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 advance.



Epidemic threshold
The problem is addressed in



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> Peng, Chengbin, Xiaogang Jin, and Meixia Shi. "Epidemic threshold and immunization on generalized networks." Physica A: Statistical Mechanics and its Applications 389.3 (2010): 549-560. [Link](http://www.researchgate.net/publication/233918441_PHYSA12267_published/file/9fcfd50ced21b2efdb.pdf)
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An epidemic will become extinct if and only if the spectral radius of matrix $X$ is smaller than 1.


