” video about my “research” in vampire population dynamics that supposedly proved that vampires could not exist. The host discussed my blog post like it was real research. I was disheartened by this. I would like to engage teenage girls about operations research and analytics but without the vampires. It’s only fun if it gets people more engaged with real research, science, and engineering.

A positive example is the one entitled “Vampire Ecology: Twilight vs. Buffy” on a science blog that argues that vampires could exist by linking to another tongue-in-cheek paper that takes human predation on vampires into consideration as a form of vampire population control. It doesn’t seem to take my post too seriously (phew).

I have given a lot of though to increasing scientific literacy in the general public for the last decade. The far-too-serious coverage of my silly vampire post did not dissuade me from engaging the public about my research. Instead, it encouraged me to be more intentional with how I communicate scientific principles to the public and motivated me to discuss real scientific issues with the general public as much as I can. I have blogged about some of my public talks and have appeared in the media many times. I have found that a lot of people are receptive to science and engineering research, especially if it seems relevant to their lives. I try to stick to applications of operations research, analytics, and industrial engineering in the public sector.

I’ve been encouraged by the discussion of real scientific principles during the COVID-19 pandemic. It’s been a positive side effect of a serious pandemic. I hope the public’s interest in science continues.




Auf Wiedersehen Wisconsin, Hallo Deutschland!

This post is about my first month getting situated on my sabbatical in Germany. The second month was consumed by the spread of COVID-19, and I had to return to the U.S. in less than two months. This post should be entitled “Auf Wiedersehen Deutschland, Hallo Wisconsin!” but I will save the COVID-19 related experiences for another post.

My two few days was solely dedicated to moving into my apartment. In my last post, I wrote about how I preemptively placed an order for IKEA furniture a week before leaving the US so that the order would be delivered right after we arrived. This plan worked perfectly. Additionally, it felt satisfying to put together our furniture and get settled into our new home. We also made a couple of trips to grocery stores to stock up on necessary food and water. We brought a few tote bags to carry our groceries home. Luckily, we have two discount grocery stores within half a mile of our apartment. I highly recommend taking children to a grocery store and letting them pick out a bunch of new food to try, even familiar favorites like Oreos. We made an effort to try new things up front, and that helped us discover a few favorite new foods early on. Everything was new, and this was more difficult for the children than it was for me, but we enjoyed sampling the many types of pastries, cakes, and chocolates available in Germany.

Eventually we picked up a few things to make life in Germany easier: a water cooker for boiling water, tupperware, a hair dryer and curling iron, a big European drying rack, dishes and kitchenware, notebooks, and a mop.

Thanks to the documentation from the Department of State, who manages the Fulbright, and the RWTH Aachen Welcome Center, I printed out all the documents we needed. I had a huge binder full of documents. But the paperwork was overwhelming. The children were almost entirely shielded from the bureaucracy, and I was relieved that the stress of moving to Germany fell on my shoulders. Bureaucratic tasks for adjusting to German life included:

  • Resident registration and related tasks
  • Immigration
  • Cultural integration / education of the children and related accounts (all day care, bus tickets)
  • University registration as a visiting professor
  • Public transportation and train accounts
  • Bank account
  • Cell phone SIM card activation

Registering with the citizen’s office in Germany was a pain. This was something everyone must do within two weeks of when they move to a new apartment, not just newcomers to Germany. Appointments can be made but there is a one month wait, so the only way to register is to arrive at the office at 7:30am with the children, pull a ticket (like at the DMV in the US), and wait. On our first try, the central computer for our region of Germany used to register everyone in the region went down, and we had to come back the next day first thing in the morning.

After registering our address, we applied for a residence permit with the immigration office. Again, the children came with me. We could complete this step at the university, and it was very easy. We had all of our papers ready and brought passport photos that we printed out ahead of time at a drugstore in the US. The photos needed in Germany are basically the same but cut to a smaller size. The person working in the office cut our photos to a smaller size, so we did not have to come back with new photos.

I needed a bank account in Germany for my Fulbright. It was easy to set up an account with the N26 bank.

Once the children were in school, paperwork was never-ending. One week I hypothesized that German bureaucracy is like a marathon because I had hit the “wall” the previous week. Another time it felt like a series of nesting dolls, with each task leading to another smaller sub-task that I could not see earlier. I thought I had checked something off my list (registering the children for a bus pass), but then I had to take the form to a city office. That didn’t end the registration process. Instead, I was told to wait for something to arrive in the mail, fill it out, and return it. Finally, it seemed like an Escape Room. You keep think you’re getting close to figuring out how to complete the paperwork, but then there is another puzzle (or puzzling form) to figure out. I did better once I accepted that I needed to devote a few hours per week to paperwork. Sadly, we had to return to the US just after I figured everything out and was ready to settle into my research routine.

Getting used to public transportation was another step in the adjustment process. I find it easy to navigate with public transportation. I used two apps: google maps and the AVV app to find real-time bus route information in Aachen.

News media

I discovered some news sources. I subscribed to the Local (expat friendly German news in English) and Deutsche Welle. I listened to the weekly podcast/radio program “Inside Europe” from the Deutsche Welle. “Slow German” was great for practicing German. Paying for a subscription to the Local was well worth it, especially to keep on track of COVID-19 news.

I was glad I had these items with me

  • Buffs/multi-functional headwear The knock-off versions are about $1 apiece and have a lot of uses.
  • A deck of cards and/or Uno cards.
  • Travel yoga mat.
  • Smartphone apps: google maps with downloaded maps of places we visited, google translate, and public transportation apps
  • First aid kit: thermometer, band-aids, Tylenol, children’s Tylenol, children’s Advil, Neosporin, Aquaphor, and Hydrocortisone. I used all of these items, and each time I was relieved I didn’t have to make a special trip to the store. I’m proud to say I had no mishaps with the electric bread slicer in my apartment.

First days were fun too

The first few days were full of fun discoveries. We discovered the endless supply of chocolate at area stores. We also purchased an annual zoo pass for the zoo that was a quarter mile walk from our apartment. We visited 3-5 times per week, usually for a long stroll through the zoo by each of our favorite animals.

In my next sabbatical post, I will blog about how I got started with sabbatical planning.

Read more blog posts about my 2020 sabbatical here and see my tweets using the “PunkRockORinGermany” hashtag on twitter. I had to end my trip to Germany early due to the COVID-19 outbreak.


COVID-19 is a pandemic that requires systems thinking and solutions

I was on the INFORMS Resoundingly Human to talk about COVID-19 and first responders. You can listen here:

In the podcast, I discuss supply chains, rationing resources, and disaster planning, and I note how everything old becomes new again. For example, the US is not experiencing its first N95 mask shortage. Systems concepts are important for understanding how to prepare for and respond to a pandemic.

In this post, I want to dig deeper into systems concepts. I wrote a quick primer on systems thinking and explain why systems concepts are important for understanding the COVID-19 pandemic.

What is a system?

A system is a set of things—people, vehicles, basketball teams, hospital beds, or whatever—interconnected in such a way that they produce their own pattern of behavior over time.

Here are three examples:

(1) A car is just a vehicle. A collection of cars can be a traffic jam.

(2) A single ventilator can be used to treat a patient. A hospital’s collection of beds and ventilators are available for treating patients. When a surge of patients require these resources, they may have to wait and queue for these limited resources.

(3) An N95 mask protects first responders from infectious disease when they treat patients. A supply chain of personal protective equipment (PPE) can have delays and shortages, leading to first responders not having the N95 masks they need at any given moment.

How is systems engineering relevant to COVID-19?

COVID-19 is absolutely a medical challenge. It is also a systems challenge that require systems thinking and systems solutions. In systems, decisions are not made in isolation, but rather, decisions are interrelated.

My discipline is operations research: the science of making decisions using advanced analytical methods. Systems require a series of decisions to operate effectively with or without patient surges in a pandemic. Operations research provides the analytical tools required to design and operate systems more effectively and efficiently.

In systems there are many trade-offs and complicated interactions. Here are examples of how systems engineering is important now:

(1) If a first responder does not have adequate personal protective equipment (PPE) such as latex gloves and N95 masks, they are at higher risk from acquiring COVID-19. If they do, they will not be able to treat patients in the coming months, thereby reducing the number of first responders (a critical resource) in the future. This informs how responders should treat patients and ration resources now.

(2) Surges in COVID-19 cases may lead to more patients requiring ventilators than are available in hospitals. This could lead to rationing and painful choices that would not be considered without a patient surge.

Systems concepts will continue to be important in the future. Here is a third example:

(3) One person who gets a vaccine has immunity. If enough people receive vaccines or have immunity from previously having had the disease, we can achieve herd immunity and eliminate person-to-person transmission of the disease even among those who do not have immunity. With herd immunity, the benefits are greater than the sum of its parts.

What can systems thinking tell us about the fatality rate for COVID-19?

It depends. We know that it depends on age, gender, and co-morbidities. The fatality rate is not an exogenously given number, but rather it is a function of the resources available for treating patients, which is endogenous to the system. The fatality rate for COVID-19 is a systems concept. If the number of infected individuals is low enough so that hospitals can handle the surge and give every patient the treatment they require, the fatality rate will be lower (relatively speaking. In absolute terms it will still be too high). The fatality rate will be a lot higher if hospitals are over capacity and have to ration beds and ventilators.

How are my personal decisions related to healthcare systems in the COVID-19 pandemic?

The resources in our healthcare system are being stretched to the limit. The resources include personnel (physicians, nurses, first responders), hospital beds, ventilators, and personal protective equipment. When there are not enough resources to give every COVID-19 patient the best treatment they require, physicians will have to ration resources and make tough choices. Our efforts to delay the second wave as long as possible and to reduce the number of people who require medical treatment will save lives. Flattening the curve is a systems concept aimed at reducing painful tradeoffs and complicated interactions.

How can we prevent the next wave?

Preventing the next wave of any infectious disease is a numbers game. I do not know how to practice medicine but I know how to crunch numbers. The key is to lower the overall transmission rate. The best way to lower the transmission rate varies according to the disease, but there are some basic principles for preventing a disease outbreak from becoming another wave of a pandemic. Best practices include better hygiene practices such as washing your hands and your mobile phones with soap and water, and covering your cough. Limiting the number of people you come in contact with reduces the opportunities for transmission. All those trips to the store to buy extra toilet paper increase one’s chance of contracting COVID-19.

What can we do to prepare for a second wave?

A second wave in a prolonged pandemic is not going to be easy for many of us. I use mathematical models and analytics in my research, and I find them to be useful in my everyday life. My research tells me that I make better decisions with better information and that I should use limited resources wisely. When I think about what it means to apply these principles to my decisions in a pandemic, I realized I can achieve both of these goals by gathering up to date information and following instructions from official, trusted sources such as local and state governments, local police and emergency medical service departments, and the Centers for Disease Control and Prevention. I plan to use the official sources to limit what I think about, worry about, and do in any upcoming waves of the pandemic. We are all inundated with conflicting information and advice from many sources, and it is taking its toll and potentially leading us to make unsafe choices such as making repeated trips to grocery stores to stockpile items we do not need.


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