Eigenvectors in model analysis Say I have a model with five ODEs. This is a model based on virus, host cells and immune response somewhat similar to the model in here:

I want to know what I can learn about the system through eigenvectors. I know that from eigenvalues I can learn about the stability. But, what do eigenvectors tell about the system?
Say I have eigenvalue $D$ and the corresponding eigenvector, $V$ as follows.
 
What can be determined about the system using eigenvectors?
 A: I assume that the eigenvalues and -vectors you are talking about, are obtained through a linearisation of the dynamics at a particular fixed point $\tilde{y}$.
The linearised differential equation would thus have the shape:
$$\dot{y}=A(y-\tilde{y}),$$
where $A$ is the matrix whose eigenvalues ($D_1$, $D_2$, …) and eigenvectors ($V_1$, $V_2$, …) you are looking at.
The solutions to such a differential equation are known to be of the form:
$$ \tilde{y} + c_1 e^{D_1 t} V_1 + c_2 e^{D_2 t} V_2 + …$$
where the $c_i$ are constants determined by the initial conditions.
As you already know, this tells you something about stability. For example, if all eigenvalues are negative¹, solutions in the vicinity of $\tilde{y}$ will converge to $\tilde{y}$. The eigenvectors tell you along which direction this happens, i.e., how the solutions converge to or diverge from $\tilde{y}$. Practically, almost all initial conditions have components corresponding to each eigenvector (i.e., $c_i ≠ 0\;∀i$) and thus the eigenvector corresponding to the largest eigenvalue¹ dominates the dynamics, as the corresponding term grows fastest or shrinks slowest, respectively. This is most prominent, if your largest eigenvalue is much larger than the next one¹.
For example, if your largest eigenvalue $D_1$ is positive and real and all others are negative¹, you do not only know that $\tilde{y}$ is not a stable fixed point, but also that in which direction the instability is directed. More specifically you can state that for initial conditions in the vicinity of $\tilde{y}$ you will get a solution of the form $\tilde{y} + c_1 e^{D_1 t}V_1$. So, in your application, if an uninfected state is a fixed point (which is unstable in terms of your differential equations, but practically stable due to the fact that virus numbers are discrete), you can describe how it would initially react to an infection with a small number of viruses.

¹ in the real part
