Waning and boosting: on the dynamics of immune status.

The aim is to describe the distribution of immune status (as captured by antibody level) on the basis of a within-host submodel for continuous waning and occasional boosting. Inspired by Feller's fundamental work and the more recent delay equation formulation of models for the dynamics of physiologically structured populations, we derive, for given force of infection, a linear renewal equation. The solution is obtained by generation expansion, with the generation number corresponding to the number of times the individual became infected. Our main result provides a precise characterization of the stable distribution of immune status.

Linking the seroresponse to infection to within-host heterogeneity in antibody production.

A recently published model for the serum antibody response to infection appeared well suited for use in statistical analyses of longitudinal serological data. The published model assumed exponential decay with fixed rates for pathogen and serum antibody kinetics, ignoring any within-host heterogeneity in the seroresponse. A bi-exponential model shows that there is rapid initial decay followed by a prolonged period of persistent low serum antibody concentrations. We propose a small modification of the decay model that greatly increases its flexibility by allowing for non-exponential antibody decay. The modified model produces power functions that may be interpreted as a mixture of exponential decay curves, with a mixing distribution representing the relative contribution of many centres of antibody production to the serum antibody concentration. Fitting the power function decay model to observed longitudinal data for pertussis shows improved goodness of fit compared to the exponential decay model, with estimates for the shape parameter (r=2.2; 95% CI (1.7-2.8)) that differ from exponential shape (r=1). The power function decay model predicts more persistent antibody concentrations in the long term (symptomatic threshold reached >30 years after infection) which, when used in biomarker studies, will lead to lower estimates of seroconversion rates compared to exponential antibody decay.

A two-phase within-host model for immune response and its application to serological profiles of pertussis.

We present a simple phenomenological within-host model describing both the interaction between a pathogen and the immune system and the waning of immunity after clearing of the pathogen. We implement the model into a Bayesian hierarchical framework to estimate its parameters for pertussis using Markov chain Monte Carlo methods. We show that the model captures some essential features of the kinetics of titers of IgG against pertussis toxin. We identify a threshold antibody level that separates a large increase in antibody level upon infection from a small increase and accordingly might be interpreted as a threshold separating clinical from subclinical infections. We contrast predictions of the model with observations reported in the literature and based on independent data and find a remarkable correspondence.

On the formulation of epidemic models (an appraisal of Kermack and McKendrick).

The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.