An Infection Spreading Model on Binary Trees

An important and ongoing topic of research is the study of infectious diseases and the speed at which these diseases spread. Modeling the spread and growth of such diseases leads to a more precise understanding of the phenomenon and accurate predictions of spread in real life. We consider a long-range infection model on an infinite regular binary tree. Given a spreading coefficient α > 1, the time it takes for the infection to travel from one node to another node below it is exponentially distributed with specific rate functions such as 2−kk−α or 1 αk , where k is the difference in layer number between the two nodes. We simulate and analyze the time needed for the infection to reach layer m or below starting from the root node. The resulting time is recorded and graphed for different values of α and m. Finally, we prove rigorous lower and upper bounds for the infection time, both of which are approximately logarithmic with respect to m. The same techniques and results are valid for other regular d-ary trees, in which each node has exactly d children where d > 2.