In this paper a Luenberger-type state observer for a class of nonlinear systems with multiple delays is presented. Sufficient conditions are provided to guarantee practical stability of the error dynamics. An exponential decay of the observation error dynamics is assured using Lyapunov-Krasovskii functionals and the feasibility of Linear Matrix Inequalities. Also, a time-delay SIR compartmental epidemiological model is presented. Time delay corresponds to the transition rates between compartments. The model considers that a part of the recovered population becomes susceptible again after a period following recovery. Three time-delays are incorporated due to the exchange of individuals between the population compartments: $\tau_{1,2,3}$, for the dead-times of recovery, immunity loss and incubation, respectively. It is shown that the effective reproduction number of the delay model depends on the rate of the susceptible population which became infected but after a period starts to be infectious and the fraction of the infectious recovered after a time-delay. For the resulting delay model, the problem of state estimation is addressed. To illustrate the efficiency and performance of the proposed observer, we apply the developed results to the COVID-19 pandemic. The observer can estimate the compartmental populations of Susceptible $S(t)$ and Recovered $R(t)$ from only the availability of real data of the Infectious compartmental population $I_r(t)$. The $I_r(t)$ confirmed data used for the state estimation are from a 55-day window with the highest impact on the Mexican population, as reported by the World Health Organization (WHO).