## 1 Introduction

With a view to basic epidemiological parameters such as incidence, prevalence and mortality of a disease, it has been proven useful to consider so-called state models or compartmental models (Fix and Neyman, 1951; Chiang, 1968). The model used here is also termed illness-death model (Kalbfleisch and Prentice, 2002). It consists of the three states *Normal*, *Disease*, *Death* and the transitions between the states. *Normal* means non-diseased with respect to the disease under consideration. The numbers of persons in the *Normal* and *Disease* state are denoted as *S* (susceptibles) and *C* (cases), respectively. The transition intensities (synonymously: rates) are called as shown in Figure 1: *i* is the incidence rate, *m*_{0} and *m*_{1} are the mortality rates of the non-diseased and diseased persons, respectively. In general, the intensities depend on calendar time *t* and age *a* with the mortality *m*_{1} also depending on the duration *d* of the disease.

When the rates do not depend on calendar time *t*, the model is called *time-homogeneous*. Then, with the additional assumption that there is no dependence of *m*_{1} on the duration, Murray and Lopez (1994) have considered a system of ordinary differential equations to relate the changes of the numbers of healthy and diseased persons with the rates of the inflows and outflows of the corresponding states:

Age *a* plays the role of temporal progression. The linear system in Equation ((1)) looks relatively harmless, but the impression is misleading. Mostly, only the age-specific mortality of the general population is well-known, and rate *m*_{1} is epidemiologically accessible as relative risk. Then, the system becomes nonlinear. Once the functions *S* and *C* are known, the age-specific prevalence can be calculated.

The benefits of these equations are twofold. First, for smooth incidence and mortality rates plus an initial condition, the age profile of the numbers of patients or the prevalence is uniquely determined. To state it clearly, the ‘forces’ incidence and mortality uniquely prescribe the prevalence – not only qualitatively but in these quantitative terms. In this, we speak of the *forward problem*: we deduce the effect, namely the number of diseased persons, from the causes (the forces) Second, the reverse way means deducing from the numbers of diseased persons to the incidence. This is the *inverse problem* as we infer from the effect to the cause.

This paper is structured as follows: in the next section, we describe the illness-death model of Figure 1 in terms of a new system of two stochastic differential equations (SDEs). As an application, in section 3, we solve a forward problem to estimate the age-specific prevalence of systemic lupus erythematosus (SLE) in England and Wales from published data. This allows the calculation of the mean age at onset of SLE, the mean duration and the burden of SLE in terms of the number of diseased persons. These values are compared with the empirically measured values.