A simplified model for the analysis of COVID-19 evolution during the lockdown period in Italy
Authors: Simeone Roberto 
Published: June 2020

A simplified model applied to COVID-19 cases detected and officially published by the italian government [1], seems to fit quite well the time evolution of the disease in Italy during the period feb-24th - may-19th 2020.

The hypothesis behind the model is based on the fact that in the lockdown period the infection cannot be transmitted due to social isolation and, more generally, due to the strong protection measures in place during the observation period. In this case a compartment model is used and the interactions between the different compartments are simplified. The sample of cases detected is intended as a set of individuals susceptible to infection which, after being exposed and undergoing the infection, were isolated (’treated’) in such a way they can no longer spread the infection.

The values obtained are to be considered indicative.

The same model has been applied both to the data relating to Italy and to some regions of Italy (Lombardia, Piemonte, Lazio, Campania, Calabria, Sicilia, Sardegna), generally finding a good response and indicatively interesting values (see chap. 5).

The only tuning parameter is the ‘incubation period’ τ that, together with the calculated growth rate κ of the exponential curve used to approximate the early stage data.

Conclusions

A simplified compartmental model that uses only the incubation period and the exponential growth rate as parameters is applied to the COVID-19 data for Italy in the lockdown period finding a good fitting.

Revision History

This section summarize the history of revisions.

Revision # 1

Errata corrige in section 1 (Introduction): the equations that summarize the relationship between the parameters were wrong. This revised version contains the correct equations at page 2.

The synchronization criteria is updated. No need to use a threshold different to the one used to determine the growth coefficient. The results are now updated with the synchronization point near to the 20% of the maximum value of the cases detected per day:

Modifications in section 4 (Model results for Italy). It is appropriate to use an exponential function instead of a logistic function to find the growth rate in the initial phase. Section 4 and the results are now updated.

Some non-substantial corrections in the descriptive part.

Revision # 2

Errata corrige in the system differential equation 6: in the the derivative of S were reported a wrong additional term N. Now the equation 6 is correct.

Revision # 3

New approach to detect the exponential rate and new concept for the transfer coefficients.

Exponential rate:

The old criteria was oriented to the growth of the cases: yΔt = y0 * ekΔt thus: y0 + Δy = y0 * ekΔt. The exponential growth rate was then: k = log(1 + Δy/y0)/Δt.

The new criteria is oriented to the growth of the differences Δy = ekΔt − 1 obtaining: k = log(1 + Δy)/Δt.

Transfer coefficients:

The new approach is based on the following assumptions:

αSE = kekδ [day−1]: this coefficent is supposed to be the variation of the exponential growth per unit of time (δ = 1 day).

αEI = 1/τ [day−1] where t is the incubation period (this assumption is not changed).

αIT = kδ/T [day−1] this coefficent is supposed to be proportional to the ratio kδ/τ.

The constant δ =1 [day] represent the unit variation in time.

Basic reproduction number:

With the above assumptions, the basic reproduction number become:

Revision summary:

The old approach, although adapting well to the data, presented several inconsistencies in the parameters, and in particular on the relationship between and k.

In this revision the new approach still shows a good fit to the data and shows congruent relationships between the parameters.