Replicating the model presented by NeherLab.

The content of this repo are explained there.

Optimising the model parameters

Overview

Parameters fall into 2 categories:

• Disease specific: those are independent from where the disease arises (at least in first approximation). Whether someone is in France or Germany, the level of care is essentially identical and fatality rates (proportion of death amongst ICU patients) is assumed identical.

• Country-specific: For example how many infected individuals at the beginning of the model, what do the mitigation profile looks like...

The optimisation of the model is assessed by the only certain measure: how well the model assesses the number of deaths (using RMSE). Other published figures are not useful. For example, the number of confirmed cases is only an unknown portion of the actual total number of cases. The optimisation process is split into 2 consecutive steps:

• Optimise disease parameters by modifying them to achieve the lowest possible loss across all countries simultaneously (i.e. summing all the losses of all the countries). During this, the country-specific parameters remain constant.

• Optimise each individual country while keeping the disease parameters constant.

Some results

Those results took less than 10 minutes on my old laptop (Thinkpad x260).

In summary, the results of the optimisation gives an r₀ slightly lower than Neherlab's moderate value of 2.7. This is however offset by a hug number of non-identified asymptomatic and symptomatic individuals (by a factor of about 40). The number of days assumed in each state is more or less identical to the Neherlab values.

Country	r₀	tₗ	tᵢ	tₕ	tᵤ	γₑ	γᵢ	γⱼ	γₖ	δₖ	δₗ	δᵤ	Asymp	Sympt	mv0	mv1	mv2	mv3	mv4	mv5	mv6	mv7	mv8	mv9
France	2.493	5.795	5.597	3.179	13.933	0.231	1.073	2.048	0.487	1.175	1.216	0.833	40.808	44.067	0.992	1.020	0.197	0.549	0.963	0.387	0.539	0.899	0.560	1.092
Germany	2.493	5.795	5.597	3.179	13.933	0.231	1.256	2.048	0.487	1.175	1.216	0.822	46.740	43.408	1.042	0.896	0.172	0.552	0.904	0.375	0.509	0.974	0.523	1.100
Italy	2.493	5.795	5.597	3.179	13.933	0.231	1.167	2.048	0.487	1.175	1.216	0.848	42.731	42.805	0.996	0.965	0.173	0.597	1.063	0.374	0.464	0.850	0.616	1.234
Spain	2.493	5.795	5.597	3.179	13.933	0.231	1.192	2.048	0.487	1.175	1.216	0.868	47.949	40.095	0.917	0.957	0.184	0.519	0.938	0.383	0.474	0.957	0.582	1.078
Switzerland	2.493	5.795	5.597	3.179	13.933	0.231	1.237	2.048	0.487	1.175	1.216	0.857	46.312	42.157	1.080	0.890	0.193	0.554	1.083	0.377	0.555	0.857	0.625	1.048
UK	2.493	5.795	5.597	3.179	13.933	0.231	1.277	2.048	0.487	1.175	1.216	0.819	45.160	41.339	0.961	0.921	0.167	0.515	0.994	0.385	0.495	0.871	0.551	1.149
USA	2.493	5.795	5.597	3.179	13.933	0.231	1.201	2.048	0.487	1.175	1.216	0.807	43.230	39.758	1.074	0.965	0.178	0.517	0.953	0.374	0.460	0.985	0.625	1.224

Legend:

• r₀: average number of people infected by a symptomatic individual
• tₗ tᵢ tₕ tᵤ: average number of days as asymptomatic, symptomatic, hospitalised (severe) and in ICU (critical).
• γₑ γᵢ γⱼ γₖ: multiplier of the infectiousness. γᵢ is the infectiousness of a symptomatic individual and therefore set at 1. The other multiplier apply to asymptomatic, severe (when out of hospital) and critical (when out of hospital)
• δₖ δₗ δᵤ: multiplier of the fatality rate as compare to someone in ICU (δᵤ set at 1) for critical individuals in normal hospital beds or out of hospital.
• Asymp Sympt: The data provides a number of confirmed cases at the start of the model. Those are multipliers to get the number of asymptomatic and symptomatic individuals (only a portion of them are actually confirmed).
• mv...: Multiplier of r₀ reflecting the effectiveness of mitigation measures at weekly intervals.

Source code

It is organised in 4 files, plus the runMe.jl that provides results.

• COVID-19-model.jl sets a number of constants and the differential equations.

• COVID-19-utils.jl is a single function that is probably provided in the standard librabries...

• COVID-19-run-model.jl sets the initial default value of the parameters to be optimised, calculates solutions and corresponding losses.

• COVID-19-data.jl updates data from the repos, populates dictionaries

The optimisation is done with the BlackBoxOptim.jl. It is just magic...