The population can be modelled by a two$-$dimensional continuous$-$time Markov chain

$($Xt

$,$ Yt$)$ where Xt

is the number of individuals in state A and Yt

is the number of

individuals in state B$,$ at time t$.$ The states then consist of $($i$,$ j$),$ i$,$ j $=0,1,...$

individuals.

Given

Xt $=$ i: Number of individuals in state A

Yt $=$ j: number of individuals in state B

Let current state of $($Xt

$,$ Yt$)$ be $($i$,$ j$)$; i$.$e$.($Xt $=$ i$,$ Yt $=$ j$),$ and assume one cell change

in a split of a moment at a time.

Starting with state $($i$,$ j$)$ we have

$($i $1,$ j $+1)\backslash$alpha $($i$,$ j$)\backslash$beta

$->$

$($i $+2,$ j $1)$

i$.$e$.$ A $->$ B : $($i$,$ j$)->($i $1,$ j $+1)$ at rate $\backslash$alpha ; for each conversion of type A cell

From B we have $($i $+2,$ j $1)$ for each conversion of type B cell.

We don$$t know how many cells will convert at any given moment t$,$ but there is a

range of possible states, Xi$($t$)$ at a rate i$\backslash$alpha and Yj $($t$)$ at a rate j$\backslash$beta $.$

So if we start with $1$ cell, of type A say, then this will develop into a branching process

on the continuous time spectrum.

So there could be a scenario like this:

$(1,0)\backslash$alpha $->(0,1)\backslash$beta

$->$

$(2,0)"$

$\backslash$alpha $->(1,1)($if one type A converts$)$

$2\backslash$alpha $->(0,2)($if two type A convert$)$

Continuing in this manner, now for each of the $(1,1)$ and $(0,2)$ states we get more

possible branches over time:

$(1,1)->$

$"$

$\backslash$alpha $->(0,2)($when $1$ A$-$type cell splits$)$

$\backslash$beta

$->$

$(2,1)($when $1$ B$-$type cell split$)$

$(0,2)->$

$"$

$\backslash$beta

$->$

$(2,1)($when $1$ B$-$cell converts$)...$

$2\backslash$beta

$->$

$(4,0)($when both B$-$type cells split$)...$

The process of conversions and splits goes on with time.

Note that in the diagram each state state is based on the assumption that one cell

converts in any split moment of a second, and the conversions are independent of each

other. The resulting diagram gets complex very quickly as the conversions occur and

develop into a $$tree$-$like$$ network.

Also, note that the continuous$-$time aspect is a result of the fact that the conversions

occur in splits of seconds.