Epidemic spread on networks is one of the most studied dynamics in network science and has important implications in real epidemic scenarios. Nonetheless, the dynamics of real epidemics and how it is affected by the underline structure of the infection channels are still not fully understood. Here we apply the susceptible-infected-recovered model and study analytically and numerically the epidemic spread on a recently developed spatial modular model imitating the structure of cities in a country. The model assumes that inside a city the infection channels connect many different locations, while the infection channels between cities are less and usually directly connect only a few nearest neighbor cities in a two-dimensional plane. We find that the model experience two epidemic transitions. The first lower threshold represents a local epidemic spread within a city but not to the entire country and the second higher threshold represents a global epidemic in the entire country. Based on our analytical solution we proposed several control strategies and how to optimize them. We also show that while control strategies can successfully control the disease, early actions are essentials to prevent the disease global spread.

网络上的流行病传播是网络科学中研究最多的动态之一，并且在实际流行病场景中具有重要意义。尽管如此，实际流行病的动态以及它如何受到感染渠道的下划线结构的影响仍未完全了解。在这里，我们应用易感感染恢复模型，并在最近开发的模拟一个国家城市结构的空间模块化模型上对流行病传播进行分析和数值研究。该模型假设在一个城市内部，感染通道连接了许多不同的位置，而城市之间的感染通道较少，通常只直接连接二维平面上的几个最近邻城市。我们发现该模型经历了两次流行病转变。第一个较低的阈值代表一个城市内的局部流行病，但未传播到整个国家，第二个较高的阈值代表整个国家的全球流行病。基于我们的分析解决方案，我们提出了几种控制策略以及如何优化它们。我们还表明，虽然控制策略可以成功控制疾病，但早期行动对于防止疾病全球传播至关重要。