 Discrete-time multi-state diagnosis model

With a one-step early state, early diagnosis probability $a$, and non-early diagnosis probability $b$ per subsequent step, mean infections $h_i$ yield early diagnoses $e_k=a{h}_{k-1}$ and late diagnoses $n_k=b(1-a){\sum }_{i\le k-2}{h}_i(1-b{)}^{k-i-2}$. The early intensity fraction is $e_k/(e_k+n_k)$ when positive. Initial infections and endpoint conventions determine the index limits.