In a community in which A people are susceptible to a particular virus, the rate of growth of the spread of the virus was jointly proportional to the number of people who hadbcaught the virus and the number of susceptible people who have not caught it. If 10% of those susceptible had the virus initially and 25% had been infected after 3 weeks, what percent of those susceptible had been infected after 6 weeks.
If I read this right, then if y is the number of people who have the virus,
dy/dt = ky(A-y)
dy / y(A-y) = k dt
now finish it off. I wouldn't expect to have a nice closed formula for y=f(t)