Complex sentence: During most of 1984 and 1985, Hudson kept his illness a secret while continuing to work and, at the same time, travel to France and other countries seeking a cure, or at least treatment to slow the progress of the disease.
Split into simpler sentences: During most of 1984 and 1985, Hudson kept his illness a secret. He continues to work and, at the same time, traveled to France and other countries seeking a cure, or at least treatment to slow the progress of the disease.