Abstract. We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of nodes. A phase diagram is presented, showing a second order phase transition from a connected to a disconnected phase. We study both the canonical formulation, where the size is large but fixed, and the grand canonical formulation, where the size is sampled from a discrete distribution, and show their equivalence in the thermodynamical limit. We also compute analytically the spectral density, which consists of a discrete set of isolated eigenvalues, representing short cycles, and a continuous part, representing cycles of diverging size.

Background: In early May 2020, following social distancing measures due to COVID-19, governments consider relaxing lock-down. We combined individual clinical risk predictions with epidemic modelling to examine simulations of riskbased differential isolation and exit policies. Methods:We extended a standard susceptible-exposed-infected-removed (SEIR) model to account for personalised predictions of severity, defined by the risk of an individual needing intensive care if infected, and simulated differential isolation policies using COVID-19 data and estimates in France as of early May 2020. We also performed sensitivity analyses. The framework may be used with other epidemic models, with other risk predictions, and for other epidemic outbreaks.Findings: Simulations indicated that, assuming everything else the same, an exit policy considering clinical risk predictions starting on May 11, as planned by the French government, could enable to immediately relax restrictions for an extra 10% (6 700 000 people) or more of the lowest-risk population, and con- * Corresponding author

Background: In early May 2020, following social distancing measures due to COVID-19, governments consider relaxing lock-down. We combined individual clinical risk predictions with epidemic modelling to examine simulations of riskbased differential isolation and exit policies. Methods: We extended a standard susceptible-exposed-infected-removed (SEIR) model to account for personalised predictions of severity, defined by the risk of an individual needing intensive care if infected, and simulated differential isolation policies using COVID-19 data and estimates in France as of early May 2020. We also performed sensitivity analyses. The framework may be used with other epidemic models, with other risk predictions, and for other epidemic outbreaks. Findings: Simulations indicated that, assuming everything else the same, an