Mathematics of Epidemics on Networks by István Z. Kiss Joel C. Miller & Péter L. Simon

Author:István Z. Kiss, Joel C. Miller & Péter L. Simon
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof.

Simple calculation shows that the values given in the statement determine a steady state. The investigation of its stability is based on the linearisation of the right-hand side of the differential equations at the disease-free steady state. Differentiating the right-hand side of the differential equation of [S k ] c with respect to [S l ] c (with l ≠ k) yields zero since [SI] c  = 0 in this equilibrium. Differentiating with respect to [S k ] c yields −γ for the same reason. Differentiating the right-hand sides of the differential equations of [SI] c , [SS] c and [II] c with respect to [S k ] c yields again zero. Differentiating the right-hand sides of the differential equations of [SI] c , [SS] c and [II] c with respect to these variables yields the matrix

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