### Epidemic Modelling

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

### bioNRICH

bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your study of biology at A-level and university.

### Mixed up Mixture

Can you fill in the mixed up numbers in this dilution calculation?

# The Genes of Gilgamesh

##### Stage: 4 Challenge Level:

It is important to realise in this question that the cross is not like a normal Mendelian cross, but that all previously accumulated genetic infortmation is inherited.

It would be impossible to create an offspring who was two-thirds G and one-third M. This can be explained as follows: each successive generation doubles the number of possible 'parts' that it can be made out of. For example the first generation is purely G or M, whereas the second generation has two 'parts' - it can be GG, MM or GM. Additionally, the third generation has four parts and can be GGGG, GGGM, GGMM, GMMM or MMMM. Thus, the number of 'parts' is clearly $2^n$ where n is an integer.

In order to be able to be composed on one-third M, we are essentially asking if there is a value of n such that $\frac{2^n}{3}$ is an integer. It can be seen that there is no value of n to make this true because $2^n$ generates numbers which are divisible only by the prime number 2, but by no other primes. Because 3 is a prime number, this means that the expression can never yield an integer.

We are looking to find a composition which is within 1% of $\frac{1}{3}$. which is equivalent to the range $\frac{99}{300}$ - $\frac{101}{300}$ which is $0.33 - 0.33\dot{6}$.

Using trial and error:
$\frac{3}{8} = 0.375$
$\frac{5}{16} = 0.313$
$\frac{11}{32} = 0.348$
$\frac{21}{64} = 0.328$
$\mathbf{\frac{43}{128} = 0.3359}$

Thus, 128 = 2$^7$, and so there need to be a minimum of 8 generations. Can you draw out the crossing scheme to create this final progeny?