Epidemiologists are modelling the spread of an infectious disease in a town. The following assumptio

Question

Epidemiologists are modelling the spread of an infectious disease in a town. The following assumptions are made:

At Week $0$, there are $500$ new infections.
In each subsequent week, the number of new infections is $60\%$ the number of new infections in the previous week, due to ongoing community transmission.

Let $u_n$ denote the number of new infections in Week $n$, for $n \geq 0$, $n \in \mathbb{Z}$.

Write down the value $u_1$ and verify that $u_2 = 180$.[2]
If $S_n$ denotes the total cumulative number of new infections after $n$ weeks, show that $S_n = 1250(1 - 0.6^{n+1})$.[2]

The local healthcare facility has a capacity of $1240$ beds. You may assume that each infected person occupies one bed, and that none are discharged during the period being considered.

Find the smallest value of $n$ such that the total number of infections up to and including Week $n$ first exceeds $1240$.[1]

The healthcare facility later increases its capacity to $1280$ beds.

Based on the given model, comment on whether this new capacity is sufficient to accommodate all infected persons over a prolonged period of time.[2]

Accepted Answer

(i) $u_1=300,u_2=180$ (ii) $S_n=1250(1-0.6^{n+1})$ (iii) $n=9$ (iv) Sufficient under the model; limiting total $1250<1280$.

2025 MI P1 Q7

2025 MI P1 Q7

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