Next generation matrix for virus dynamic model

Question

I am reading an article on next generation matrix to find basic reproduction number found here
Because I am working with within host models I tried to use the method discussed in that article to the simple viral dynamic model given by

$\begin{alignat}{1} \dot{x} &= λ - dx - βxv,\\ \dot{y} &= βxv - ay, \\ \dot{v} &= ky - uv, \end{alignat}$

For this system I know that I don't want to go through these steps to find $R_o$, but I just want to try the steps for next generation matrix.

So, as the "infection subsystem" (those equations of the ODE system that describe the production of new infections and changes in state among infected individuals.) I selected,
$\begin{alignat}{1} \ \dot{y} &= βxv - ay, \\ \dot{v} &= ky - uv, \end{alignat}$

Then linearized this infected subsystem around infection free steady state to gain
$\begin{alignat}{4} \dot{y} &=~& - ay &~+~& \frac{λβ}{d} v, \\ \dot{v} &=& ky &~-~& uv, \end{alignat}$
Then have to decompose this as $T+\Sigma$ "where $T$ is the transmission part, describing the production of new infections, and $\Sigma$ is the transition part, describing changes in state (including removal by death or the acquisition of immunity)"

So, $T= \begin{pmatrix} 0 & \frac{λβ}{d} \\ k & 0 \end{pmatrix}$

and $\Sigma= \begin{pmatrix} -a & 0 \\ 0 & -u \end{pmatrix}$

Then $K_L=-T\Sigma^{-1}=\begin{pmatrix} 0 & \frac{λβ}{d} \\ \frac{k}{a} & 0 \end{pmatrix}$

Then, $R_0$ will be the dominant eigenvalue of $K_L$ which I get as $\sqrt{{\beta \lambda k\over adu}}$.

But, for this system $R_0$ should be $\beta \lambda k\over adu$

Can someone please tell me what I am doing wrong to get an incorrect $R_0$.

Also does it hugely matter how I select the infection subsystem. If I include a component that shouldn't belong to this system then would it impact the final results?

Also, can someone point to any resources which discusses next generation matrix for within host models as all the articles that I found discusses population level models like SEIR.

Answer 1

I think this has something to do with the free virus model having two stages (infected cells and then free virus). I think if you find the square of the next generation matrix you have made and then find the dominant eigenvalues you will get the correct answer currently you are estimating an average over the virus and the cell, but really you just want the average per cell.