The phrase "to continue the spread" means to keep spreading or increasing something (such as information, disease, or influence) without stopping. Full definition
Of course, because the person feels better, they get back to having fun and continue the spread of the virus around the ship. (travelinsurancereview.net)
My friend continued spreading the news, and now there was a different energy in the space. (ktmade.com)
«If they did that, the virus would be compromising its own ability to exit the cell and continue spreading infection,» he noted. (sciencedaily.com)