Spread and control of medical rumors in a social network: A generalized diffusion model with a highly asymmetric network structure

Medical rumors have become a threat to modern society. To study the spread and control of rumors, nonlinear differential equations modeling with the well-mixed assumption is commonly used. However, this approach ignores the underlying network structure which plays an important role in information spreading. We establish a generalized differential equations model to study the spread and control of medical rumors in a highly asymmetric social network. In our model, each node represents a group of people and a "weighted" and "directed" network describes the communications between these nodes. This network can be generated from real-world data by community detection algorithms. We provide methods to numerically calculate the final size of a rumor in each node and its derivatives with respect to each parameter. With these methods, if the government has resources to influence the parameters subject to certain constraints or cost functions, one can obtain the optimal resources allocation easily through nonlinear programming algorithms. We show that the implications on the government's resources allocation from the well-mixed special case in the literature or conventional wisdom may become inapplicable in the general situation. Therefore, the underlying network should not be ignored. Because the final size of a medical rumor is not always the best measure of its damage, we extend our results to a wide class of objectives and show that different objectives result in very different implications. While the lack of a rule of thumb may sound negative, our flexible framework provides a powerful workhorse for interested parties to work out the details in their specific situations. Finally, we provide a sufficient condition for no outbreak of rumors. This condition can serve as a heuristic that a government with abundant resources can use to prevent the outbreak of rumors.