The insurer has received around 500 travel insurance claims related to the coronavirus so far and paid out around 500,000 pounds ($644,750), its executives said.
Tulloch said the coronavirus epidemic "presents a new uncertainty in 2020".
The Aviva insider took over a year ago and has shied away from radical change to the business, which offers life and general insurance and has operations in Europe, Canada and Asia as well as Britain.
Operating profit rose 6% to 3.2 billion pounds ($4.12 billion), slightly above forecasts, on strong performance across the board, though profits in the UK life business dipped.
"There is still much to do," Tulloch told a media call.
"We will improve business performance further...with discipline on expenses and underwriting."
KBW analysts said the results were "positive", reiterating their market perform rating on the stock. Aviva's shares rose 4.3% to 365 pence per share at 0845 GMT, at the top of the FTSE 100.
Chief financial officer Jason Windsor told the call the insurer's investments did not have a large exposure to equity markets, which suffered a correction last week.
The company said it would pay a dividend of 30.9 pence per share, up 3% and in line with forecasts.
By Carolyn Cohn