COVID-19 country spikes have been reported at varying temporal scales as a result of differences in the disease-driving factors. Factors affecting case load and mortality rates have varied between countries and regions. We investigated the association between socio-economic, weather, demographic and health variables with the reported cases of COVID-19 in Eswatini using the maximum likelihood estimation method for count data. A generalized Poisson regression (GPR) model was fitted with the data comprising 15 covariates to predict COVID-19 risk in the whole of Eswatini. The results show that the variables that were key determinants in the spread of the disease were those that included the proportion of elderly above 55 years at 98% (95% CI: 97–99%) and the proportion of youth below the age of 35 years at 8% (95% CI: 1.7–38%) with a pseudo R-square of 0.72. However, in the early phase of the virus when cases were fewer, results from the Poisson regression showed that household size, household density and poverty index were associated with reported COVID-19 cases in the country. We then produced a disease-risk map of predicted COVID-19 in Eswatini using variables that were selected by the regression model at a 5% significance level. The map could be used by the country to plan and prioritize health interventions against COVID-19. The identified areas of high risk may be further investigated to find out the risk amplifiers and assess what could be done to prevent them.