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The Swans, led by interim manager Alan Curtis in the Italian's absence, picked up a shock victory over Arsenal at the Emirates Stadium last week and saw off Norwich City on Saturday to lie nine points clear of the relegation zone.
"With the gaffer not being here we've had to step up a little bit and try and make it smoother on game day and in training," Williams told British media.
"The gaffer's not been well and hopefully the two wins have cheered him up from his hospital bed.
"Before he went into hospital he set us up for the games... Hopefully he will be back in Swansea soon, fit and well and back on the training pitch."
Midfielder Gylfi Sigurdsson, who netted his sixth goal in 10 games against Alex Neil's men on Saturday, highlighted the respect Curtis enjoys in the dressing room.
"Alan has done a great job again. All the boys call him 'the legend' and there is a reason for it," Sigurdsson said.
Swansea, who are 16th in the table, travel to face 14th-placed Bournemouth in the league on Saturday.
