We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbours are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.

翻译：我们研究了以下在社交网络中的疾病传播模型。初始状态下，所有个体要么被感染，要么健康。随后，在离散的时间步中，疾病在网络上从感染者向健康个体传播，其规则为：健康个体当且仅当其足够数量的直接邻居已被感染时才会被感染。我们将社交网络表示为图。受当前疫情中现实限制条件（特别是社交距离和物理距离要求）的启发，我们仅考虑可表示为几何交图的网络。我们证明，即便是在单位圆盘图上，找到能使疾病传播至整个网络的初始感染者最小顶点集在计算上已是困难的。因此，为获得一定的算法结果，我们将研究重点放在更简单的几何图类上，例如区间图和网格图。