In this talk, I will introduce state structures in mathematical models for infectious diseases. The state is a measure of infectivity of an infected individual in epidemic models or the intensity of viral replications in an infected cell for in-host models. In modelling, a state structure can be either discrete or continuous.In a discrete state structure, a model is described by a large system of coupled ordinary differential equations (ODEs). The complexity of the system often poses a serious challenge for the analysis of system dynamics. I will show how such a complex system can be viewed as a dynamical system defined on a transmission-transfer network (digraph), and how a graph-theoretic approach to Lyapunov functions developed by Guo-Li-Shuai can be applied to rigorously establish the global dynamics.In a continuous state structure, the model gives rise to a system of nonlinear integro-differential equations with a nonlocal term. The mathematical challenges for such a system include a lack of compactness of the associated nonlinear semigroup. The well-posedness and dissipativity of the semigroup is established by directly verifying the asymptotic smoothness. An equivalent principal spectral condition between the next-generation operator and the linearized operator allows us to link the basic reproduction number R0 to a threshold condition for the stability of the disease-free equilibrium. The proof of the global stability of the endemic equilibrium utilizes a Lyapunov function whose construction is informed by the graph-theoretic approach in the discrete case.