Memory effects and self-excited oscillations in deterministic epidemic models with intrinsic time delays

The simplest delay differential equation describing the dynamics of non-lethal infectious diseases in a fixed-size population is extended to include the incubation period, as an additional delay parameter. It is observed that these types of deterministic models consist of one delay differential equation, whereas standard SIR and SEIR models consist of two and three ordinary differential equations, respectively. The extended model presents interesting peculiarities as, for example, initial oscillatory patterns in the curve counting the infectious individuals. A comparison of the doubly delayed differential equation with the standard SEIR model is made. It is argued that self-sustained oscillations, which are intrinsic properties of models with time delay, have to be taken into account in designing optimal epidemic containment strategies.