Someone argues: "The epidemic threshold condition is that the reproduction number…

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

Many Parts, One Behaviour, Ch Epidemics on networks

Problem

Someone argues: "The epidemic threshold condition is that the reproduction number R0R_0 exceeds one, and R0R_0 is the transmissibility TT times the mean number of contacts, so on any network R0=T⟨k⟩R_0 = T\langle k\rangle. For a configuration-model network with degree density p(k)∝k−γp(k) \propto k^{-\gamma}, γ=3.0\gamma = 3.0, minimum degree kmin⁡=3k_{\min} = 3, cutoff kmax⁡=200k_{\max} = 200, and transmissibility T=0.1T = 0.1, one only needs ⟨k⟩\langle k\rangle." Using the continuum approximation p(k)=(γ−1)kmin⁡γ−1k−γp(k) = (\gamma-1)k_{\min}^{\gamma-1}k^{-\gamma} for kmin⁡≤k≤kmax⁡k_{\min} \le k \le k_{\max}, ((a)) compute ⟨k⟩\langle k\rangle and hence the claimed R0=T⟨k⟩R_0 = T\langle k\rangle; ((b)) compute the correct network reproduction number R0=T(⟨k2⟩−⟨k⟩)/⟨k⟩R_0 = T(\langle k^2\rangle - \langle k\rangle)/\langle k\rangle; ((c)) say which conclusion about an epidemic is right, and find the error in the quoted argument.

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