The novel coronavirus (COVID-19) pandemic is causing devastating demographic, social, and economic damage globally. Understanding current patterns of the pandemic spread and forecasting its long-term trajectory is essential in guiding policies aimed at curtailing the pandemic. This is particularly important in regions with weak economies and fragile health care systems such as West Africa. We formulate and use a deterministic compartmental model to (i) assess the current patterns of COVID-19 spread in West Africa, (ii) evaluate the impact of currently implemented control measures, and (iii) predict the future course of the pandemic with and without currently implemented and additional control measures in West Africa. An analytical expression for the threshold level of control measures (involving a reduction in the effective contact rate) required to curtail the pandemic is computed. Considering currently applied health control measures, numerical simulations of the model using baseline parameter values estimated from West African COVID-19 data project a 67% reduction in the daily number of cases when the epidemic attains its peak. More reduction in the number of cases will be achieved if additional public health control measures that result in a reduction in the effective contact rate are implemented. We found out that disease elimination is difficult when more asymptomatic individuals contribute in transmission or are not identified and isolated in a timely manner. However, maintaining a baseline level of asymptomatic isolation and a low transmission rate will lead to a significant reduction in the number of daily cases when the pandemic peaks. For example, at the baseline level of asymptomatic isolation, at least a 46% reduction in the transmission rate is required for disease elimination. Additionally, disease elimination is possible if asymptomatic individuals are identified and isolated within 5 days (after the incubation period). Combining two or more measures is better for disease control, e.g., if asymptomatic cases are contact traced or identified and isolated in less than 8 days, only about 29% reduction in the disease transmission rate is required for disease elimination. Furthermore, we showed that the currently implemented measures triggered a 33% reduction in the time-dependent effective reproduction number between February 28 and June 26, 2020. We conclude that curtailing the COVID-19 pandemic burden significantly in West Africa requires more control measures than those that have already been implemented, as well as more mass testing and contact tracing in order to identify and isolate asymptomatic individuals early.