Delay versus age-of-infection - Global stability

We consider an SIR model of disease transmission which has been formulated with delay in order to give a fixed duration of infectiousness. A Lyapunov functional that has worked for similar models does not work here. Instead, the model is shown to be a consequence of an age-of-infection model for which the same class of Lyapunov functionals has worked, resolving the global stability. In this paper, we consider an SIR model of disease transmission that was presented and studied in [5]. The variables S, I and R denote the size of the susceptible, infectious and recovered classes, respectively. The model includes mass action incidence and a fixed duration s in the infectious class. Thus, the rate of new infections at time t is bS(t)I(t). The per capita death rate is l so that the fraction of individuals in the infectious class which survive for time s is e Àls. Infectious individuals which survive for time s cease to be infectious, entering the recovered class. Thus, the rate at which individuals leave the infectious class due to recovery at time t (having been infected at time t À s) is e Àls bS(t À s)I(t À s). In addition, it is assumed that new individuals enter the population, into the susceptible class, at the constant rate l. This gives the model equations [5]