Erythropoiesis is a multi$-$stage process through which bone marrow stem cells turn into red blood cells. The

completion of each stage of differentiation is marked by cell replication, where a cell at differentiation state $($i$)$

splits at a rate of ki into two daughter cells that are each at differentiation state $($i$+1).$ The dynamics of this

multi$-$stage process for N representative differentiation states can be described as the followingdxi$/$dt$=1-$k$1$x$1$

dX$2/$dt$=2$k$1$x$1-$k$2$X$2=2k_{i-1}{x}_{i-1}-k_i{x}_i$

$\frac{d{x}_N}{dt}=2k_{N-1}{x}_{N-1}-k_N{x}_N$

Where $t$ represents time $($in days$),x_i$ is the number of cells in differentiation state $($i$),$ and I is the rate at which

bone marrow stem cells commit to the erythropoiesis process.

Assume that all the red blood cells that are formed by this process enter the blood stream, where they serve their

purpose and die at the rate of $200$ billion cells per day. Since the number of red blood cells in a healthy

individual remains relatively constant, the balance on the blood stream results in the following equation:

$\frac{d{x}_{\text{B}\text{l}\text{o}\text{o}\text{d}}}{dt}=2k_N{x}_N-200{10}^9=0$

Answer the following questions:

$($a$)$ Analyze the steady state condition for the system of ODEs describing the process of erythropoiesis.

Express the steady state level of cells in each compartment $(x_i^{SS})$ in terms of I and $k_i.$

$($b$)$ What is the total number of stem cells per day $($I$)$ that need to commit to the erythropoiesis process in

order to produce the required $200$ billion red blood cells per day? Assume that the stem cells undergo a

total of $10$ stages of differentiation $(N=10)$ and that the initial conditions and transition rate constants

are as follows: Xi$(0)=0$cells'for i$=1,2,...$N

ki$=2\mathrm{.}2$day$^-1$;$,$for i$=1,2,....$N Show clearly how you calcuate the value of I.

$($c$)$ Develop a Matlab code for the system of ODEs describing the erythropoiesis process. Use

parameter values and initial conditions as described in $($b$).$ Show the time profiles $($i$.$e$.,$ dynamics$)$ for all

N stages of the differentiation process during a period of two weeks in the same plot.