Effective communication of modelling results to policy and decision makers has been a longstanding challenge in times of crises. This communication takes many forms - visualisations, reports, presentations - and requires careful consideration to ensure accurate maintenance of the key scientific messages. Science-to-policy communication is further exacerbated when presenting fundamentally uncertain forms of science such as infectious disease modelling and other types of modelled evidence, something which has been understudied. Here we assess the communication and visualisation of infectious disease modelling results to national COVID-19 policy and decision makers in 13 different countries. We present a synthesis of recommendations on what aspects of visuals, graphs, and plots policymakers found to be most helpful in their COVID-19 response work. This work serves as a first evidence base for developing guidelines on the communication and translation of infectious disease modelling into policy.