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# The Timescale of Burden for the Delta Variant

The age of a car can be measured in years. It can also be measured in miles.

How should we measure the age of an epidemic? I recently led a paper, currently in review, on a potentially useful way to measure time in pandemics. The paper is based on a simple premise: as more people get infected, more people should be immune and transmission should decrease. "Herd Immunity Thresholds" - how many people need to get infected for transmission to reach a certain value (Rt=1) - are just one point on a curve defining how cumulative incidence affect transmission, from high rates of transmission when everybody is susceptible to no transmission when everyone is immune. If we can find this continuous relationship between infections & immunity, we can try to estimate when epidemics will end and how bad unmitigated epidemics can be. However, while transmission can be measured neatly by how fast cases or ICU admissions grow, it's hard to measure how many people have been infected. Not all cases are ascertained because not everybody shows up to the doctor to get tested, not all tests of infected patients come back positive, and not all previously infected patients seroconvert. Early in a pandemic, as we saw with COVID, the total number of people who've been infected can be difficult to estimate. Justin Silverman, Nathaniel Hupert & I estimated how many people had been infected in March, 2020 by counting how many extra people were visiting their doctor with influenza-like illness, but estimates aren't certainties. In the Timescale of Burden paper, we show that you don't have to be certain about cumulative incidence to study how infections reduce transmission. Any quantity of "burden" due to disease that we believe is proportional to cumulative incidence can do. Burden can be cases, hospitalizations, ICU admissions, deaths, cumulative google searchers for "loss of smell" and more. Any measure of cumulative burden can serve as a clock for measuring how much "time" has elapsed in an epidemic. By analyzing how transmission changes with burden, we can use early & less-mitigated outbreaks to estimate how bad later, less-mitigated outbreaks can be. The Timescale of Burden provides a neat way to visually and statistically compare epidemics & estimate the burden at which we reach an effective herd immunity threshold.

For the COVID-19 pandemic prior to February 1, 2021, we used deaths per-capita due to COVID as a useful measure of burden. Deaths per-capita were a nice way to compare outbreaks because our treatments didn't dramatically improve outcomes for hospitalized patients, and vaccines were not widely administered to reduce the probability of infection & the probability of severe outcomes given infection. If the infection fatality rate is similar across regions, then the fatality rate should be proportional to infections. More fatalities, more infections, less transmission - that's how I monitored prior outbreaks in the Summer, Fall, and Winter of 2020.

Right now, though, in the Summer of 2020, the Delta variant is causing cases to rise US states and most European countries. Unfortunately, we can't use the old tricks for this new virus. Vaccines are effective at reducing disease severity, and vaccination rates vary widely across regions & age groups. Infection fatality rates can be reasonably assumed to vary widely across US states during the Delta wave, so we need a new measure of burden to compare, monitor & forecast these outbreaks. We have early archetypal Delta-induced outbreaks in the UK, Portugal, Russia, and South Africa epidemics, all of which are (as of today) below a prior peak, and thus define entire curves of transmission decreasing as more people were infected, and with a good measure of burden we can use these early archetypes to forecast later outbreaks such as the later-starting and epidemics across US states whose cases are still rising. The figure below shows one way I've tried to update this method for the Delta variant by defining the "relative size of the Delta wave" as a (possibly) more comparable measure of burden. We can't use cumulative cases directly because case ascertainment rates vary considerably across regions based on testing availability, access to care, and more. (A) I measure the cumulative cases in the "Delta wave" relative to the amount of cases accumulated in the "Reference" period between August 1, 2021 to March 1, 2021. (B) Since the cumulative caseload of the Reference wave is constant & the Delta wave is strictly increasing, this measure of burden can serve as a timescale of burden that might help us estimate when cases might peak. Below, the trajectories of early European and US archetypes are plotted, and the current state of most US states (some were excluded due to changing day-of-week reporting that lead to erratic estimates of growth rates using the same negative binomial state space model as in the paper). The line of best fit through our latest observations of US states

This approach based on relative wave sizes has some limitations. If the various regions had a similar cumulative incidence during the reference period (an admittedly big assumption) and case ascertainments that may vary across regions haven't varied much since August 2020, then this relative size of the delta wave is proportional to cumulative incidence. I think the first assumption - that outbreaks had a similar cumulative incidence during the reference period - is probably the most egregious and adds considerable variation to the relationship between the x-axis & cumulative Delta incidence.

This measure of burden is going to the trash pile, but other metrics of burden may prove useful for comparing Delta outbreaks. I've worked on & am still working on others that are not yet in the trash heap, and hope I get to share them on here when they're ready! Science!!!