On The Convexity Of The Effective Reproduction Number

In this study we analyze the evolution of the effective reproduction number, $R$, through a SIR spreading process in heterogeneous networks; Characterizing its decay process allows to analytically study the effects of countermeasures on the progress of the virus under heterogeneity, and to optimize their policies. A striking result of recent studies has shown that heterogeneity across nodes/individuals (or, super-spreading) may have a drastic effect on the spreading process progression, which may cause a non-linear decrease of $R$ in the number of infected individuals. We account for heterogeneity and analyze the stochastic progression of the spreading process. We show that the decrease of $R$ is, in fact, convex in the number of infected individuals, where this convexity stems from heterogeneity. The analysis is based on establishing stochastic monotonic relations between the susceptible populations in varying times of the spread. We demonstrate that the convex behavior of the effective reproduction number affects the performance of countermeasures used to fight a spread of a virus. The results are applicable to the control of virus and malware spreading in computer networks as well. We examine numerically the sensitivity of the Herd Immunity Threshold (HIT) to the heterogeneity level and to the chosen policy.