A research team is modeling the spread of a highly contagious pathogen in a densely packed, fully connected community where the homogeneous mixing assumption holds. 

Because they are modeling the disease beyond its initial early stages, they cannot use the simple exponential approximation. Instead, they use the exact SI model equation:

$$i(t) = \frac{i_0 \exp(\beta t)}{1 - i_0 + i_0 \exp(\beta t)}$$

At the start of the observation ($t = 0$), exactly $10\%$ of the population is infected ($i_0 = 0.1$). After a certain number of days $t$, the infection's exponential growth factor reaches $e^{\beta t} = 9$.

What is the fraction of infected individuals $i(t)$ in the population at this time?

a) $10.0\%$

b) $33.3\%$

c) $50.0\%$

d) $81.8\%$

e) None of the above.


Original idea by: Matheus de Oliveira Saldanha

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