Epidemic models describe the establishment and spread of infectious diseases. Among them, the SIR model is one of the simplest, involving exchanges between three compartments in the population, that represent respectively the number of susceptible, infective and recovered individuals. The issue of state estimation is considered here for such a model, subject to seasonal variations and uncertainties in the transmission rate. Assuming continuous measurement of the number of new infectives per unit time, a class of interval observers with estimate-dependent gain is constructed and analyzed, providing lower and upper bounds for each state variable at each moment in time. The dynamical systems that describe the evolution of the errors are monotonous. Asymptotic stability is ensured by appropriate choice of the gain components as a function of the state estimate, through the use of a common linear Lyapunov function. Numerical experiments are presented to illustrate the method.