Illness-death model: statistical perspective and differential equations

Abstract

The aim of this work is to relate the theory of stochastic processes with the differential equations associated with multistate (compartment) models. We show that the Kolmogorov Forward Differential Equations can be used to derive a relation between the prevalence and the transition rates in the illness-death model. Then, we prove mathematical well-definedness and epidemiological meaningfulness of the prevalence of the disease. As an application, we derive the incidence of diabetes from a series of cross-sections.

Appendices

Appendix 1

From Eq. (5) it follows:

$$\begin{aligned} 1 - \pi (\tau ) = \frac{P_{00}(0,\tau )}{P_{00}(0,\tau )+P_{01}(0,\tau )}. \end{aligned}$$

(11)

For brevity, we write $\pi = \pi (\tau ), P_{0j} = P_{0j}(0, \tau ),$ and $q_{jk} = q_{jk}(\tau ), ~j,k \in S$. Taking the derivative of $\pi (\tau )$ with respect to $\tau$ yields

In (*) the properties $q_{11} = -q_{10} - q_{12}$ and $q_{00} = -q_{01} - q_{02}$ have been used.

Appendix 2

We show that $P_{00}(0, \tau ) + P_{01}(0, \tau ) > 0$ for all $\tau \ge 0.$ It holds

Again, in (*) the properties $q_{11} = -q_{10} - q_{12}$ and $q_{00} = -q_{01} - q_{02}$ have been used.

For the moment, we assume that $M := \sup _{\tau \ge 0} \{ q_{02}(\tau ), q_{12}(\tau ) \}$ is finite ($M < \infty$). With $y(\tau ) := P_{00}(0, \tau ) + P_{01}(0, \tau )$ we have following differential inequality $y^\prime \ge -M \, y$ with initial condition $y(0) = 1.$ By defining $h := y^\prime + M \,y \ge 0$ we find that

$$\begin{aligned} y(\tau ) = \exp \left( -M \, \tau \right) \, \left\{ 1+\int _0^\tau h(\alpha ) \, \exp \left( M \, \alpha \right) \mathrm {d} \alpha \right\} . \end{aligned}$$

From $h(\alpha ) \ge 0$ it follows that $P_{00}(0, \tau ) + P_{01}(0, \tau ) \ge \exp \left( - M \, \tau \right) > 0 \text { for all } \tau \ge 0,$ which we wanted to prove.

The question arises under which condition we have $M < \infty$. If $q_{02}$ and $q_{12}$ are continuous (which is reasonable for real diseases) and we apply the illness-death model up to a maximum age $\omega$, when all people in the considered population have died, then we may conclude that $M < \infty$.