This problem builds up three results in sequence

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems

Many Parts, One Behaviour, Ch Epidemics on networks

Problem

This problem builds up three results in sequence, each resting on a stated approximation. ((a)) The well-mixed SIR model obeys

\dot s = -\beta s i

\dot i = \beta s i - \mu i

, with transmission rate

\beta

(per contact, per unit time) and recovery rate

\mu

(per unit time). By linearising at the disease-free state

s = 1

i = 0

, find the growth rate

\sigma

of a small seed and show that it changes sign at

R_0 = \beta/\mu = 1

. ((b)) A pathogen has

\beta = 0.15\,\mathrm{day^{-1}}

per contact. Derive the transmissibility

T

of one edge (the probability that an infected node infects a given neighbour before it stops being infectious) for: (i) a fixed infectious period

\tau = 6\,\mathrm{days}

; (ii) an infectious period that is exponential with mean

6\,\mathrm{days}

, i.e. constant recovery rate

\mu = 1/6\,\mathrm{day^{-1}}

. Give both values of

T

to three significant figures. ((c)) Consider a five-node network: one hub joined to four rim nodes, and the four rim nodes also joined in a cycle (rim node 1 to rim node 2 to rim node 3 to rim node 4 back to rim node 1), so each rim node has degree

3

and the hub has degree

4

. For an SIS infection with effective spreading ratio

\lambda = \beta/\mu

on this fixed (quenched) network: (i) derive the degree-class (annealed) threshold

\lambda_{\rm c} = \langle k\rangle/\langle k^2\rangle

for this degree sequence; (ii) derive the quenched threshold

\lambda_{\rm c} = 1/\kappa_1

by finding the leading eigenvalue

\kappa_1

of the adjacency matrix directly, using the four-fold rotational symmetry of the rim to reduce the eigenvalue problem to two coupled equations. Quote both thresholds to three significant figures and say which approximation each rests on.

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