A COVID‐19 model for local authorities of the United Kingdom

Abstract We propose a new framework to model the COVID‐19 epidemic of the United Kingdom at the local authority level. The model fits within a general framework for semi‐mechanistic Bayesian models of the epidemic based on renewal equations, with some important innovations, including a random walk modelling the reproduction number, incorporating information from different sources, including surveys to estimate the time‐varying proportion of infections that lead to reported cases or deaths, and modelling the underlying infections as latent random variables. The model is designed to be updated daily using publicly available data. We envisage the model to be useful for now‐casting and short‐term projections of the epidemic as well as estimating historical trends. The model fits are available on a public website: https://imperialcollegelondon.github.io/covid19local. The model is currently being used by the Scottish government to inform their interventions.

Here, N (µ, σ) denotes a normal distribution with mean µ and standard deviation σ. We say that X follows a positive half normal distribution N + (0, σ) if X ∼ |Y |, where Y ∼ N (0, σ). We link our observed deaths mechanistically to transmission as in Flaxman et al. (2020). We use a previously estimated COVID-19 specific distribution of times from infection to death π, as detailed in Verity et al. (2020); Walker et al. (2020). We assume the distribution of times from infection to death π (infection-to-death) to be the convolution of an infection-to-onset distribution (π )  and an onset-to-death distribution (Verity et al., 2020): π ∼ Gamma(5.1, 0.86) + Gamma(17.8, 0.45). ( Similar to generation distribution, π is further discretized via π s = s+0.5 s−0.5 π(τ )dτ for s = 2, 3, ..., and π 1 = 1.5 0 π(τ )dτ , where π(τ ) is the density of π. For estimating time varying infection fatality ratio (IFR, probability of death given infection) we use the estimates from Riley et al. (2020) as a starting point and then a bi-weekly change is estimated using the infections survey data from Office for National Statistics (2020) and REACT Study (2020).
The expected number of deaths d w , in a given week w, is given by the following discrete sum: where t is the total number of days till week w, i τ is the number of new infections on day τ and where π W is the transformation of discretized π to a weekly lag. We also observe weekly cases C w from 1 st June 2020 w ∈ {w(t = 2020-06-01), . . . , w(n)}. Similar to weekly deaths, weekly cases are modelled using a positive real-valued function c w = E[C w ] that represents the expected number of cases identified in testing for COVID-19. Again, the weekly cases C w are assumed to follow a negative binomial distribution but with mean c w and variance c w + c 2 w ψ2 , where ψ 2 follows a positive half normal distribution, i.e.
We assume the lag between an infection to be identified as a case, π i2c , is zero for first three days and then has an equal chance over next 10 days, i.e. π i2c = c 0, 0, 0, repeat( 1 10 , 10) .
We link the observed weekly cases and estimated daily infections using an estimated parameter infection ascertainment ratio (IAR), which is defined as the number of reported cases divided by the true number of infections (including both symptomatic and asymptomatic infections).
The expected number of cases c w , in a given week w is given by the following discrete sum: where, again, where t is the total number of days till week w, and i τ is the number of new infections on day τ . We also observe daily infections from ONS I ons t in our model. ONS infections are modelled using a positive real-valued function i ons t = E[I ons t ] that represents the expected number of infections given by the ONS infection survey. Daily infections from ONS I ons t are assumed to follow a normal distribution but with mean i ons t and standard deviation given by the 95% CI in ONS infection survey, i.e.
The expected number of ONS infections i ons t , on a given day t is given : We also observe total infections from REACT I react t,total in our model. REACT infections are modelled using a positive real-valued function i react t,total = E[I react t,total ] that represents the expected total number of infections given by the REACT. Total infections from REACT I react t,total are assumed to follow a normal distribution but with mean i react t,total and standard deviation given by the 95% CI in REACT survey, i.e.
The expected number of total infections i react t,total , on a given day t is given : We parametrise R t,m with a random effect for each week of the epidemic as follows where f (x) = 2 exp(x)/(1+exp(x)) is twice the inverse logit function and w(t) is a weekly random walk (RW) process, that captures variation between R t in each subsequent week. Following Liu et al. (2020), the prior distribution for R 0 was chosen to be We assume that seeding of new infections begins 30 days before the day after a state has cumulatively observed 10 deaths. From this date, we seed our model with 6 sequential days of an equal number of infections: i 1 = · · · = i 6 ∼ Exponential( 1 τ ), where τ ∼ Exponential(0.03). These seed infections are inferred in our Bayesian posterior distribution.
The weekly effect is modelled as a weekly random walk process, centred around 0 with variance σ that, starts on the first day of its seeding of infections, i.e. 30 days before a total of 10 cumulative deaths have been observed in this state. The RW process starts with 1 = 0, raw ∼ N (0, 1) (17) The prior for σ , the variance of RW process , is chosen as σ ∼ N + (0, .2). The conversion from days to weeks is encoded in w(t). Every 7 days, w is incremented, i.e. we set w(t) = (t − t start )/7 + 1, where t start is the first day of seeding. We estimated parameters independently for each local authority. Fitting was performed with the R package epidemia (Scott et al., 2021), written in the probabilistic programming language Stan (Carpenter et al., 2017) using an adaptive Hamiltonian Monte Carlo (HMC) sampler. Now for running our model not all observations are available at all levels. Hence we take a three stage approach to estimate the parameters as robustly as possible.
In first stage, Figure 1,we fit a model to entire of England, where we have all observations available, i.e., cases, deaths, ONS infections, REACT attack rate (total cumulative population infected). This means while fitting all four observations are used for inferring parameters namely via the observation process defined in Eq (1), Eq (5), Eq (9), and Eq (12). This step is very crucial as it is the only place we have an observation for infections. Hence, we estimate if r(w) and iar(w) only in this step, in all other steps we use provide tight priors around the estimates from this step.
In second stage, Figure 1, we fit individual models to all regions in England, as well as to all nations in the UK. The observations used in this stage are deaths and cases, hence the likelihood of model in this stage is calculated by adding Eq (1) and Eq (5). As stated earlier the if r(w) and iar(w) used in Eq (1) and Eq (5) are provided strong priors centred around the values estimated in first stage.
Finally, in our last stage, Figure 2, we fit individual models to all local authorities (LTLAs in England, local authorities in all other nations of the UK). As in second stage the likelihood of model is only composed of Eq (1) and Eq (5). Additionally in this stage we use the R t of the region (region for LTLAs in England and nations for local authorities in other nations) as a covariate for R t of the local area in addition to the weekly random walk.