Need some help / ideas to proceed. Stuck for a while on this.

In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{max}(A)$ where $\lambda_{max}(A)$ is the largest eigenvalue of the adjacency matrix $A$, thus implying that the outbreak is basically dependent on the spectral radius of the underlying contact graph.

Say, now, there's a new kind of virus that spreads by a different mechanism (i.e., no longer purely based on the existence of a link between two nodes). For instance, say each time it infects a specific type of people (nodes) (e.g., male/female, young/old, asian/european/american/african etc.) and then it mutates and infect different kind of people (nodes) next time instance.

For this, I come up with a matrix, say $X$, that records the probability of the virus infecting each node (i.e., $x_{ij}$ = probability of node $j$ being infected if $i$ is the infectant). It is a non symmetric square hollow matrix with positive values for all off-diagonal elements in the matrix.

With this infection probability matrix $X$ in mind, I'm struggling on how to derive the epidemic threshold. Is it still $1/\lambda_{max}(X)$ by analogy of the conventional epidemic theory? Any idea, suggestions, hints or pointers will be appreciated.

Secondly, if I have two of such matrices $X_1$ and $X_2$ essentially describing the behaviour of two viruses, how do I compare them using their corresponding probability matrix, $X$?

Thank you in advanced.