Preventive and modelling approaches to address the COVID-19 pandemic have been primarily based on the age or occupation, and often disregard the importance of heterogeneity in population contact structure and individual connectivity. To address this gap, we developed models based on Erdős-Rényi and a power law degree distribution that first incorporate the role of heterogeneity and connectivity and then can be expanded to make assumptions about demographic characteristics. Results demonstrate that variations in the number of connections of individuals within a population modify the impact of public health interventions such as lockdown or vaccination approaches. We conclude that the most effective strategy will vary depending on the underlying contact structure of individuals within a population and on timing of the interventions.