Discrete-time multi-state model

A discrete-time model describes individual transitions between states at successive steps. State occupancy counts, first-entry counts and transition counts are distinct quantities, especially when a state has a self-loop. With independent individuals and fixed transition probabilities, destination counts conditional on the source count have a multinomial distribution.

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.