Fig 1.
Illustration of the inference of Bayesian data assimilation system for time-varying parameter estimation. The latent state Xt includes the variables and parameters of an epidemic model to be estimated. The epidemiological observation is denoted as Ct, and is linked to the latent state via the observation function. For each time step, the estimation of the latent state p(Xt|C1:t) is constantly updated according to ongoing reported observations using sequential Bayesian updating with forward filtering and backward smoothing. (A) Forward filtering at each time step. The posterior state estimation p(Xt−1|C1:t−1) estimated from previous step t−1 is transformed as the prior p(Xt|C1:t−1) for the current step t, calculated from the state transition model as detailed in the Method section. Together with the likelihood p(Ct|Xt) obtained from epidemiological observation at the current step, the posterior of the current step p(Xt|C1:t) is estimated. At the same time, as shown in (B), backward smoothing is used to compute , taking account of all the observations C1:T up to the time T by applying a Bayesian smoothing method (see the Methods section for more informaiton).
Fig 2.
Validation experiment of the DARt system on simulated data.
First, the ground-truth Rt sequence is synthetic using piecewise Gaussian random walk split by several abrupt change points. The sequence of incident infection jt is simulated based on a renewal process parameterised by the synthetic Rt. The observation process includes applying a convolution kernel that represents the probabilistic observation delay to obtain the expected observation and adding Gaussian noise that represents the reporting error to obtain the noisy ‘real’ observation Ct. The inputs (in grey) to the DARt system are the distributions of generation time, observation kernel and simulated noisy observation Ct. The system outputs are the estimated
, estimated
and change indicator
. These outputs are compared with the synthetic Rt, jt and the time of abrupt changes. Also, the observation function is applied to the estimated
to compute the estimated observation
with uncertainty, which is compared to the ‘real’ observation.
Fig 3.
(A) Synthetic Rt, simulated jt and Ct curves. (B) shows the comparison of the synthetic Rt (in red) with estimated Rt curves from DARt, EpiEstim and EpiNow2. (C) shows the estimated Mt from DARt to indicate sharp changes of Rt. (D) shows the simulated jt, jt from DART, and jt from EpiNow2. (E) compares the distributions of estimated Ct from DARt and EpiNow2 with the simulated Ct curve with 95% CrI. (F) compares the DARt estimated Rt results with and without smoothing.
Fig 4.
Epidemic dynamics in Wuhan (A), Hong Kong (B), United Kingdom (C) and Sweden (D). The top row of each subplot shows the number of daily observations (in yellow), the estimated daily observations (in blue) and the estimated daily infections (in green). The middle row compares the DARt Rt estimation (in black) with the EpiEstim results (in blue) and EpiNow2 (in yellow). The distributions of all estimated Rt are with 95% CrI. The bottom row shows the probabilities of having abrupted changes (Mt = 1) (in green bars).
Fig 5.
Three components of DARt inference model: state transition model, observation function and sequential Bayesian update module with two phases (forward filtering and backward smoothing).
The latent state that can be observed in Ct are defined as Xt = <Rt*,Jt*,Mt*> where Rt* is the instantaneous reproduction number, Mt* is a binary state variable indicating different evolution patterns of Rt*, is a vectorised form of infection numbers jt, t* indicates the most recent infection that can be detected at time t is from the time t* due to observation delay, and Tφ is the length of the vector Jt* such that Ct is only relevant to Jt* and jt*+1 only depends on Jt* via the renewal process.
Fig 6.
Illustration of the hierarchical transition process and observation process.
The most recent infection that can be observed by Ct is at the time t* = t−d where d is a constant determined by the distribution of observation delay. Suppose Tφ is the length of the vector such that Ct is only relevant to Jt* and jt* only depends on Jt*−1 via the renewal process. Therefore, Tφ≥max(Tw, TH−d+1). The case that Tφ = TH−d+1 is depicted in this figure.