The specific form of F and G in Eqs. ͑29a͒ and ͑29b͒ is complicated and is therefore presented in Appendix B. We numerically integrate the complete stochastic system of transformed equations of the SEIR model ͓Eqs. ͑20a͒-͑20c͔͒ along with the reduced system of equations that is found using the stochastic normal form coordinate trans form ͓Eqs. ͑29a͒, ͑29b͒, ͑B1a͒, and ͑B1b͔͒. The complete system is solved for U, V, and W, while the reduced system is solved for X 1 = V and X 2 = W. In the latter case, U is computed using the center manifold equation given by Eq. ͑28a͒. As before, once the values of U, V, and W are known, we compute the values of S, Ē , and Ī using the transformation given by Eqs. ͑7a͒-͑7c͒. We shift S, Ē , and Ī, respectively, by S 0 , E 0 , and I 0 to find the values of S, E, and I. 5. ͑Color online͒ Time series of the fraction of the population that is infected with a disease I, computed using the complete stochastic system of transformed equations of the SEIR model ͓Eqs. ͑20a͒-͑20c͔͒ ͑red solid line͒, and computed using the reduced system of equations of the SEIR model that is found using the stochastic normal form coordinate transform ͓Eqs. ͑29a͒, ͑29b͒, ͑B1a͒, and ͑B1b͔͒ ͑blue dashed line͒. The standard deviation of the noise intensity used in the simulation is i = 0.0005, i =4,5,6. The time series is shown for ͑a͒ t =0 to t = 40 and for ͑b͒ t =40 to t = 100. computed using the complete stochastic system of transformed equations of the SEIR model ͓Eqs. ͑20a͒-͑20c͔͒ with the time series of I computed using the reduced system of equations of the SEIR model that is found using the stochastic normal form coordinate transform ͓Eqs. ͑29a͒, ͑29b͒, ͑B1a͒, and ͑B1b͒ ͔. Figure 5͑a͒ shows the initial transients, while Fig. 5͑b͒ shows the time series after the transients have decayed. One can see that there is an excellent agreement between the two solutions. The initial outbreak is successfully captured by the reduced system. Furthermore, Fig. 5͑b͒ shows that the reduced system accurately predicts recurrent outbreaks for a time scale that is orders of magnitude longer than the relaxation time. This is not surprising since the solution decays exponentially throughout the transient and then remains close to the lower-dimensional center manifold. Since we are not looking at periodic orbits, there are no secular terms in the asymptotic expansion, and the result is valid for all time. Additionally, any noise drift on the center manifold results in bounded solutions due to sufficient dissipation transverse to the manifold. The cross correlation of the two time series shown in Fig. 5 is approximately 0.98, which quantitatively suggests there is an excellent agreement between the two solutions.