30.05.2020

Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures

Modeling InfectiologyTransversal
Gatto M et al
Proc Natl Acad Sci USA

Main result

Using Pontryagin's principle of maximum a first order optimality system is written, which is solved using a forward-backward sweep method. Finally, using concrete and realistic data, numerical results are given.

Takeaways

In this paper the authors consider a model of the evolution of the epidemic based on ordinary differential equations. They use the theory of optimal control to find the best strategy to be followed by public authorities to control the transmission of the virus while waiting for a vaccine. This optimal strategy consists of a control with an intensity that increases sharply at the beginning and then decreases steadily. The authors show the superiority of this strategy over strategies often suggested by epidemiologists, such as the constant control strategy, the so-called "lock-down" strategy, and the strategy of constant control.

Strength of evidence Weak

The model distinguishes for each of the states (E), (A), (I) and (R) two degrees of severity: severe (s) and "not severe" (or "mild" (m)). The goal of the paper is to find an optimal control c*(t) which minimizes an appropriate cost function; this optimal control is computed by numerically solving a system of optimality obtained using the Pontryagin maximum principle. This optimal strategy consists of a control with an intensity that increases sharply at first and then decreases steadily. The authors show the superiority of this strategy compared to strategies often suggested by epidemiologists, such as the constant control strategy, the so-called lock-down strategy, or also cyclic strategies which consist in alternating minimum and maximum values for the control function. I think that the results obtained will be very useful for epidemiologists.

Objectives

The purpose of the paper is to find an optimal control c*(t) that minimizes an objective function that takes into account the number of deceased persons and a total cost associated with the implementation of isolation measures.

Method

The authors consider a compartimental model to describe the dynamics of the COVID-19 epidemic in a population where individuals may belong to one of the following six states: susceptible (S), latent, i.e. infected but asymptotic and non-infectious (E), asymptomatic but infectious (A), symptomatic infectious (I), recovered (R) and deceased (D). The evolution of these populations is described by a system of ordinary differential equations (ODE).
An important parameter in these equations is what can be called "control effort" c(t) which represents the percentage reduction in transmission that is due to government isolation measures at time t.
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