Three-state model.

<p>Living individuals of the population aged a at calendar time t are either in state A or state B. The respective numbers are S(t, a) and C(t, a). Individuals may change states according to the transition rates.</p

Smoking initiation and cessation rates.

<p>Modelled smoking initiation (red) and cessation (green) rates of the male German population in the year 1920. Note that this is only one possibility to model the prevalence of smoking shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118955#pone.0118955.g005" target="_blank">Fig. 5</a>.</p

Difference stratified by age.

<p>Mean (top) and standard deviation (bottom) of the difference <i>D</i> between the left and the right-hand side of Equation (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118955#pone.0118955.e004" target="_blank">3</a>) stratified by age <i>a</i>.</p

Learning about the OC.

<p>Rate of learning about the period of highest fertility during the ovulatory cycle (in units 1000 per person-year) in Egyptian women stratified by year and age.</p><p>Learning about the OC.</p

Prevalence of dementia.

<p>Age-specific prevalence of dementia in the simulated data (crosses, with 95% confidence bounds) and published values (diamonds, [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118955#pone.0118955.ref012" target="_blank">12</a>]).</p

Change Rates and Prevalence of a Dichotomous Variable: Simulations and Applications

<div><p>A common modelling approach in public health and epidemiology divides the population under study into compartments containing persons that share the same status. Here we consider a three-state model with the compartments: <i>A</i>, <i>B</i> and <i>Dead</i>. States <i>A</i> and <i>B</i> may be the states of any dichotomous variable, for example, <i>Healthy</i> and <i>Ill</i>, respectively. The transitions between the states are described by change rates, which depend on calendar time and on age. So far, a rigorous mathematical calculation of the prevalence of property <i>B</i> has been difficult, which has limited the use of the model in epidemiology and public health. We develop a partial differential equation (PDE) that simplifies the use of the three-state model. To demonstrate the validity of the PDE, it is applied to two simulation studies, one about a hypothetical chronic disease and one about dementia in Germany. In two further applications, the PDE may provide insights into smoking behaviour of males in Germany and the knowledge about the ovulatory cycle in Egyptian women.</p></div

Summary statistics of the difference <i>D</i>.

<p>Summary of the difference <i>D</i> between the left and the right-hand side of Equation (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118955#pone.0118955.e004" target="_blank">3</a>) in Simulation 1.</p><p>Summary statistics of the difference <i>D</i>.</p

Prevalence of smoking in German men.

<p>Age-specific prevalence of occasional or regular smoking in the male German population in 1998. The dark and light grey columns indicate the surveyed and the modelled prevalence, respectively.</p

Incidence of dementia.

<p>Comparison of the age-specific incidence of dementia: input used to simulate the data (line) and the reconstructed values (crosses).</p

Forgetting about the OC.

<p>Rates of forgetting about the period of highest fertility during the ovulatory cycle (in units 1000 per person-year) in Egyptian women stratified by year and age.</p><p>Forgetting about the OC.</p