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Epidemiological modelling and its use to manage COVID-19

Insights into mechanistic models, by the DYNAMO team

Over the next few weeks, we will present some key elements of epidemiological modelling through short educational articles. These articles will help you to better understand and decipher the assumptions underlying the epidemiological models that are currently widely used, and how these assumptions can impact predictions regarding the spread of pathogens, particularly SARS-CoV-2. The objective is to discover the advantages and limitations of mechanistic modelling, an approach that is at the core of the DYNAMO team's work. The examples of models will be inspired by models used in crisis, but sometimes simplified to make them accessible.

#11 - Analysis of the sensitivity of the COVID19 model to parameter variations

In this post, we analyze the sensitivity of the model presented in article #6, with the values of parameters estimated in article #8. We simultaneously varied by 10% all the model parameters (table) except the previously estimated date of virus introduction (22/01), as well as the start (16/3) and end (11/5) dates of containment fixed as observed. We used here a FAST (Fourier Amplitude Sensitivity Test) sampling protocol, and the "sensitivity" package under the R software.

Table of model parameters

Parameter

Definition

Values (min-max)

β

Basal transmission rate (d-1)

1.48 (1.332-1.628)

σ

Factor reducing the excretion of Ip, Ia, Ips (%)

0.47 (0.423-0.517)

v1

Residual contacts of Iss (%)

0.25 (0.225-0.275)

v2

Residual contacts of H & ICU (%)

0.25 (0.225-0. 275)

ε

Latency exit rate (d-1)

0.30 (0.270-0.330)

γp

Exit rate from state Ip (j-1)

0.66 (0.600-0.733)

f

[pa, pps, pms, pss]

Variation factor

Probability of being Ia, Ips, Ims or Iss (sum = 1)

1.00 (0.900-1.100)

[f*[0.3,0.2,0.3], 1-f*0.8]

γ

Exit rate from states Ia, Ips, Ims, Iss (j-1)

0.43 (0.391-0.478)

pICU

Probability to enter an intensive care unit after Iss

0.25 (0.225-0. 275)

gH

Hospital exit rate (d-1)

0.07 (0.060-0.073)

µH

Proportion of H that will eventually recover (%)

0.90 (0.89-0.91)

γICU

Intensive care exit rate (d-1)

0.05 (0.045-0.055)

µICU

Proportion of ICU that will eventually recover (%)

0.70 (0.67-0.73)

c1

Residual contact during lockdown

0.10 (0.09-0.11)

c2

Residual contact after lockdown (distancing)

0.50 (0.450-0.550)

For illustration purposes, we focus on three outputs of the model: the number of cases at the date of the epidemic peak (PeakValue), the date of the peak (PeakDate), and the cumulative number of deaths over the simulated period (CumDeaths). The Figure below shows that the number of cases at the peak and the cumulative number of deaths over the period are outputs governed by the same Parameters, essentially the basal transmission rate (b) and residual ISS (s) contacts. Thus, the contribution of ISS prevails in the dynamics of infection. Next comes the duration of the latency phase (1/ε). The date of the peak is influenced by ε, as well as the duration in the infectious (1/γ), the duration in the intensive care unit (1/γICU) and residual contact during containment (c1). Parameters explaining less than 10% of the variation in an output are generally considered non-influential.

Indices de sensibilité (FAST) pour les trois sorties analysées

It is therefore clear that not all outputs of the same model are necessarily influenced by the same parameters, and that the relevant outputs should be analysed before interpreting the predictions of a model. Furthermore, the results also indicate that the date of the epidemic peak is not very sensitive to interactions between parameters. On the contrary, the two other outputs are strongly influenced by interactions between the parameters involved in the transmission capacity of ISS individuals, which should be better specified to improve the accuracy of the model. Finally, some parameters do not seem to impact these three outputs. If the range of variation tested is indeed relevant (10% could be a little low for some parameters) and if only these outputs are of interest, then there is no need to work towards better estimating these parameters, or the model can even be simplified.

Noteworthy, the data used in article #8 are not sufficient to estimate the values of the model's key parameters accurately enough, with confidence intervals of more than 10% around the likely values. More data are therefore required before going any further in an accurate way!