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We describe the behaviour of solutions of a scalar Delay Differential Equation (DDE) with delay that periodically switches between two constant values. Such an equation arises naturally from structured vector populations involved in a range of vector-borne diseases spreading in a periodically varying environment. We examine if and how the two different time lags and the switching time influence the existence and patterns of periodic solutions. We pay particular attention to the patterns involving multi-cycles within the prime period of the periodic solutions.

Multi-cycle Periodic Solutions of a Differential Equation with Delay that Switches Periodically