Estimate recovered/infectious for SIR model

Question

I like to fit a SIR model to COVID-19 data of UK. If i understood it correctly i need the numbers of I (infected) people (and not the reported case daily case data) which i would calculate as followed:

I(t) = C(t)-D(t)-R(t)

with C(t)=confirmed cases at time t; D(t)=deaths at time t and R(t)=recovered at time t Does the computation of I(t) makes sense? Next problem would be that UK doesn't provide reliable data for the numbers of recovered. I tried to estimate them as a function of confirmed cases and deaths:

R(t) = C(t-10)-D(t)

with C(t-10)= confirmed cases from 10 days ago (estimated duration of being infectious from COVID-19) and D(t) with deaths at time t. If I use the two equations i get negative numbers for the infected peple at some t's. So there has to be a mistake which i can't figure out. Here the beginning of the numbers (they are cumulative):

date	Confirmed	Deaths
1	374	1
2	429	2
3	483	2
4	630	3
5	889	7
6	1301	7

Answer 1

Your computation $I(t) = C(t)-D(t)-R(t)$ makes sense. But it is not very reliable* because the numbers for $D(t)$ and $R(t)$ are not published and if they would be published then it would be based on vague observations (if at all).

If i understood it correctly i need the numbers of I (infected) people (and not the reported case daily case data)

No, you do not (necessarily) need 'the numbers of I (infected) people'.

Yes indeed, with the SIR model you typically model/compute/solve $I(t)$ based on some differential equation

\begin{array}{rccl} \overbrace{I(t)^\prime}^{\text{net change}} &=& \overbrace{ \beta \frac{S}{N} I(t)}^{\text{rate of new cases}} - \overbrace{\gamma I(t)}^{\text{rate of recovery}}\\ \end{array}

What most tables show is not $I(t)$ but the number of daily new cases. For this you can compare this with $\beta \frac{S}{N} I(t)$.

Alternatively, you can also compare the cumulative daily new cases. This should equal $I(t) + R(t)$ (or other letters if your model has more).

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*I am saying that $D(t)$ and $R(t)$ are not reliable, but the same is true for $C(t)$. The number of confirmed cases is not the same as the actual number of people that got sick. Due to inaccurate measurements, this number can differ several orders from the real figure, and this difference may not need to be constant in time (due to changes in measurements and reporting).