Asymptotic convergence in delay differential equations arising in epidemiology and physiology

Abstract. In this paper we investigate the presence of oscillations and simple dynamics in x'(t) = -νx(t) + f(x(t))g(x(t-τ)). Specifically, we prove that the global attraction towards a nontrivial equilibrium is reduced to the nonexistence of solutions of a certain system of inequalities. Our results cover situations in which the global attraction critically depends on the delay. From an applied point of view, this equation naturally appears in many biological situations, e.g., epidemic models with awareness and the production of platelets. In epidemiology, it is becoming clear that the different types of population awareness and the time delays of individuals' responses to available information about the disease play a critical role in its spread. Using our theoretical results, we describe qualitative properties of the behavioral responses that prevent the presence of sustainable oscillations in the number of infected individuals. Regarding physiological models, we discuss the in uence of some biological parameters in certain anomalies in the production of platelets.