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Browne was part of the Ireland team beaten 3-0 at Wembley after goals from Harry Maguire, Jadon Sancho and Dominic Calvert-Lewin.
Testing since that match has revealed the Preston North End man has contracted COVID-19, however.
Browne, who contested a game-high 13 duels, has had "no close contacts", FA Ireland reported on Friday.
"The rest of the staff and squad tested negative ahead of Sunday's UEFA Nations League [game] against Wales in Cardiff," a short statement read.
While Ireland visits Wales then hosts Bulgaria in the UEFA Nations League, England goes to Belgium before playing Iceland at Wembley.
The Iceland fixture had been in some doubt because of the visitor's prior match in Denmark, where a latest coronavirus outbreak has led to tightening of travel restrictions.
But Iceland was granted exemption on Friday (AEDT), allowing the game to go ahead.