The spread of an infection, a contagion, meme, emotion, message and various other spreadable objects have been discussed in several works. Burning and firefighting have been discussed in particular on static graphs. Graph burning simulates the notion of the spread of "fire" throughout a graph (plus, one unburned node burned at each time-step); graph firefighting simulates the defending of nodes by placing firefighters on the nodes which have not been already burned while the fire is being spread (started by only a single fire source). This article studies a combination of firefighting and burning on a graph class which is a variation (generalization) of temporal graphs. Nodes can be infected from "outside" a network. We present a notion of both upgrading (of unburned nodes, similar to firefighting) and repairing (of infected nodes). The nodes which are burned, firefighted, or repaired are chosen probabilistically. So a variable amount of nodes are allowed to be infected, upgraded and repaired in each time step. In the model presented in this article, both burning and firefighting proceed concurrently, we introduce such a system to enable the community to study the notion of spread of an infection and the notion of upgrade/repair against each other. The graph class that we study (on which, these processes are simulated) is a variation of temporal graph class in which at each time-step, probabilistically, a communication takes place (iff an edge exists in that time step). In addition, a node can be "worn out" and thus can be removed from the network, and a new healthy node can be added to the network as well. This class of graphs enables systems with high complexity to be able to be simulated and studied.

翻译：感染、传染病、模因、情感、消息及其他可传播对象的传播已在多项研究中进行探讨。其中，图燃烧与图灭火在静态图上的讨论尤为深入。图燃烧模拟了“火势”在图中的传播概念（此外，每个时间步有一个未燃烧节点被点燃）；图灭火则模拟了通过在未燃烧节点上部署消防员来防御节点（火势仅从单一火源开始传播）。本文研究了图类上灭火与燃烧的组合模型，该图类是时序图的一种变体（泛化形式）。节点可能从网络“外部”被感染。我们提出了两种机制：升级（针对未燃烧节点，类似灭火）与修复（针对已感染节点）。被燃烧、灭火或修复的节点以概率方式选择，因此每个时间步可允许感染的节点数量、升级及修复的节点数量均为变量。在本文提出的模型中，燃烧与灭火过程并行进行，我们引入该机制以帮助学界研究感染传播与升级/修复机制之间的对抗关系。所研究的图类（用于模拟上述过程）是时序图类的变体——在每个时间步中，通信以概率方式发生（当且仅当该时间步内存在边时）。此外，节点可能因“损耗”而从网络中移除，同时新健康节点也可加入网络。该图类使具有高复杂度的系统得以被模拟与研究。