A recent parameter identification technique, the local lagged adapted generalized method of moments, is used to identify the time-dependent disease transmission rate and time-dependent noise for the stochastic susceptible, exposed, infectious, temporarily immune, susceptible disease model (S E I RS) with vital rates. The stochasticity appears in the model due to fluctuations in the time-dependent transmission rate of the disease. All other parameter values are assumed to be fixed, known constants. The method is demonstrated with US influenza data from the 2004-2005 through 2016-2017 influenza seasons. The transmission rate and noise intensity stochastically work together to generate the yearly peaks in infections. The local lagged adapted generalized method of moments is tested for forecasting ability. Forecasts are made for the 2016-2017 influenza season and for infection data in year 2017. The forecast method qualitatively matches a single influenza season. Confidence intervals are given for possible future infectious levels. Keywords Compartment disease model · Stochastic disease model · Local lagged adapted generalized method of moments · Time-dependent transmission rate Mathematics Subject Classification 60H10 · 60P10 · 92D30

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We estimate and analyze the time-dependent parameters: transmission rate, symptomatic recovery rate, immunity rate, infection noise intensities, and the effective reproduction number for the United States COVID-19 cases for the period 01/22/2020-02/25/2021 using an innovative generalized method of moments estimation scheme. We assume the disease-dynamics is described by a stochastic susceptible-exposed-infected-recovered-susceptible (SEIRS) epidemic model, where the infected class is divided into the asymptomatic infected, and symptomatic infectious classes. Stochasticity appears in the model due to fluctuations in the disease’s transmission and recovery rates. The disease eradication threshold is derived from the reproduction number. The estimated parameters are used to model the disease outbreak’s possible trajectories. Our analysis reveals that current interventions are having positive effects on the transmission and recovery rates. The analysis is demonstrated using the daily United States COVID-19 infection and recovered cases for the period: 01/22/2020-02/25/2021.

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