A Mathematical Intersection Between Epidemiology and Chemistry

Early during the coronavirus pandemic, researchers in many fields got a crash course in epidemiology. More specifically, they got a crash course in how to apply the math behind the SIR model, which describes how fast a contagious condition might spread through a population before becoming endemic.

The SIR model divides a population into three categories, the Susceptible, the Infectious, and the Recovered (or Removed). Once basic data about the rates of infection and recovery are determined, the model can simulate how many people will fall within each of these categories at different points of time. Here's a primer introducing that basic math, in which we featured the following 22-minute video from Numberphile's Brady Haran and Ben Sparks on how to build your own SIR model from scratch using the online GeoGebra application:

Although each of the individual equations for each component of the SIR model involves relatively simple relationships, their interactions lead to much more complex math. Math that cannot be done simply by plugging numbers into an algebraic formula. Instead, the SIR model's math requires serious computing power to apply numerical methods, running thousands or millions of calculations to progressively reach reasonably accurate, but still not exact solutions.

That's why the press release for a paper recently published in the International Journal of Chemical Kinetics caught our attention. Its authors recognized part of the SIR model's math developed by epidemiologial pioneers W. O. Kermack and A. G. McKendrick is identical to the math used to describe the progress of an autocatalytic reaction in chemistry. Here's the Kermack-McKendrick integral, which has no direct solution:

In this equation, So and Io represent initial values for the number of Susceptible and Infectious portions of the popuation, R represents the Recovered (or Removed) portion of the population as a function of time (t), while the Greek letter lambda (λ) represents the ratio of the rate of spread among susceptible population to the rate of recovery. The letter e is Euler's constant.

That was math for which chemists James Baird, Douglas Barlow and Buddhi Pantha had developed the next best thing to a direct solution. They derived an approximate algebraic formula for quickly solving the Kermack-McKendrick integral with a small margin of error. Here's their simplified formulation:

Better still, they identified where their simplified formulation will work best:

In this report, a description is given of an accurate approximation to the Kermack-McKendrick integral which in turn can be used to determine values for R(t), I(t) and S(t) in the SIR epidemic model. The result is shown to be effective for situations where 1.5 ≤ Ro ≤ 10 with no need to numerically compute an integral.

The press release better describes how their formulation meshes with the chemistry of autocatalytic reactions:

Dr. Baird presented the model in May at the Southeastern Theoretical Chemistry Association meeting in Atlanta.

"The World Health Organization could program our equation into a hand-held computer," Dr. Baird says. "Our formula is able to predict the time required for the number of infected individuals to achieve its maximum. In the chemical analog, this is known as the induction time."

The formula is capable of predicting the number of hospitalizations, death rates, community exposure rates and related variables. It also calculates the populations of susceptible, infectious and recovered individuals, and predicts a clean separation between the period of onset of the disease and the period of subsidence....

"The rate of infection initially accelerates until it reaches a point where the infection rate is balanced by the recovery rate of infected individuals, at which point the number of infected people peaks and then starts to decay," he says.

That mechanism reminded him of the mechanism that governs an autocatalytic reaction.

"I subsequently learned that the mathematical description of the spread of infectious diseases was first described by Kermack and McKendrick," Dr. Baird says.

"When I read their paper, I realized that their mechanism was exactly the same as that of an autocatalytic reaction, where a catalyst molecule combines with a reactant molecule to produce two catalyst molecules," he says. "The rate of production of catalyst molecules accelerates until it is balanced by the rate of decay of the catalyst to form the product."

And that's how the algebraic formula that can quickly approximate the solution to the Kermack-McKendrick integral for the epidemiological SIR model with minimal computing power came to be published in a chemistry journal.

References

James K. Baird et al, Analytic solution to the rate law for a fundamental autocatalytic reaction mechanism operating in the "efficient" regime, International Journal of Chemical Kinetics (2022). DOI: 10.1002/kin.21598.

James K. Baird, Douglas A. Barlow, Buddhi Pantha. A Solution for the Principle Integral of the Kermack-McKenrick Epidemiological Model. [Preprint (PDF)]. DOI: 10.31224/2264. April 2022. [This second paper is an ungated preprint that focuses on the epidemiological application of the authors' approximation of the Kermack-McKendrick integral.]

How to Use a Measuring Tape to Assess Your Health Risk

How much health risk do you have from carrying too much mass around your midsection?

That question arises because studies point to the Waist-to-Height Ratio (WHtR) as a better indicator of early health risk than the Body Mass Index (BMI). As a general rule of thumb, if the circumference of your waist is greater than half your height, you have an elevated risk for developing chronic conditions like hypertension, diabetes mellitus, hypercholesterolemia, joint and low back pains, hyperuricemia, and obstructive sleep apnea syndrome.

The Waist-to-Height Ratio is also reported to be better than BMI in predicting heart attacks, especially for women, with higher ratios corresponding to higher risk.

That sounds like good bit of information to have, so we've built a tool to calculate your Waist-to-Health Ratio. Since you probably already know your height, the hard part will be finding out your waist circumference. Here's a video showing how to measure it.

Once you've done that for yourself, you're ready to go. If you're accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version of the tool. Here it is:

Waist and Height Measurements
Input Data Values
Waist Circumference
Height

Waist-to-Height Ratio
Calculated Results Values
Waist-to-Height Ratio
Risk Level

In using the tool, be sure to use the same units of measurement for both waist circumference and height. You'll get accurate results so long as you don't start mixing and matching inches and centimeters together....

According to documents leaked in February 2022, starting in July 2022, U.S. Air Force personnel will have their Waist to Height Ratio assessed. Individuals with waists that measure at more than half their height will be reassessed six months later, with those whose waists exceed that threshold at the later measurement date separated from service. "Seperated from service" meaning "discharged from the Air Force". Here's the chart the Air Force will be using to make that determination:

The thresholds shown on this chart for low, moderate, and high risk are those we've built into the tool's feedback. We've also made a point of giving the answer to the same two-decimal place results as would be used by Air Force medical personnel in their assessments, so there are no surprises for what to expect.

References

Margaret Ashwell and Sigrid Gibson. Waist-to-height ratio as an indicator of ‘early health risk’: simpler and more predictive than using a ‘matrix’ based on BMI and waist circumference. BMJ Open 2016:6:3010159. [DOI: 10.1136/bmjopen-2015-010159 | NIH: PDF Document]. 14 March 2016.

Sanne A.E. Peters, Sophie H. Bots and Mark Woodward. Sex Differences in the Association Between Measures of General and Central Adiposity and the Risk of Myocardial Infarction: Results From the UK Biobank. Journal of the American Heart Association. Vol. 7, No. 5. [DOI: 10.1161/JAHA.117.008507]. 28 February 2018. American Heart Association. Waist size predicts heart attacks better than BMI, especially in women. [Online Article]. 28 February 2018.

Darsini Darsini, Hamidah Hamidah, Hari Basuki Notobroto, and Eko Agus Cahyono. Health risks associated with high waist circumference: A systematic review. Journal of Public Health Research. Vol. 9, No. 2: Papers from the 4th International Symposium of Public Health (4th ISOPH), Brisbane, Australia. 29-31 October 2019. [DOI: 10.4081/jphr.2020.1811 | NIH: PDF Document]. 2 July 2020.

ShapeFit. Waist to Height Ratio Calculator - Assess Your Lifestyle Risk. [Online Article and Tool]. 31 March 2015.

What’s the Substitute for Sugary Soft Drinks?

Imagine this scenario. Public health advocates campaign for your city to impose a tax on sugary beverages. They claim it will improve the public's health through fighting obesity by making soda and other soft drinks made with sugar more costly to buy, forcing budget-minded consumers to substitute much lower calorie containing beverages. Your city's politicians, always happy to get more tax revenue, go along with their scheme. How do you think consumers of sugary soft drinks in your city will respond?

If you answered they will drink more calorie-laden alcohol-based beverages, you're right!

The latest proof that consumers substitute beer and liquor for sugar-sweetened soft drinks comes to us from Seattle. In December 2017, the city imposed a unique \$0.0175 per ounce tax on beverages containing calories from sugar, but not on beverages made with non-calorie-laden sweeteners. For example, consumers buying a two-liter bottle of Coca-Cola would pay an additional tax of \$0.35 that consumers of the same size bottle of Diet Coke or Coke Zero would not.

At first glance, you might think consumers of Sugar-Sweetened Beverages (SSB) would choose to switch to the sugar-free versions of their previously preferred soft drink or to water to avoid having to pay so much more for it.

But that's not what happened according to a peer-reviewed study published in PLOS ONE, which found that the tax "induces substitution to alcoholic beverages". More specifically, the consumers preferred substitute wasn't sugar-free beverages. It was beer, whose sales rose by 7% relative to those of the demographically similar city of Portland, Oregon, which didn't impose a soda tax:

There was evidence of substitution to beer following the implementation of the Seattle SSB tax. Continued monitoring of potential unintended outcomes related to the implementation of SSB taxes is needed in future tax evaluations.

How many competent public health advocates do you suppose would push for new or expanded soda taxes knowing that real life consumers are more likely to shift to alcohol-based beverages with equivalent levels of calories instead of water or low-calorie sugar-free soft drinks? Not only do they miss any benefit from reducing calories consumed among the public, higher alcohol consumption comes with the "higher risk of motor accidents/deaths, liver cirrhosis, sexually transmitted diseases, crime and violence, and workplace accidents" to the public's health.

Then again, if you're a long-time reader of Political Calculations, you could have easily predicted that from our analysis of what happened to alcohol sales in Philadelphia after that city's soda tax went into effect.

Image credits: Coca-Cola Photo by Omar Elmokhtar Menazeli on Unsplash. Miller High Life Photo by Waz Lght on Unsplash.

References

Lisa M. Powell, Julien Leider. Impact of the Seattle Sweetened Beverage Tax on substitution to alcoholic beverages. PLOS ONE 18 January 2022. DOI: 10.1371/journal.pone.0262578.

Baylen Linnekin. Study: Seattle's Soda Tax Has Been Great for... Beer Sales? Reason. [Online Article]. 12 February 2022.

How Did COVID-19 Change U.S. Life Expectancy?

The reports of how COVID-19 changed U.S. life expectancy are grim.

The pandemic crushed life expectancy in the United States last year by 1.5 years, the largest drop since World War II, according to new Centers for Disease Control and Prevention data released Wednesday. For Black and Hispanic people, their life expectancy declined by three years.

U.S. life expectancy declined from 78.8 years in 2019 to 77.3 years in 2020. The pandemic was responsible for close to 74 percent of that overall decline, though increased fatal drug overdoses and homicides also contributed.

“I myself had never seen a change this big except in the history books,” Elizabeth Arias, a demographer at the CDC and lead author of the new report, told The Wall Street Journal.

The figures for just COVID-19's impact on U.S. life expectancy are roughly in line with the CDC's preliminary estimates from February 2021, which was based on the then-available data through the first half of 2020.

Unfortunately, the CDC's estimates are rather misleading. Dr. Peter Bach, the director of the Center for Health Policy and Outcomes at the Memorial Sloan Kettering Cancer Center, ran some back of the envelope calculations after the CDC released its preliminary estimates and came up with very different results.

The CDC reported that life expectancy in the U.S. declined by one year in 2020. People understood this to mean that Covid-19 had shaved off a year from how long each of us will live on average. That is, after all, how people tend to think of life expectancy. The New York Times characterized the report as “the first full picture of the pandemic’s effect on American expected life spans.”

But wait. Analysts estimate that, on average, a death from Covid-19 robs its victim of around 12 years of life. Approximately 400,000 Americans died Covid-19 in 2020, meaning about 4.8 million years of life collectively vanished. Spread that ghastly number across the U.S. population of 330 million and it comes out to 0.014 years of life lost per person. That’s 5.3 days. There were other excess deaths in 2020, so maybe the answer is seven days lost per person.

No matter how you look at it, the result is a far cry from what the CDC announced.

We built the following tool to do Bach's math, which checks out. You're welcome to update the figures with improved data or to replace them with other countries' data if you want to see the impact elsewhere in the world. If you're accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version of the tool.

COVID-19 Factors Affecting National Life Expectancy
Input Data Values
Number of COVID-19 Deaths in a Year
Estimated Average Years of Life Lost per COVID-19 Death
National Population

Change in National Life Expectancy
Calculated Results Values
Estimated Years of Life Lost for All COVID Deaths
Years of Life Lost per Person Due to COVID-19
Days of Life Lost per Person Due to COVID-19

So why is the CDC's estimate of the change in life expectancy estimate so different? As Bach explains, it is not because of either the data or the math, but rather, it is because of the CDC's assumptions in doing their math:

It’s not that the agency made a math mistake. I checked the calculations myself, and even went over them with one of the CDC analysts. The error was more problematic in my view: The CDC relied on an assumption it had to know was wrong.

The CDC’s life expectancy calculations are, in fact, life expectancy projections (the technical term for the measure is period life expectancy). The calculation is based on a crucial assumption: that for the year you are studying (2019 compared to 2020 in this case) the risk of death, in every age group, will stay as it was in that year for everyone born during it.

So to project the life expectancy of people born in 2020, the CDC assumed that newborns will face the risk of dying that newborns did in 2020. Then when they turn 1, they face the risk of dying that 1-year-olds did in 2020. Then on to them being 2 years old, and so on.

Locking people into 2020 for their entire life spans, from birth to death, may sound like the plot of a dystopian reboot of “Groundhog Day.” But that’s the calculation. The results: The CDC’s report boils down to a finding that bears no relation to any realistic scenario. Running the 2020 gauntlet for an entire life results in living one year less on average than running that same gauntlet in 2019.

Don’t blame the method. It’s a standard one that over time has been a highly useful way of understanding how our efforts in public health have succeeded or fallen short. Because it is a projection, it can (and should) serve as an early warning of how people in our society will do in the future if we do nothing different from today.

But in this case, the CDC should assume, as do we all, that Covid-19 will cause an increase in mortality for only a brief period relative to the span of a normal lifetime. If you assume the Covid-19 risk of 2020 carries forward unabated, you will overstate the life expectancy declines it causes.

In effect, the CDC's assumption projects the impact of COVID-19 in a world in which none of the Operation Warp Speed vaccines exist seeing as they only began rolling out in large numbers in the latter half of December 2020.

When the CDC repeats its life expectancy exercise next year, its estimates of the change in life expectancy should reflect the first year impact of the new COVID-19 vaccines, which will make for an interesting side by side comparison. Especially when comparisons of pre-vaccine case and death rates with post-vaccine data already look like the New Stateman's chart for the United Kingdom:

The Magic Percentage for Useful COVID Herd Immunity

What percentage of the population needs to be vaccinated to usefully reduce the risk of dying from COVID-19?

We're going to do a back-of-the-envelope calculation to estimate the answer to that question using Arizona's high quality COVID data in general, and the state's data for COVID-related hospital admissions and deaths in particular.

We're also going to build off our previous analysis that synchronized Arizona's figures for the number of positive COVID infection test results, hospitalizations, and deaths according to the approximate date of initial SARS-CoV-2 coronavirus exposure for the Arizonans who became infected and experienced these pandemic-related events. The following chart shows these three streams of data using a logarithmic scale, covering the period from 15 March 2020 through 30 April 2021.

We've annotated the chart to indicate two periods of "noise" in the data for deaths, which came into play when the daily number of COVID-related deaths of Arizonans dropped into the single digits. Because of the small numbers involved, having a relatively small change in the daily number can have an outsize effect on the appearance of the overall trend, which accounts for the "noisy" short-term trough that was recorded in mid-September 2020 and the short-term spike in late March 2021. We've added the dotted lines to these areas of the chart to indicate what the overall pattern would look like without the short term noise in the data.

Now to the bigger question. We're going to focus on the ratio of deaths to hospital admissions because these events represent the most serious classes of COVID infections. In Arizona, 75% of COVID-related deaths have occurred among the state's senior population, Age 65 or older. This same demographic has accounted for 46% of COVID-related hospital admissions in the state.

These figures confirm seniors are disproportionately vulnerable to both these outcomes if they become infected by the SARS-CoV-2 coronavirus. This fact is why this portion of the state's population was targeted for early COVID vaccinations once the vaccines became available.

Because the incidence of COVID-related deaths in concentrated in Arizona's senior population, we should see a sustained decline in the ratio of COVID deaths to hospital admissions corresponding to roughly when the population Age 65 or older achieved effective herd immunity. We can then identify what percentage of the state's elderly population had been received at least one vaccine dose at that point in time, which in turn, will give us a reasonable indication of what percentage of the population needs to be vaccinated for COVID to reduce its risk of death.

The next chart graphically shows the results when we combine these points of data together.

We find at least 55% of the population would need to have received at least one dose of the COVID-19 vaccines to provide the benefit of reduced risk of death from becoming infected by the coronavirus. That's the percentage of the Age 65 and older population of Arizonans who had been vaccinated as of 28 February 2021, which marks the point in time at which COVID-related deaths in the state began to plunge as a result of the Operation Warp Speed vaccination programs.

That's the low end for our estimate, because it does not consider the portion of the senior population who would have obtained natural immunity from having become infected with the SARS-CoV-2 coronavirus and who recovered from it. As of 28 February 2021, Arizona's senior population accounted for 109,897 known COVID infections, about 13.4% of the state's total at that time. Added to the 696,559 Age 65 or older Arizonans who had received at least one COVID vaccination dose at that date would put the high end of the estimate at 64%.

That upper level figure would explain why public health officials have set a target of 70% of the population for COVID vaccinations, but it seems strange they are not giving more weight to the potential contribution of natural immunity in achieving that goal. If they did, they could focus their limited resources for providing COVID vaccination more effectively.

Looking at Arizona's data, we would say the magic percentage for vaccinations to achieve useful COVID herd immunity is somewhere between 55% and 64% of the population. That's because there is almost certainly a good amount of overlap between those who have recovered from COVID and those who have been vaccinated. It would be more beneficial and less wasteful for public health officials to target the COVID vaccines to those who have not developed any antibodies to SARS-CoV-2 coronavirus infections.

References

Arizona Department of Health Services. COVID-19 Data Dashboard: Vaccine Administration. [Online Database]. Accessed 10 June 2021.