We study a model of disease transmission with continuous agestructure for latently infected individuals and for infectious individuals. The model is very appropriate for tuberculosis. Key theorems, including asymptotic smoothness and uniform persistence, are proven by reformulating the system as a system of Volterra integral equations. The basic reproduction number R 0 is calculated. For R 0 < 1, the disease-free equilibrium is globally asymptotically stable. For R 0 > 1, a Lyapunov functional

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... s used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present. Finally, some special cases are considered. 2000 Mathematics Subject Classification. Primary: 34K20, 92D30; Secondary: 34D20. Age-structured systems are well-suited to modelling tuberculosis (as well as other applications, such as antibiotic resistance [2]). Infectious tuberculosis is a deadly disease if not treated. However, an individual may have latent tuberculosis for months, years or even decades before the disease becomes infectious. The risk per unit time of activation appears to be higher in the early stages of latency than in later stages; see [1] , where low dimensional ODE models have been used to study this phenomenon, with the global analysis provided in [17] . In [18] , an staged progression ODE model with an arbitrary number of infectious stages in considered. As stated above, though, the ODE nature of the model puts limitations on the distribution of waiting times in the exposed population. By including the duration a spent in the exposed class, we are able to model the risk of activation as a function of a, allowing more generality in the distribution of waiting times or latency periods. Similarly, the distribution of waiting times in the infectious class is made general by allowing the exit rate to be a function of the time spent in that class. ODE models including [5, 9, 10, 14, 16] have included a version of infectionage dependent infectivity by using progression through multiple infectious stages. However, since the distribution of waiting times in each stage is exponential, there would be individuals in the first class for arbitrarily large times and others who have progressed to the final stage in arbitrarily small times. Thus, the ODE staged progression models give only a weak approximation of infection-age dependent infectivity. Continuous age-structure in the infectious class allows the infectivity to truly be a function of the duration spent in the class. Furthermore, it allows the elevated death rate due to disease to depend on the duration for which one has been infectious. Until recently [15, 19] , full global stability results for continuous age-structure models were lacking. A key goal in this paper is to treat a continuous age-structure model from start to finish, including the global stability. The global stability approach used here is related to that used in [15, 19, 20, 21, 22] . Other aspects of the analysis follow the techniques laid out in the new book [25] . In that book (and in [15]), an SI model of disease transmission, with continuous age-structure for the infectives is studied; that is, a scalar age-structured variable is used. In [15] , the SI model is reformulated as a non-densely defined Cauchy problem in order to study the asymptotic smoothness and persistence. The current approach is closer to that found in [25] . The SEI model considered here includes continuous age-structure for both the exposed and the infectious classes; that is, a two-dimensional age-structured variable is used. Thus, the application of the methods in [25] requires some care. On the other hand, we hope that the calculations here help to demonstrate the usefulness of the techniques given in [25] . In [24] , an SEI model with continuous age-structure for the infectious class was studied. The model was reformulated as an infinite delay differential equation with most of the analysis, including asymptotic smoothness and persistence, performed in [24] . The global analysis appeared in [19] . That system is a special case of the one studied here. 2. Model equations. Based on disease status, a population is divided into three classes: susceptible, exposed or infectious. The number of susceptibles at time t is given by S(t). In order to model the time-course development of the disease within