Co-infections by multiple pathogens have important implications in many aspects of health, epidemiology and evolution. However, how to disentangle the non-linear dynamics of the immune response when two infections take place at the same time is largely unexplored. Using data sets of the immune response during influenza-pneumococcal co-infection in mice, we employ here topological data analysis to simplify and visualise high dimensional data sets.

We identified persistent shapes of the simplicial complexes of the data in the three infection scenarios: single viral infection, single bacterial infection, and co-infection. The immune response was found to be distinct for each of the infection scenarios and we uncovered that the immune response during the co-infection has three phases and two transition points. During the first phase, its dynamics is inherited from its response to the primary (viral) infection. The immune response has an early shift (few hours post co-infection) and then modulates its response to react against the secondary (bacterial) infection. Between 18 and 26 h post co-infection the nature of the immune response changes again and does no longer resembles either of the single infection scenarios.

多种病原体的共同感染对健康、流行病学和进化的许多方面具有重要影响。然而，当两种感染同时发生时，如何解决免疫反应的非线性动力学在很大程度上尚未被探索。使用小鼠流感-肺炎球菌混合感染期间的免疫反应数据集，我们在这里采用拓扑数据分析来简化和可视化高维数据集。

我们在三种感染场景中确定了数据单纯复合体的持久形状：单一病毒感染、单一细菌感染和共同感染。我们发现每种感染情况下的免疫反应都是不同的，并且我们发现共同感染期间的免疫反应具有三个阶段和两个过渡点。在第一阶段，其动态继承自其对原发（病毒）感染的反应。免疫反应发生早期转变（共感染后几小时），然后调节其反应以对抗继发（细菌）感染。共感染后 18 至 26 小时之间，免疫反应的性质再次发生变化，不再类似于任何一种单一感染情况。