During the current global Covid-19 pandemic affected countries invested significant amounts of resources into containment strategies for this infectious disease. Most of the controlling measures can be classified into two groups: (i) testing and identification of infected individuals, (ii) social distancing measures to reduce the transmission probabilities. Both types of measures impose some type of cost (e.g. economic losses due to temporary closures) and/or are restricted through capacity constraints (e.g. limited number of test-kits per day). Hence important question arise for the optimal timing, magnitude and combination of different measures. We propose an extension of the established SIR epidemiological model with the status of quarantine and different courses of the disease. Then we consider the model to be distributed across a network (e.g. different regions, or different social groups) and model the diffusion of the disease between the different groups. The groups can be heterogeneous with respect to size and population density (implying different patterns of population interaction) as well as other parameters such as mortality. Furthermore the level of interaction between groups can differ, which may lead to a requirement that group-specific policies may need to be taken towards controlling the disease.