Let be the number of infecteds, and the population. Then you are told the rate of increase in the number of infecteds is:

You are also told that when that:

and:

Hence at we have:

or

Because is a constant the DE is of variables seperable type so:

so:

or rearranging and exponentiating:

Now the condition that when allows us to determin that .

Now putting in the value of we have:

.

Finally after 12 days we have:

.

Then if we let be the proportion infected we have:

,

which we may solve for to get

RonL