* NZ set for 4th day of gains

* Alumina Ltd among top gainers

* Aussie mining sub-index falls up to 0.8%

July 16 (Reuters) - Australian shares were flat on Thursday, as worries over a surge in COVID-19 cases in the country offset cheer over a potential vaccine which sent shares up around the globe.

The S&P/ASX 200 index fell 0.07% to 6,048.7 points by 0147 GMT.

Australian state premiers on Wednesday warned of stricter mobility restrictions if a COVID-19 outbreak was not brought under control. There have been 10 days of triple digit gains in the country as of Wednesday.

Meanwhile, trial of an experimental COVID-19 vaccine by biotech start-up Moderna Inc showed encouraging results on Tuesday, sending U.S. stocks sharply higher on Wednesday.

Among top gainers were shares of Alumina Ltd, which were also the most heavily traded by volume after its joint venture with Alcoa Corp, AWAC, achieved record quarterly daily alumina production.

The mining sub-index fell up to 0.8%, with sector heavyweights BHP Group , Rio Tinto and Fortescue Metals Group in the red.

The gold sub-index fell 1.3% led by Dacian Gold Ltd , down 7.6%, followed by Red 5 Ltd, losing 4.4%.

Across the Tasman Sea in New Zealand, consumer prices fell by the most in four years last quarter as most of the economy was closed to curb the spread of COVID-19.

The benchmark index opened 0.5% higher, but pared gains later in the day. However, the index was set for a fourth day of gains.

The top percentage losers on the index were Investore Property Ltd, down 4.1%, followed by Tourism Holdings Ltd, losing 1.7%. (Reporting by Arundhati Dutta in Bengaluru; Editing by Shailesh Kuber)