Acta Appl Math DOI 10.1007/s10440-014-9899-7 On the Stability of a SEIR Reaction Diffusion Model for Infections Under Neumann Boundary Conditions F. Capone · V. De Cataldis · R. De Luca Received: 13 December 2013 / Accepted: 28 February 2014 © Springer Science+Business Media Dordrecht 2014 Abstract This paper deals with a reaction-diffusion SEIR model for infections under homo- geneous Neumann boundary conditions. The longtime behaviour of the solutions is analyzed and, in particular, absorbing sets in the phase space are determined. By using a peculiar Lya- punov function, the nonlinear asymptotic stability of endemic equilibrium is investigated. Keywords Stability · Epidemic models · Reaction-diffusion system · Absorbing sets · Direct Lyapunov method 1 Introduction Population dynamic has attracted the interest of many authors in the past as nowadays (see, for instance, [1–24] and the references therein). Among them, infection disease has been deeply studied. In some models, the host population is supposed to be divided into three disjoint classes (see, for example, [2–9, 19]): S(t), the individuals susceptible to infection, I(t), the infective individuals and R(t), the removed ones. Other models [11–14, 18] take into account of a further class, E(t), the individuals exposed to infection (i.e. infected but not infective). In this paper, we want to reconsider the model introduced in [12], where a SEIR reaction-diffusion system having the incidence rate g(S,I) = KIS(1 + αI), K,α = const.> 0 (1.1) has been analyzed under mixed boundary conditions. In honor of Professor Salvatore Rionero, in the occasion of his 80th birthday. F. Capone · V. De Cataldis (B ) · R. De Luca Department of Mathematics and Applications ‘R. Caccioppoli’, University of Naples Federico II, Complesso Universitario Monte S. Angelo, Via Cinzia, 80126 Naples, Italy e-mail: valentina.decataldis@unina.it F. Capone e-mail: fcapone@unina.it R. De Luca e-mail: roberta.deluca@unina.it