We all bleed the same color

In “Mathematical Models in Biology”, Leah Edelstein-Keshet presents a model describing the number of circulating red blood cells (RBC’s). It assumes that the spleen filters out and destroys a fraction of the cells daily while the bone marrow produces a amount proportional to the number lost on the previous day:

\begin{cases} R_{n+1} = (1-f)R_n+M_n\\ M_{n+1} = \gamma f R_n\end{cases}(1)

where

R_n - number of RBC’s in circulation on day n,

M_n - number of RBC’s produced by marrow on day n,

f - fraction of RBC’s removed by the spleen,

\gamma - numer of RBC’s produced per number lost.

What would be the cell count on the n^{th} day?


Observe first that (1) is equivalent to

R_{n+2} = (1-f)R_{n+1}+M_{n+1}\quad\quad\quad(2)

where

M_{n+1} = \gamma f  R_n.\quad\quad\quad(3)

Let n = -1,

M_0=\gamma f R_{-1} \implies R_{-1} = \frac{M_0}{\gamma f}.\quad\quad\quad(4)

Substituting (3) into (2) yields

R_{n+2} = (1-f)R_{n+1}+\gamma f R_{n}.

We proceed to solve the following initial-value problem using ‘solve_rec‘ (see “Solving Difference Equations using Omega CAS Explorer“):

\begin{cases} R_{n+2}=(1-f)R_{n+1}+\gamma f R_{n}\\ R_{0}=1, R_{-1} = \frac{1}{\gamma f}\end{cases}

Evaluate the solution with f=\frac{1}{2}, g=1, we have

R_n = \frac{4}{3} + \frac{(-1)^{n+1}2^{-n}}{3}.\quad\quad\quad(5)

Plotting (5) by ‘plot2d(4/3 + (-1)^(n+1)*2^(-n)/3, [n, 0, 10], WEB_PLOT)’ fails (see Fig. 1) since plot2d treats (5) as a continuous function whose domain includes number such as \frac{1}{2}.

Fig. 1

Instead, a discrete plot is needed:

Fig. 2

From Fig. 2 we see that R_{n} converges to a value between 1.3 and 1.35. In fact,

\lim\limits_{n \rightarrow \infty}  \frac{4}{3} + \frac{(-1)^{n+1}2^{-n}}{3} = \frac{4}{3}\approx 1.3333....

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