Corona virus / Covid-19
Copyright © 2020 Jiri Kriz, www.nosco.ch

Estimating number of recovered people

Health organizations usually do not report the number of recovered people or the reports are wrong. I show how to estimate this number.

Let us use the following notation:

I(t)
The total cumulative number of people that have been infected till time t.
D(t)
The total cumulative number of people that have died till time t.
R(t)
The total cumulative number of people that have recovered till time t.
A(t)
The number of people with currently active disease at time t,
A(t) = I(t) - R(t) - D(t)

Suppose that at time t + Δt all activelly ill people either recover or die. Let us denote:

ΔD
The number of actively ill people that die during time Δt,
ΔD = D(t + Δt) - D(t)
ΔR
The number of actively ill people that recover during time Δt,
ΔR = R(t + Δt) - R(t)

We have

A(t) = ΔD + ΔR
I(t) - R(t) - D(t) = D(t + Δt) - D(t) + R(t + Δt) - R(t)
(1)
R(t + Δt) = I(t) - D(t + Δt)

The last equation can also be written as:

R(t) = I(t - Δt) - D(t)
(2)

Eq. (1) can be also derived by the following consideration. All people who have been infected till time t will be recovered till time t + Δt except those who die till t + Δt. So, we obtain again Eq. (1) and hence Eq. (2):

I(t) = R(t + Δt) + D(t + Δt)
R(t) = I(t - Δt) - D(t)

I assume that the recovery time Δt is 20 days for Covid-19 and use the estimation:

R(t) = I(t - 20) - D(t)
(3)

Let us note that the estimated R(t) is not necessarily a monotonically increasing function. In fact, it can happen that R(t) decreases. Consider:

R(t+1) - R(t) = [I(t-20+1) - I(t-20)] - [D(t+1) - D(t)]
(4)

It can happen that the increase of infections from t-20 to t-20+1 is small or even 0 and actually less than the increase of deaths from t to t+1 such that R(t+1) becomes smaller than R(t). I tolerate this behaviour and interpret it in the sense that a person that was considered recovered dies later.

References