Methodology

COVID-19 Model Structure

We have developed a compartmental model based on a general SEIR model structure which has been widely used for modelling the coronavirus pandemic to date (Peng et al., Chuynet, Lin et al., Yang et al.). The model tracks the proportion of the population that falls into the following classes:

  1. Susceptible individuals (S)
  2. Individuals who are protected as a result of public health measures such as lockdown policies (R1)
  3. Exposed individuals who have been infected but are not yet infectious (E)
  4. Infectious individuals who are asymptomatic (IA)
  5. Infectious individuals who are presymptomatic (IP)
  6. Infectious individuals who have mild symptoms (IM)
  7. Infectious individuals who have severe symptoms and are therefore hospitalized (IH, the “H” state)
  8. Infectious individuals who are hospitalized and are in critical condition requiring an ICU and/or a ventilator (IC, the “ICU” state)
  9. Individuals who die from COVID-19 disease (D)
  10. Individuals who recover from COVID-19 disease (R2)
  11. Individuals who are quarantined (contact tracing) (Q)

A diagram outlining the relationship between these classes is shown in Figure 1. The model equations describing the movement in and out of each of the 11 classes follow the diagram.

Figure 1. COVID-19 SEIR epidemic model structure. Arrows between compartments demonstrate the possible transitions between disease states and the accompanying variables refer to the transition rates which are detailed in Table 1. Dashed arrows are used to show the dependence of transmission on the infectious classes.

Parameters

Parameter definitions and proposed values are given in Table 1.

The parameters can be defined as a distribution from which we can sample values to generate prediction confidence intervals using a Monte-Carlo sampling approach. Table 1 lists the prior parameter values used to initiate the model, and we use Bayesian updating methods to refine these initial values with evolving data on the epidemic to obtain posterior values sequentially. We use data on cumulative confirmed cases and deaths from the COVID-19 disease, as reported daily for US counties in the Johns Hopkins University (JHU) database, in running the sequential calibration of our model. Although we begin with 50,000 samples of parameter vectors drawn randomly from the prior values, and use a Relative RMSE procedure to select 200 best fitting models to the joint cumulative case and death data for generating outputs of interest.

The effectiveness of lockdown in a county can be approximated by using data on movement reductions, as provided in the unacast toolkit. Updates on these movements as lockdowns are eased will be used in conjunction with changes in cases and deaths to forecast the impact of various lockdown lifting measures. Similarly, data on rates at which implemented surveillance and testing regimes are able to detect infectious cases will be used to model the impact of testing and isolation on mitigating virus transmission as lockdowns are lifted. The impacts of various social distancing measures are modelled via variable reductions in the transmission rate at this stage.

Table 1. Model parameters definitions and values. Note: we are still reviewing literature to understand how we can use existing data to estimate transition rates.

Parm Definition Prior range Units/notes Published values References
β Infection transmission rate 0.1428 – 1.5 Estimated as R0*gamma in SIR model R0 = 2-6
β = 0.6 – 1.7
Peng, Lin, Read, MMWR
L Lockdown ratio 5-10 Ratio of population under lockdown to susceptible population
α Rate of entering lockdown 2.0 (fixed) Controls how quickly lockdown is enforced, linked to lockdown ratio and lambda Model structure proposed by Peng Peng
λ Rate of leaving lockdown = alpha/L Model structure proposed by Jiwei Jiwei
σ Rate of moving from exposed class to infectious class 0.16 – 0.5 1/σ is the latent period; assumed 2-6 days 2.2-6 days latent period Sanche
p Proportion of exposed who become asymptomatic 0.25 – 0.5 CEBM, Qiu
γA Recovery rate of asymptomatic cases 0.125 – 0.33 1/γA is the infectious period; assumed 3-8 days 3-14 days infectious period, most <= 8 days Sanche, Read, Lipsitch, Prem, Peak, Maier
γM Recovery rate of cases with mild symptoms 0.125 – 0.33 1/γM is the infectious period; assumed 3-8 days 3-14 days infectious period, most <= 8 days Sanche, Read, Lipsitch, Prem, Peak, Maier
γH Recovery rate of cases with severe symptoms requiring hospitalization 0.125 – 0.33 1/γH is the infectious period of severe cases; assumed 3-8 days 3-14 days infectious period, most <= 8 days Sanche, Read, Lipsitch, Prem, Peak, Maier
γC Recovery rate of cases with severe symptoms requiring intensive care 0.125 – 0.33 1/γC is the infectious period; assumed 3-8 days 3-14 days infectious period, most <= 8 days Sanche, Read, Lipsitch, Prem, Peak, Maier
γQ Recovery rate of quarantined cases 0.125 – 0.33 1/γC is the infectious period; assumed 3-8 days 3-14 days infectious period, most <= 8 days Sanche, Read, Lipsitch, Prem, Peak, Maier
δ1 Rate of moving from presymptomatic class to mild symptomatic 0.1 – 1 1/time from start of infectious period to illness onset; assume 1-10 days Latent period: 2-6 days; Incubation period: 2-12 days Li, Ynag, Backer, Guan, Sanche
δ2 Rate of moving from mild case to hospitalized class 0.06 – 0.25 1/time from illness onset to hospitalization; assume 4-15 days 4-15 days Li, Zhou, Wang
δ3 Rate of moving from hospitalized class to ICU 0.09 – 1 1/time from hospitalization to ICU; assume 1-11 days Illness onset to ICU: 6-15 days; Illness onset to hospital: 4-15 days Li, Zhou, Wang
m Mortality rate of ICU class 0.08 – 0.25 1/time from ICU to death 4-12 days Zhou
ε Proportion of symptomatic cases that are not reported 0.1 – 0.3 Assume 10-30% of symptomatic cases are not getting tested Diagnostic rate of symptomatic 1/9 -1/3; Diagnosis rate overall 25% Jiwei, Roda
X1 Proportion of mild cases that progress to hospital 0.05 – 0.3 5-30% of mild cases are hospitalized 5%, 20.7-31.4% Verity, CDC MMWR
X2 Proportion of hospital cases that progress to ICU 0.2 – 0.3 20-30% of hospitalized cases require an ICU 26-30% Verity, Zhou, Wang, Wu
X3 Proportion of ICU cases that die 0.2 – 0.8 Proportion of ICU cases that die Zhou, Wu, Yang, and Tom’s input
z1 Proportion of asymptomatic cases that are quarantined through contact tracing Fixed value, scenario-based Ex. 50%, 75%, 90%, depends on strength of contact tracing efforts
z2 Proportion of presymptomatic cases that are quarantined through contact tracing Fixed value, scenario-based Ex. 50%, 75%, 90%, depends on strength of contact tracing efforts
z3 Proportion of mild symptomatic cases that are quarantined through contact tracing Fixed value, scenario-based Ex. 50%, 75%, 90%, depends on strength of contact tracing efforts
n1 Proportion of quarantined cases that require hospitalization 0.05 – 0.3 Assumed equal to mild symptomatic cases, see x1
κ Rate of moving from quarantined class to hospital 0.125 – 0.33 1/γH is the infectious period of severe cases; assumed 3-8 days 3-14 days infectious period, most <= 8 days Sanche, Read, Lipsitch, Prem, Peak, Maier
d Reduction in transmission due to social distancing, face masks, etc. 0.15 – 0.42 Masks are 58-85% effective Brienen

References