A mathematical model is proposed for the spread of an epidemic disease of agedependent infectivity through an asexual population with spatial heterogeneity, assuming that some individuals recover from the disease with temporary immunity, another part recover with permanent immunity, and the last part recover with no immunity. The demographic changes such as births and deaths due to natural causes and the chronological age of individuals are not taken into account. The model is based on a system of partial integro-differential equations including a differential equation to describe the evolution of individuals who have recovered with temporary immunity. The existence and uniqueness of the globally defined solution is proved, and its long-time behaviour is studied.