Periodic solutions and chaotic attractors of a modified epidemiological SEIS model

We consider a generalized SEIS (susceptible, exposed, infectious, and susceptible) model where individuals are divided into three compartments: S (healthy and susceptible), E (infected but not just infectious, or exposed), and I (infectious). Finite waiting times in the compartments yield a system of delay-differential or memory equations and may exhibit oscillatory (Hopf) instabilities of the otherwise stationary endemic state, leading normally to regular oscillations in the form of an attractive limit cycle in the phase space spanned by the compartment rates. In the present paper, our aim is to demonstrate that in the dynamics of delayed SEIS models, persistent chaotic attractors can bifurcate from these limit cycles and become accessible if the nonlinear interaction terms fulfill certain basic requirements, which to our knowledge were not addressed in the literature so far. Computing the largest Lyapunov exponent, we show that chaotic behavior exists in a wide parameter range. Finally, we discuss a more general system and show that a sudden falloff of the infection rate with respect to increasing infection number may be responsible for the emergence of chaotic time evolution. Such a falloff can describe mitigation measures, such as wearing masques, individual isolation, or vaccination. The model may have a wide range of interdisciplinary applications beyond epidemic spreading, for instance, in the kinetics of certain chemical reactions.