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Foakes was unable to keep wicket on Saturday due to back stiffness, with Jonny Bairstow taking the gloves. Foakes subsequently tested positive after taking a lateral flow test in the evening.
Billings has been in action for Kent in the Vitality Blast but will go straight into the playing XI and keep wicket, with England seeking five more New Zealand wickets before beginning their second innings.
England said in a statement that there had been no further cases and "the rest of the England party follows health protocols of symptom reporting and subsequent testing if required". Marcus Trescothick, the team's batting coach, tested positive on Tuesday and has been self-isolating throughout the third Test.
The news casts doubt on Foakes' participation in the India Test starting on Friday, although England said that they hoped he would be fit to play. India have also been affected by Covid, with Rohit Sharma, the Test captain, currently isolating after a positive test. The rearranged Test will provide a conclusion to last year's series, with India leading 2-1.
Billings was in London on Saturday evening ahead of Kent's County Championship match against Surrey at the Kia Oval and made a last-minute, late-night journey up to Leeds, arriving in the early hours of Sunday morning.
He will win his second Test cap in similarly hectic circumstances to the first, when he drove nine hours along Australia's east coast to Sydney and played in the final Ashes Test after injuries to Bairstow and Jos Buttler.