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Apr 10, 2020
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SIR model and Russian officials: how to smooth a curve if it “goes to a plateau” and “goes along a sinusoid”

Recently, the head of the Federal Biomedical Agency of Russia Veronika Skvortsova said that “we 04 - 10 days before reaching a plateau with this infection, after that we will hold onto a plateau for some time and go in the opposite direction, that is, in fact, the process resembles a sinusoid. ”

What epidemiological model did Veronika Igorevna use when she claimed that the process would reach a plateau - we do not know. Moreover, it is not known whether the disease will have a seasonal pattern - therefore, we do not yet know whether the “process will resemble a sinusoid.”

One of the slogans of the campaign to combat pandemic and movement # stayhome - this is "Flatten the curve", "smooth the curve." This refers to the epidemiological curve, it has no plateau.

But where, in fact, did this curve come from? Why exactly this, and not from a plateau, like Veronika Igorevna? What mathematical formula is described?

There is such a model in epidemiology, it is called SIR , from "susceptible", "infected" and "recovered" (susceptible [к патогену], infected and recovered).

Denote the number of susceptible letters S , the number of infected with the letter I and the number of recovered - R . Then the process can be described by a system of three differential equations.

How read these equations? Everything is simpler than it might seem. On the left side of the equations is the process that we want to describe. In the first equation, this is how the number of susceptible S in time t , in the second - how the number of infected I changes in time, and in the third - the same for R , the number of those recovered.

We will deal with the part on the right. The change in the number of susceptibles will depend on how many healthy people S communicated with infected people I . β is a constant that determines the likelihood for a healthy person to communicate with the patient and get sick. The number of people recovered in time will be determined by the parameter I , the constant γ will determine the likelihood of recovery. Both β and γ specify the rate of increase or decrease in the number of people in three groups. The dependence of the number of infected people on time (see equation 2, the same curve!) Will be described as the difference between the number of patients ( βIS and the number of people who recovered ( γI ).

I note separately that this is a system of equations, and individually these equations will not work as a model of the epidemic.

In the animated image above - a simulation of the development of the epidemic at different values ​​of the parameter β . Blue indicates the number of non-sick people, red indicates the number of infected people, and green indicates the number of people who have recovered. All these values ​​are indicated in total for a certain point in time (the total number of cases for the whole time is approximated by the Gampertz curve, but this is another topic).

We measure time along the abscissa axis , in this simulation, the development of the epidemic can be calculated from 0 to days) (on scales left of the graph T max 356575806 = 14 ). The ordinate axis is the number of people, and the population in 827 people (on the scales on the left S 0 356575806 = 827 - the initial number of healthy people, and i 0 356575806 - the initial number of patients). β - reflects the effectiveness of quarantine measures. The lower its value, which varies from 0 to 1, the more effective the quarantine measures and the less people become ill. γ reflects the effectiveness of the treatment (that is, the rate of recovery), its value is also in the range from 0 to 1, and the larger it is, the better.

In real life and β , and γ also changes in time, but their changes are irregular in nature, so it is difficult to take them into account.

Staying at home, we decrease the parameter β , and the curve of the infected is smoothed . By the way, note that the curve of the sick does not even remotely resemble the sinusoid - its decline, in particular, in most cases will go slower than the rise.

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