Existence and uniqueness of endemic states for the age-structured S±I±R epidemic model 1 Youngjoon Cha a , Mimmo Iannelli b , Fabio A. Milner c,* a Department of Mathematics, Sejong University, Seoul, South Korea b Department of Mathematics, University of Trento, 38050 Povo(TN), Italy c Department of Mathematics, Purdue University, West Lafayette, Mathematical Sciences Building #812, IN 47907-1395, USA Received 6 May 1997; received in revised form 5 January 1998 Abstract The existence and uniqueness of positive steady states for the age structured S±I±R epidemic model with intercohort transmission is considered. Threshold results for the existence of endemic states are established for most cases. Uniqueness is shown in each case. Threshold used are explicitly computable in terms of demographic and epidemio- logical parameters of the model. Ó 1998 Elsevier Science Inc. All rights reserved. 1. Introduction In this paper we study the existence and uniqueness of positive steady states of an age-structured S±I±R type epidemic through a population in age density equilibrium. This problem is intimately associated with the study of long-time behavior of solutions of the model, which has some important epidemiological consequences such as determining whether an outbreak of a particular disease may result in an endemic situation or the infection will die out. Thus, we consider the following system on the basic variables sa; t; ia; t; ra; t, which, respectively, denote the age-speci®c density of sus- ceptible, infective, and removed individuals: * Corresponding author. Tel.: +1-765 494 1922; fax: +1-765 494 0548; e-mail: milner@math. purdue.edu. 1 This work was supported in part by CNR Grant #94.00074.CT01. 0025-5564/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 9 8 ) 1 0 0 0 6 - 8 Mathematical Biosciences 150 (1998) 177±190