Epidemic Models with Age of Infection, Indirect Transmission and Incomplete Treatment ∗ Li-Ming Cai † Department of Mathematics, Xinyang Normal University, Xinyang 464000, PR China Maia Martcheva Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611–8105 Xue-Zhi Li Department of Mathematics, Xinyang Normal University, Xinyang 464000, PR China Abstract An infection-age-structured epidemic model with environmental bacterial infection is in- vestigated in this paper. It is assumed that the infective population is structured according to age of infection, and the infectivity of the treated individuals is reduced but varies with the infection-age. An explicit formula for the reproductive number ℜ 0 of the model is obtained. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for ℜ 0 < 1. It is also shown that if the reproduction number ℜ 0 > 1, then the system has a unique endemic equilibrium which is locally asymptotically stable. Fur- thermore, if the reproduction number ℜ 0 > 1, the system is permanent. When the treatment * This work was carried out with the support of the Natural Science Foundation of China grants 10971178 and 10911120387, US National Science Foundation grants DMS-0817789 and DMS-1220342, and University Key Teacher Foundation of Henan Province (2009GGJS-076). † Corresponding author: limingcai@amss.ac.cn, Tel: 86-376-6390708. 1