MTH6142 Complex Networks
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MTH6142 Complex Networks
Assessed Coursework 4
The reproduction number R of an epidemic spreading process taking place on a random network with degree distribution P(k) is given by
R = λ (1)
where k indicates the degree of the nodes and the average h...i indicates the average over the degree distribution, P(k).
Therefore R is the product between the infectivity λ of the virus, due to its biological fitness and the branching ratio of the network, depending on the degree distribution of the network and given by hk(k − 1)i/hki.
According to the value of R the epidemic can be in di↵erent regimes:
• If R > 1 the epidemics is in the supercritical regime: the epidemics spreads on a finite fraction of the population, resulting in a pandemics.
• If R < 1 the epidemics is in the subcritical regime: the epidemics a↵ects a infinitesimal fraction of the population and can be considered suppressed.
• If R = 1 the epidemics is in the critical regime: this is the regime that separates the previous two regimes.
Consider an epidemics with infectivity λ = 1/4. Investigate how the network topology can determine the regime of the epidemics in the following cases.
(A) Consider a Poisson network with average degree c = 3 and a Poisson
network with average degree c = 5. Calculate R and establish in which regime the epidemic process is in these networks. [1 MARK]
(B) Calculate R for a scale-free network with degree distribution P(k) = Ck −γ, minimum degree m, maximum degree K and power-law expo- nent γ = 2.5 using the continuous approximation for the degrees. [2 MARKS]
(C) Take the scale-free network considered in point (B) calculate R and establish in which regime the epidemic process is if m = 2,K = 50. [1 MARK]
2022-03-20