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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.

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  • $\begingroup$ Interesting question, but can you clarify: by "hollow matrix", do you mean one whose diagonal entries are all zero? Wikipedia gives two different meanings for the term: en.wikipedia.org/wiki/Hollow_matrix $\endgroup$
    – Tom Leinster
    Commented Jan 16, 2014 at 13:11
  • $\begingroup$ Apology. By hollow matrix, I'm referring to the matrix which has zero for all it's diagonal elements ($tr(X)=0$). $\endgroup$
    – Val K
    Commented Jan 16, 2014 at 13:29

1 Answer 1

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The problem is addressed in

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

An epidemic will become extinct if and only if the spectral radius of matrix $X$ is smaller than 1.

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  • $\begingroup$ Thank you very much for this pointer which seems very relevant indeed. One question though: It seems that the system matrix $M$ mentioned in this paper seems to be symmetric but my probability matrix $X$ is not (in fact, it has complex eigenvalues as well). Does this invalidate the theorem derived in that paper? Thank you again. $\endgroup$
    – Val K
    Commented Jan 16, 2014 at 15:15
  • $\begingroup$ @ValK The authors do not assume that the matrix is symmetric - see e.g. Section "Solution for (17) with nonsymmetric PAM" $\endgroup$
    – Waldemar
    Commented Jan 16, 2014 at 15:21

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