COVID Odyssey: NZ New Year Fear 27 Post #954~ NZ COVID-19 ~ If the total number of cases increase by r daily then new cases also increase by r

If the total number of cases increase by a factor of r daily then new cases also increase by r.
This may not be obvious to everyone. So we will prove it.

Below are case numbers for the early 2020 outbreak of COVID-19 in New Zealand.

We see that a daily increase of r = 1.4 fits the data well.

We also see that the estimated number of new cases also increases by a factor of r = 1.4.

See:

NZcases2020r

In General:

If the total number of cases increase by r daily then new cases also increase by r.

Proof

Let T[i] be the total number of cases on Day i.

Then the number of new cases each day is C{i] = T[i] – T[i-1].

WLOG (without loss of generality) we may assume T[0] = 10 cases.

We assume T[i] increases by a factor of r each day (e.g. r = 1.4).

Then T[i] = r T[i-1]

When i>1 We have

C[i]

= T[i] – T[i-1]

= rT[i-1] – rT[i-2]

= r(T[i-1] – T[i-2])

= rC[i-1]

Also

C[1]

= T[1] – T[0]

= rT[0] – T[0]

= 10r – 10

= 10(r-1).

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