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:
- The total cumulative number of people that have been infected till time t.
- The total cumulative number of people that have died till time t.
- The total cumulative number of people that have recovered till time 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:
- The number of actively ill people that die during time Δt,
ΔD = D(t + Δt) - D(t)
- The number of actively ill people that recover during time Δt,
ΔR = R(t + Δt) - R(t)
The last equation can also be written as:
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 assume that the recovery time Δt is 20 days for Covid-19 and use the estimation:
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:
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.