An exact solution would involve math not completely trivial. Though, you can get a satisfactory answer by simplifying the case, because the decay times differ a lot.

432 years, 2 microseconds, 27 days, 159,000 years.

What happens to neptunium formed from americium decay?

From 2µs and 432yr, what is the proportion of neptunium versus americium, since you wait 5 years, much more than 2µs? From 27d ans 432yr, the protactinium versus americium?

You can go on to uranium, which accumulates, and evaluate its concentration over time - this one isn't a fixed proportion over americium.

The thorium concentration is quadratic over time over the year time scale, and radium cubic, as well as the followers up to ^{209}Bi which accumulates. I'm not convinced you get one atom of ^{205}Tl within 5yr with this chain.

- Is it Berkeley.edu that puts "Americurium"?

- Some computed concentrations will be so low that initial impurity will overwhelm them.

General methods:

- If you have an electronics background, then all methods for linear systems apply: linear networks, fedback systems... Using fluence graphs, Laplace transformation and the like.

- If you have a background for linear algebra, you can write the set of linear equations with the formal s variable (Laplace transformation), solve, go the inverse Laplace.

- With 15 nodes, the general methods are not really accessible to hand solving. Either have a software that does it, and better, already knows all existing nuclides and decay modes - or go simplified methods by hand, as suggested previously.