Epidemic spread on networks can be described by continuous time Markov chains. The size of its state space blows up exponentially as the number of vertices is increased, hence several averaging methods leading to low-dimensional non-linear differential equations were derived. The approximation of the system by differential equations, containing some characteristics of the underlying graph, is one of the most important tools of investigation. In this talk we give an introduction to this field with emphasis on the mathematical methods introduced so far. Besides considering spreading processes onstatic networks we will deal with adaptive networks, when the epidemic dynamics on the network is coupled with a network which evolves in time. Moreover, we show how this approach leads to the control of the network process, a mathematical problem attracting significant research interest nowdays.
Kiss., I.Z, Miller, J.C., Simon, P.L., Mathematics of Epidemics on Networks; From Exact to Approximate Models, Springer (2017)