Differential equation models of disease transmission ∗

Most readers will already have encountered at least one ODE model of disease transmission. Here we briefly review the construction of such models and some of their predictions. For an in-depth review of these models the reader may want to consult the survey article [1]. All ODE models of disease transmission are continuous-time compartment-level models. Here we focus on models of type SI, SIS, SIR, and SEIR. The ODE models of these types have variables S, I, and, if applicable, R,E. They can be interpreted as actual numbers, expected numbers, or (actual or expected) proportions of hosts in the S-, I-, R, and E-compartments. These variables will change over time, and S(t), I(t), R(t), E(t) denote their values at a given time t. The state of the system at time t is the vector (S(t), I(t)) for SIand SIS-models, the vector (S(t), I(t), R(t)) for SIR-models, and the vector (S(t), I(t), R(t), E(t) for SEIR-models. We will ignore demographics, that is, births, immigration, emigration, and deaths from causes that are unrelated to the disease. Mathematically this means that at all times the equality