COVID-19 (Corona Virus Disease 2019) is globally spreading and the international cooperation is urgently required in joint prevention and control of the epidemic. Using the Maximum-Hasting (MH) parameter estimation method and the modified Susceptible Exposed Infectious Recovered (SEIR) model, the spread of the epidemic under three intervention scenarios (suppression, mitigation, mildness) is simulated and predicted in South Africa, Egypt, and Algeria, where the epidemic situations are severe. The studies are also conducted in Nigeria, Senegal and Kenya, where the epidemic situations are growing rapidly and the socio-economic are relatively under-developed, resulting in more difficulties in preventing the epidemic. Results indicated that the epidemic can be basically controlled in late April with strict control of scenario one, manifested by the circumstance in the South Africa and Senegal. Under moderate control of scenario two, the number of infected people will increase by 1.43–1.55 times of that in scenario one, the date of the epidemic being controlled will be delayed by about 10 days, and Algeria, Nigeria, and Kenya are in accordance with this situation. In the third scenario of weak control, the epidemic will be controlled by late May, the total number of infected cases will double that in scenario two, and Egypt is in line with this prediction. In the end, a series of epidemic controlling methods are proposed, including patient quarantine, close contact tracing, population movement control, government intervention, city and county epidemic risk level classification, and medical cooperation and the Chinese assistance.