The $R_{0}$ value of a virus refers to the average number of new infections resulting from each infected person. (a) A new virus is discovered. Extensive testing reveals that by the $n^{\text{th}}$ day, a total of 268 people were infected. In the following day, 686 people were newly infected. In the day following that, 2401 people were newly infected. Virologists propose to use a geometric progression to model the number of infections due to the virus. They propose the model: $u_{n}=aR_{0}^{\,n-1}$ where $u_{n}$ is the number of new infections on the $n^{\text{th}}$ day after the virus emerged, and $a$ is the number of people infected initially.

(a(i))
Using this model, estimate the value of $R_{0}$ .

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Worked solution

8(a(i))

1. Identify consecutive terms of the geometric progression from the given context.

From the problem, the number of new infections on consecutive days (after the $n^{\text{th}}$ day) represent consecutive terms in the geometric progression model. Let these be the $(n+1)^{\text{th}}$ and $(n+2)^{\text{th}}$ days:
$u_{n+1} = 686$
$u_{n+2} = 2401$

2. Calculate the common ratio $R_0$ by dividing the consecutive terms.

Since the model is a geometric progression with common ratio $R_0$ :
$R_0 = \frac{u_{n+2}}{u_{n+1}}$
$R_0 = \frac{2401}{686}$
$R_0 = 3.5$

Answer

$3.5$

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Worked solution

8(a(i))

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