Continuous-Time Differential-Difference Models in Population Dynamics and Epidemiology

We are interested in models of population dynamics and epidemiology that involve two phases, one active and one inactive, with exchanges between the two phases. The durations of these two phases may be finite or infinite. Depending on the phenomenon under investigation, the active/inactive phases may have different interpretations. For example, in a population model composed of immature and mature individuals, the mature phase may be considered active and the immature phase inactive. In a predator-prey model, for example, the inactive phase may represent a refuge where the prey population is protected from predators. It may also represent a resting phase during which the predator does not hunt. In epidemiology, it may represent a period of immunity due to vaccination. For a population of cells, the inactive phase may correspond to the G1/G0 phase of the cell cycle, and the active phase to the stages S/G2/M leading to cell division. One of the questions addressed by this type of model is the impact of the inactive phase on the asymptotic behavior of the solutions in the system. We will use continuous-time differential-difference models to study these problems.