Robust models of disease heterogeneity and control, with application to the SARS-CoV-2 epidemic

In light of the continuing emergence of new SARS-CoV-2 variants and vaccines, we create a robust simulation framework for exploring possible infection trajectories under various scenarios. The situations of primary interest involve the interaction between three components: vaccination campaigns, non-pharmaceutical interventions (NPIs), and the emergence of new SARS-CoV-2 variants. Additionally, immunity waning and vaccine boosters are modeled to account for their growing importance. New infections are generated according to a hierarchical model in which people have a random, individual infectiousness. The model thus includes super-spreading observed in the COVID-19 pandemic which is important for accurate uncertainty prediction. Our simulation functions as a dynamic compartment model in which an individual’s history of infection, vaccination, and possible reinfection all play a role in their resistance to further infections. We present a risk measure for each SARS-CoV-2 variant, ρV, that accounts for the amount of resistance within a population and show how this risk changes as the vaccination rate increases. ρV highlights that different variants may become dominant in different countries—and in different times—depending on the population compositions in terms of previous infections and vaccinations. We compare the efficacy of control strategies which act to both suppress COVID-19 outbreaks and relax restrictions when possible. We demonstrate that a controller that responds to the effective reproduction number in addition to case numbers is more efficient and effective in controlling new waves than monitoring case numbers alone. This not only reduces the median total infections and peak quarantine cases, but also controls outbreaks much more reliably: such a controller entirely prevents rare but large outbreaks. This is important as the majority of public discussions about efficient control of the epidemic have so far focused primarily on thresholds for case numbers.


S1 Supplemental
Proactive control reduces total infections as well as peak numbers in quarantine. Simulation with Delta variant and strict case thresholds at 25-50/100,000/14 days.

S2 Sampling in Practice
We use the following data and parameters specific to Austria • As generation interval (w t ) we will choose a discretization of a Gamma-distribution with mean 4.46 and standard deviation 2.63 cut off after day 13 as found by AGES in [1]. We use this generation interval for all variants. source: • The Austrian population is given by N = 8.932.664 (as of 2021-01-01) [2].
• We use the data on the progress of the Austrian vaccination program provided by the Austrian Ministry of Health [3]. Individuals vaccinated with an mRNA respective vector vaccine will be added to the appropriate group 14 respective 21 days after the first dose. We take the median doses of the last 7 days of available data for projecting daily administered doses into the future.
April 11, 2022 S1/S8 Boosters primarily prevent a winter outbreak due to waning. With vaccine boosting, ρ O t declines fast enough so that large winter outbreaks are preventedboth under reactive and proactive control. Case thresholds are 25-50/100,000/14 days.
• We use reported incidence provided by the WHO dashboard [4].
We assume all infections that happened in 2020 were with the wild type and we allocate the incidence of 2021 according to data provided by AGES [5] (see also Fig 15). Furthermore we scale this reported incidence to account for undetected infections based on values found in [6]. that actually get reported) of 0.42 which lies within the range of values investigated for Austria in [6].
• We use an estimate of the observed reproduction number using the R package EpiNow developed by [7]. These estimates are provided via [8].
The sampling procedure is as follows: (1) Import data and choose the parameters as above.
(2) Simulate the overlap between vaccinated and previously infected individuals to determine the group sizes S h t .
April 11, 2022 S2/S8   We modelled a modified Omega, which in addition to high immune escape (see Table 2), also has high basic reproduction number (same as Delta), while the mean of its generation interval is smaller at 3 days (CI: 1.5, 4.5). Case thresholds are 25-50/100,000/14 days. Both immunity waning and vaccine boosting are included. In contrast to all other simulations, here the import of Omega at the rate of 1/day starts later on 1.12.2021.

S3 Estimating Initial Variant Prevalence
In order to initialize the model with different variant prevalences as in the Delta simulations in Simulation and control of Delta, we use the following tentative estimation given the variant specific effective reproduction number R V e,t . For simplicity, we assume that initially all infections occur in non-interaction groups. and the only group infected with variant V will be group g = 0. Let us consider a simple exponential growth model April 11, 2022 S4/S8 for a growth rate β V > 0. We equate this to the expected incidence given in (3), that is (where for the last equation we set the necessary w m to 0) to get the following relation w t x t denotes the probability generating function of w. Thus we can recover the growth rate β V from the group reproduction number R V e,t in the following way (S2) Taking day t as a reference point we furthermore estimate the total growth rateβ of the total incidence I t using the observed reproduction number R e , preciselŷ If p V ≥ 0 denotes the prevalence of variant V on day t, that is then the estimated incidence of the previous days can be computed via Since the group specific reproduction number R V e,t is usually not known it also needs to be tenatively estimated. For this we start with assuming constant variant prevalence first. Let p Vi ≥ 0, i = 1, . . . , n such that n i=1 p Vi = 1 denote the prevalence of variant V i within the total incidence I t . Precisely I Vi t−s = p Vi I t−s and I t−s = n i=1 p Vi I t−s for s = 0, . . . , ν − 1. Using this constant initial incidence for each variant and using the observed reproduction number R e we compute a tentative initial mitigationM t using equation (S1). The variant specific reproduction number R V e,t can then be estimated in the following wayR