Freely-Moving Delay Induces Periodic Oscillations in a Structured SEIR Model

In this paper, a disease transmission model of SEIR type with stage structure is proposed and studied. Two kinds of time delays are considered: the first one is the mature delay which divides the population into two stages; the second one is the time lag between birth and being able to move freely, which we call the freely-moving delay. Our mathematical analysis establishes that the global dynamics are determined by the basic reproduction number R0. If R0 < 1, then the disease free equilibrium E1 is globally asymptotically stable, and the disease will die out. If R0 > 1, then a unique positive equilibrium E2 exists, and E2 is locally asymptotically stable when the freely-moving delay is less than the critical value. We show that increasing this delay can destabilize E2 and lead to Hopf bifurcations and stable periodic solutions. By using the normal form theory and the center manifold theory, we derive the formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to verify the theoretical analysis and some biological implications are discussed.