This paper applies Micken's discretization method to obtain a discrete-time SEIR epidemic model.The positivity of the model along with the existence and stability of equilibrium points is discussed for the discrete-time case.Afterwards, the design of a state observer for this discrete-time SEIR epidemic model is tackled.The analysis of the model along with the observer design is faced in an implicit way instead of obtaining first an explicit formulation of the system which is the novelty of the presented approach.Moreover, some sufficient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality that can be cast in the form of a LMI.The feasibility of the matrix inequality is proved, while some simulation examples show the operation and usefulness of the observer.