SIS epidemic model

Consider the SIS epidemic model on a scale-free network with degree distribution exponent \(\gamma\). In this model, \(\lambda\) represents the spreading rate of the disease, \(\lambda_c\) is the epidemic threshold, and \(i(\lambda)\) is the prevalence (fraction of infected nodes in the steady state).

Analyze the following statements:

I. For \(2 < \gamma < 3\), the epidemic threshold is zero (\(\lambda_c = 0\)), meaning that an epidemic can persist for arbitrarily small spreading rates \(\lambda > 0\).

II. For \(\gamma = 3\), the epidemic threshold is positive (\(\lambda_c > 0\)), so the disease can only persist if \(\lambda\) exceeds a finite critical value.

III. For \(3 < \gamma < 4\), the epidemic threshold is positive (\(\lambda_c > 0\)).

IV. For \(\gamma > 4\), the SIS dynamics approach the behavior predicted by homogeneous mixing models.

Which of the statements are TRUE?

A) I and III only

B) I, III and IV only

C) II, III and IV only

D) I, II and IV only

E) None of the above

Original idea by: Fernando de Facio Rossetti

Comentários

  1. Joao Meidanis13 de junho de 2026 às 07:02

    Very nice question. However, I'm concerned about III. On one hand, the book says that lambda_c is positive, for gamma between 3 and 4, for instance, in Figure 10.12. On the other hand, there is a paragraph shortly after this figure and immediately before Section 10.4 on Contact networks, where they mention that the degree-block approximation is not in fact needed, and an alternative treatment taking into account the full mathematical complexity of the stochastic problem leads to the conclusion that even for gamma > 3 the epidemic threshold vanishes. So. I guess the positivity of lambda_c depends on the depth of the analysis used.

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