Model Structure

The LMIC reports are generated using an age-structured SEIR model. The developed model is an extension of the model used in our previous report (see Report 12) and the source code for the model can be found at In this model, the infectious class is divided into different stages reflecting progression through different disease severity pathways. These compartments are:

Given initial inputs of hospital/ICU bed capacity and the average time cases spend in hospital, the model dynamically tracks available hospital and ICU beds over time. Individuals newly requiring hospitalisation (either a hospital or ICU bed) are then assigned to either receive care (if the relevant bed is available) or not (if maximum capacity would be exceeded otherwise). Whether or not an individual receives the required care modifies their probability of dying.


Version 5

We continue the extensions made in version 4, by extending \(R_t\sigma\) to consider more flexible mobility independent changes to transmission after \(pld\). Starting one week after \(pld\) we fit a shift to \(R_t\) every 14 days, which is maintained after the 14 day period. In this way we model mobility independent changes to that change every 2 weeks. The last mobility independent change is maintained for the last 4 weeks prior to the current day to reflect our inability to estimate the size of this parameter, \(R_t\rho_n\) below, due to the approximate 21 day delay between infection and death. \(R_t\) is now given by:,

\[ R_{t,i} = R_{0,i} . f(-M_{eff}.(1-M_i(t)) -M_{eff.pld}.(M_i(t) - M_{pld}) - R_t\rho_1 - R_t\rho_2\ \ ... \ -R_t\rho_n)\] where \(R_t\rho_1\) is the first mobility independent change in transmission, which starts 7 days after \(pld\), i.e. is equal to 0 before this, and is in effect for the remainder of the simulation. \(R_t\rho_2\) is the second mobility independent change in transmission, which starts 21 days after \(pld\), i.e. 2 weeks after \(R_t\rho_1\). We use strong priors, (\(N(0, 0.2)\), where \(N\) is the normal distribution with a mean of 0 and a standarad deviation of 0.2) to ensure that where mobility is still informative of transmission (where mobility is increasing transmission is also increasing) it is the mobility parameters that are driving the epidemic trajectory rather than the mobility independent changes in transmission.

Version 4

We assume that transmission can decouple further from mobility, such that after lockdown transmission can be lower than \(R_0\) at 100% mobility:

\[ R_{t,i} = R_{0,i} . f(-M_{eff}.(1-M_i(t)) -M_{eff.pld}.(M_i(t) - M_{pld}) - R_t\sigma)\] where \(M_{pld}\) is the mobility at its nadir (at post lockdown date, \(pld\)) and is only in effect after \(pld\) such that when \(t <= pld\), \(M_{eff.pld} = 0\).

\(R_t\sigma\) is a one time signmoidal shift in \(R_t\) that occurs over 30 days after post lockdown date to reflect mobility independent changes to transmission resulting from changing behaviour. \(R_t\sigma\) is thus equal to 0 prior to \(pld\) and is equal to the maximum value of \(R_t\sigma\) for all time points more than 30 days after \(pld\).

Version 3

We assume that the impact of mobility on transmission will be different after lockdown ends, with a new \(M_{eff}\) parameter, \(M_{eff.pld,i}\) estimated for after lockdown, with \(R_t\) given by:

\[ R_{t,i} = R_{0,i} . f(-M_{eff,i}.(1-M_i(t))),\ when\ t < Post\ Lockdown\ Date \] \[ R_{t,i} = R_{0,i} . f(-M_{eff.pld,i}.(1-M_i(t))),\ when\ t > Post\ Lockdown\ Date \] where the post lockdown date is inferred as when mobility is at its nadir.

Version 2

We incorporate interventions using mobility data made publically available from Google, which provides data on movement in each country and includes the percent change in visits to places of interest (Grocery & Pharmacy, Parks, Transit Stations, Retail & Recreation, Residential, and Workplaces). Similar to Version 1, we assume that mobility changes will reduce contacts outside the household, whereas the increase in residential movement will not change household contacts. Consequently, we assume that the change in transmission over time can be summarised by averaging the mobility trends for all categories except for Residential and Parks (in which we assume significant contact events are negligable). Formally, \(R_t\) (time varying reproductive number) for country \(i\) is given by:

\[ R_{t,i} = R_{0,i} . f(-M_{eff,i}.(1-M_i(t))) \] where \(f (x) = 2 exp(x)/(1 + exp(x))\) is twice the inverse logit function. \(M_i(t)\) is the average mobility trend at time \(t\) (in which 1 represnts 100% mobility (i.e. no change) and 0 represents 0% mobility) and \(M_{eff,i}\) is the mobility effect size for country \(i\). In countries in which mobility data is not available, we use a Boosted Regression Tree model, trained to government policy data from the ACAPS Government measures Dataset, to predict the change in mobility. In scenario projections going forwards we use the mean of the last 7 days mobility as the assumed mobility in the absence of changs in interventions.

Version 1

In version 1 of these reports, we incorporated the impact of interventions that have been put in place using data from the Oxford Coronavirus Government Response Tracker. We currently make assumptions about the efficacy of these interventions and so the projections should be interpreted as scenarios rather than predictions. Work is ongoing for version 2 to integrate formal statistical fitting to improve these projections. In summary, school closures are assumed to cause a 10% reduction in contacts. Work closure is assumed to cause 30% reduction in contacts. Banning of public events is assumed to lead to a 5% reduction in contacts while restrictions on movement or a lockdown is not in force. Restrictions of movement is assumed to cause an additional 37.5% reduction in contacts on top of the 40% reduction due to school and work closure, leading to a total 77.5% reduction.

Model Parameters

The parameter table below summarises the current best estimates incorporated in the package as of 2020-09-19. These will be updated as our understanding of the epidemic develops.

Parameter Value Reference
Basic reproductive number, \(R_0\) 3.0 Report 13
Mean Incubation Period 4.6 days Report 9; Linton et al.; Li et al. The last 0.5 days are included in the I_MILD and I_CASE states to capture pre-symptomatic infectivity
Generation Time 6.75 days Report 9
Mean Duration in \(I_{MILD}\) 2.1 days Incorporates 0.5 days of infectiousness prior to symptoms; with parameters below ~95% of all infections are mild. In combination with mean duration in \(I_{CASE}\) this gives a mean generation time as above
Mean Duration in \(I_{CASE}\) 4.5 days Mean onset-to-admission of 4 days from UK data. Includes 0.5 days of infectiousness prior to symptom onset
Mean Duration of Hospitalisation for non-critical Cases (\(I_{HOSP}\)) if survive 9.5 days Based on unpublished UK data
Mean Duration of Hospitalisation for non-critical Cases (\(I_{HOSP}\)) if die 7.6 days Based on unpublished UK data
Mean duration of Critical Care (\(I_{ICU}\)) if survive 11.3 days Based on UK data adjusted for censoring
Mean duration of Critical Care (\(I_{ICU}\)) if die 10.1 days Based on UK data
Mean duration of Stepdown post ICU (\(I_{Rec}\)) 3.4 days Based on unpublished UK data
Mean duration of hospitalisation if require ICU but do not receive it 1 day Working assumption
Probability of dying in critical care 50% Based on UK data
Probability of death if require critical care but do not receive it 95% Working assumption based on expert clinical opinion*
Probability of death if require hospitalisation but do not receive it 60% Working assumption based on expert clinical opinion*
Multiplier of duration of stay for LIC and LMIC settings compared to HIC 50% Working assumption based on expert clinical opinion*

*N.B. Given the substantially weaker health systems in LIC and LMIC, it is likely that disease outcomes will differ from the UK. The estimates listed above for the key parameters determining the severity outcomes are the result of a rapid expert clinical review. Eight clinical experts with experience both in treating COVID-19 patients in the UK and with previous experience in clinical practice in LIC/LMICs were asked to provide their assessment of severity outcomes in LMICs. Although there was broad consensus on these outcomes, it should be noted that there was also consensus that this is likely to be highly heterogeneous both within and between countries due to other factors that are difficult to quantify and for which data sources do not readily exist as of 2020-09-19.

Age-Specific Parameters

Age-Group Proportion of Infections Hospitalised Proportion of hospitalised cases requiring critical care Proportion of non-critical care cases dying
0 to 4 0.001 0.050 0.013
5 to 9 0.001 0.050 0.013
10 to 14 0.001 0.050 0.013
15 to 19 0.002 0.050 0.013
20 to 24 0.005 0.050 0.013
25 to 29 0.010 0.050 0.013
30 to 34 0.016 0.050 0.013
35 to 39 0.023 0.053 0.013
40 to 44 0.029 0.060 0.015
45 to 49 0.039 0.075 0.019
50 to 54 0.058 0.104 0.027
55 to 59 0.072 0.149 0.042
60 to 64 0.102 0.224 0.069
65 to 69 0.117 0.307 0.105
70 to 74 0.146 0.386 0.149
75 to 79 0.177 0.461 0.203
80+ 0.180 0.709 0.580
Source Verity et al. 2020 corrected for non-uniform attack rate in China (see Report 12) Adjusted from IFR distributional shape in Verity et al. 2020 to give an overall proportion of cases requiring critical care of ~30% to match UK data Calculated from IFR in Verity et al. 2020 corrected for non-uniform attack rate in China (see Report 12) given the 50% fatality rate in critical care.