Projecting COVID-19 Mortality as States Relax Nonpharmacologic Interventions

This simulation modeling study projects COVID-19 deaths between March 1, 2022, and December 31, 2022, in each of the 50 US states, District of Columbia, and Puerto Rico assuming different dates of lifting of mask mandates and nonpharmacologic interventions.

A2. Development Timeline eTable 1 lists major model updates and the dates on which they were introduced. eTable 1. Major model updates.

Date
Update February 2021 Vaccine rollout August 2021 Age-stratification to incorporate age-stratified vaccine data and differential mortality of age groups October 2021 Lower vaccine effectiveness due to Delta variant from August 1, 2021 December 2021 Waning (natural and vaccine-conferred) immunity January 2022 Lower vaccine effectiveness due to Omicron variant from December 1, 2021

A3. Model Overview
Our model is an extension of the traditional susceptible-infected-recovered (SIR) model, 2 which partitions a population into compartments representing mutually exclusive disease states. At any time , the variables , , , , and denote the number of people in the susceptible, exposed, infected, recovered, and deceased compartments respectively. The flow of people between compartments is assumed to obey a system of deterministic ordinary differential equations. We let Δ 1 to be compatible with data sources reporting daily data.

Age stratification
We stratify the population into two age groups, <65 years (low-risk) and ≥65 years (high-risk), with the subscript ∈ , . The total population in age group , denoted by , is assumed to be constant over the simulation period.

Vaccination
To reflect administration guidelines of the Pfizer-BioNTech and Moderna vaccines, i we stratify the disease states by vaccination status. The subscript ∈ 0, 1, 2 denote the number of vaccine doses received under the recommended two-dose regime. The third vaccine, Janssen, approved for a single-dose regime, is omitted from the model due to its accounting for only 3.7% of all administered doses in the U.S. as of October 31, 2021. ii Since there is no data on vaccination status at the time of infection, we assume doses are allocated proportionally to the susceptible and recovered compartments over the historical time horizon, iii i.e., if , and , are the actual number of first and second doses administered to age group on day , the proportion of the -dose susceptible and recovered compartments, ∈ 0, 1 , moving into the corresponding 1 -dose compartments on day is , min 1, .
The time lag of 12 days accounts for the delay between receiving a vaccine dose and the beginning of protection. 3 The implicit assumption is that a susceptible person does not become infected in the 12 days after receiving a dose. The vaccine reduces both susceptibility to infection and mortality risk. After vaccine doses, the probability of contracting the virus is reduced by 100 %, with 0 , 1; similarly, the infection fatality rate is reduced by 100 %, with 0 , 1.

Transmission
For a susceptible individual in age group who has received vaccine doses, the rate of exposure to the virus is given by where ℛ is the time-varying effective reproduction number. We model ℛ as a step function with breakpoints at the beginning of each calendar month over the historical time horizon to capture the effect of NPIs enforced during this period. The coefficients , are the elements of the contact matrix with row sums normalized to 1, so that , is the proportion of contacts per day of age group that are with age group ′. When a susceptible individual contracts the virus, they enter the exposed state and remain there for the duration of the latent period with a mean of 1/ days. After that, they transition to the infected state and remain there for the duration of the infectious period with a mean of 1/ days. Finally, the infected individual will either die with probability , 1 , where is the baseline infection fatality rate for age group , or recover with probability 1 , .

Waning immunity
An individual who has recovered from natural infection ( , ) enjoys a period of natural immunity with a mean of 1/ days before transitioning back into the susceptible state ( , ). A fully vaccinated susceptible individual ( , is protected for the duration of vaccine-conferred immunity with a mean of 1/ days before transitioning back into the partially susceptible state ( , ). Finally, since individuals with natural immunity who are subsequently vaccinated have been reported to exhibit "unusually potent immune responses", 4 a fully vaccinated recovered individual ( , ) is assumed to possess two 'layers' of immunity, shedding first their natural immunity then their vaccine-conferred immunity.

Booster shots
It is assumed that, once vaccinated, an individual will never shed their immunity completely (within the time frame of the simulation), and a fully vaccinated individual who has shed their vaccine-conferred immunity is indistinguishable from a partially vaccinated individual. Thus, the model differentiates between the subpopulation that is willing to receive booster shots and the subpopulation that is unwilling to be vaccinated. Fully vaccinated individuals wane into the partially vaccinated state and are 'boosted' back into the fully vaccinated state.

Differential equation formulation
In summary, our model is described by the following system of equations, where has been dropped for notational simplicity:

A4. Calibration and Numerical Solution
We calibrate the model to historical daily incident deaths. The system of ordinary differential equations is solved numerically using Euler's method (R package deSolve). 16 The calibration method is generalized simulated annealing (R package GenSA) with the sum of squared errors as the objective function. 17

A5. Forecasting
We make forecasts by allowing the model to continue running past the historical time horizon. Diagnosed cases and hospital and ICU occupancy are not accounted for in the SEIR model. We estimate these in a post-processing step as follows.

Diagnosed cases
We assume the future diagnosis rate remains at the latest estimated value, i.e., the number of incident diagnosed cases on the last day of data divided by the number of incident total cases on the last day of data.

Hospital and ICU bed occupancy
We back-calculate hospital and ICU bed occupancy from incident deaths assuming an average time to death from hospital and ICU admission of 16 and 10 days respectively. 20 Starting in August, we forecast occupancy data provided by the U.S. Department of Health and Human Services. Note that the data does not include all hospitals in any given state so our forecasts do not estimate the total demand for hospital and ICU beds.