The impact of the power law exponent on the behavior of a dynamic epidemic type process

• Authors: Adrian Ogierman;Robert Elsässer
• Venue: Proceedings of the twenty-fourth annual ACM symposium on Parallelism in algorithms and architectures
• Year: 2012

Epidemic processes are widely used to design efficient distributed algorithms with applications in various research fields. In this paper, we consider a dynamic epidemic process in a certain (idealistic) urban environment modeled by a complete graph. The epidemic is spread among $n$ agents, which move from one node to another according to a power law distribution that describes the so called attractiveness of the corresponding locations in the urban environment. If two agents meet at some node, then a possible infection may be transmitted from one agent to the other. We analyze two different scenarios. In the first case we assume that the attractiveness of the nodes follows a power law distribution with some exponent less than 3, as observed in real world examples. Then, we show that even if each agent may spread the disease for f(n) time steps, where f(n)=O(log n) is a (slowly) growing function, at least a small (but still polynomial) number of agents remains uninfected and the epidemic is stopped after a logarithmic number of rounds. In the second scenario we assume that the power law exponent increases to some large constant, which can be seen as an implication of certain countermeasures against the spreading process. Then, we show that if each agent is allowed to spread the disease for a constant number of time steps, the epidemic will only affect a polylogarithmic number of agents and the disease is stopped after (\log log n)O(1) steps. Our results explain possible courses of a disease, and point out why cost-efficient countermeasures may reduce the number of total infections from a high percentage of the population to a negligible fraction.