Digital Herd Immunity and COVID-19

Vir Bannerjee Bulchandani; Saumya Shivam; Sanjay Moudgalya; S L Sondhi

The effects of these parameters on the growth of the epidemic (or R) are studied using a simple branchingprocess model, where all infectious individuals are either symptomatic (S) or asymptomatic (A), and either on the app-based contact tracing network (C) or not (N). In an an uncontrolled setting, all types of infectious cases are assumed to proliferate with R 0 = 3, which is a reasonable estimate [18] for COVID-19 1 . Suppose that the outbreak starts from a single infected individual at time t = 0. In our discrete time (generational) model, this person infects R 0 other people at time t = 1, and each new infection is allotted one of the categories {CA, CS, N A, N S} randomly, with probabilities that are determined by the values of θ and φ. Individuals infected at the beginning of each generation are assumed not to infect anyone else after that generation has elapsed. Whenever a symptomatic person on the contact network (CS) is encountered during this branching process, the contact network is triggered, and all people connected by the network through past and present infections are placed in quarantine. As discussed earlier, since pre-symptomatic infections are common for COVID-19, our model includes the possibility that a CS person infects R S people by the time they trigger the contact network. As a consequence, non-CS people in the same generation also infect the next generation before the activation of the contact network (see Fig. 1c ). A few time-steps of the model are illustrated explicitly in Fig. 1 , together with the implementation of recursive contact tracing via removing connected components of the contact graph. Different combinations of the parameters θ, φ and R S lead to an effective reproduction number R distinct from the bare reproduction number R 0 , and we expect epidemic growth to be suppressed whenever R < 1.