NAIROBI, June 17 (Reuters) - COVID-19 cases rose by over 20% week-on-week in nearly two dozen African countries and progress on vaccinating Africans is proceeding slowly, with just 0.79% of people on the continent fully vaccinated, senior health officials said on Thursday. "Africa is in the midst of a full blown third wave...We’ve seen in India and elsewhere how quickly COVID-19 can rebound and overwhelm health systems," Matshidiso Moeti, WHO regional director for Africa, told a news conference. New cases are up nearly 30% in the past week and deaths are up by 15%, she said, with five countries -- South Africa, Tunisia, Zambia, Uganda and Namibia -- accounting for 76% of the new cases. The continent of 1.3 billion people has not yet been battered with an emergency during the pandemic on the scale of that recently facing India, but officials including Moeti continue to warn that such a catastrophe could hit. African nations lack sufficient numbers of hospital beds and oxygen supplies, and would be even more overwhelmed than India if cases surged in a similar way, the Africa Centres for Disease Control and Prevention warned in April. Africa has reported over 5 million COVID-19 cases, representing 2.9% of total cases globally, according to the Africa CDC. The Africa CDC"s chief John Nkengasong told reporters in a separate news conference on Thursday that amid quickly rising cases in many countries, progress on vaccination campaigns is generally slow. Four African countries have not begun vaccination campaigns, but the WHO said on Thursday that Tanzania plans to request to join the COVAX global vaccine-sharing facility. This was the latest sign of the country"s change of tack following the death of its COVID-19 sceptic president in March. read more The aim of COVAX is to help poorer countries vaccinate their people. Both Moeti and Nkengasong stressed that African nations carrying out vaccinations at a relatively good pace were running out of their stocks. Five African countries -- Sao Tome and Principe, Morocco, Kenya, Cote d"Ivoire, Ghana, and Libya -- have used 90% or more of their vaccine stocks, according to the WHO. Nkengasong welcomed the Group of Seven’s pledge of 1 billion doses of the COVID-19 vaccine, but said news should be put in the right context. “The one billion doses are not here now. We are hoping they will be here soon.” (This story adds dropped word “people” in headline)