Diseases with chronic stage in a population with varying size Maia Martcheva a, * , Carlos Castillo-Chavez b a Department of Mathematics, Polytechnic University, Brooklyn, NY 11201, USA b Department of Biological Statistics and Computational Biology, Cornell University, Ithaca, NY 14853, USA Received 6 July 2002; received in revised form 13 September 2002; accepted 24 September 2002 Abstract An epidemiological model of hepatitis C with a chronic infectious stage and variable population size is introduced. A non-structured baseline ODE model which supports exponential solutions is discussed. The normalized version where the unknown functions are the proportions of the susceptible, infected, and chronic individuals in the total population is analyzed. It is shown that sustained oscillations are not possible and the endemic proportions either approach the disease-free or an endemic equilibrium. The expanded model incorporates the chronic age of the individuals. Partial analysis of this age-structured model is carried out. The global asymptotic stability of the infection-free state is established as well as local asymptotic stability of the endemic non-uniform steady state distribution under some additional condi- tions. A numerical method for the chronic-age-structured model is introduced. It is shown that this nu- merical scheme is consistent and convergent of first order. Simulations based on the numerical method suggest that in the structured case the endemic equilibrium may be unstable and sustained oscillations are possible. Closer look at the reproduction number reveals that treatment strategies directed towards speeding up the transition from acute to chronic stage in effect contribute to the eradication of the disease. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Chronic stage; Variable infectivity; Disease-age structure; Hepatitis C; Variable population size; Difference scheme; Numerical methods; Sustained oscillations 1. Introduction The impact of a chronic stage on the disease transmission and behavior in an exponentially growing or decaying population is the focus of this paper. The framework is applied to the case of Mathematical Biosciences 182 (2003) 1–25 www.elsevier.com/locate/mbs * Corresponding author. Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801, USA. Tel.: +1-718 260 3294; fax: +1-718 260 3660. E-mail addresses: mayam@duke.poly.edu (M. Martcheva), cc32@cornell.edu (C. Castillo-Chavez). 0025-5564/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0025-5564(02)00184-0