Probabilistic Modeling of Antibody Kinetics Post Infection and Vaccination: A Markov Chain Approach

Understanding dynamics of antibody levels is crucial for characterizing time-dependent response to immune events: either infections or vaccinations. The sequence and timing of these events significantly influence antibody level changes. Despite extensive interest in the recent years and many experimental studies, the effect of immune event sequences on antibody levels is not well understood. Moreover, disease or vaccination prevalence in the population are time-dependent. This, alongside the complexities of personal antibody kinetics, makes it arduous to analyze a sample immune measurement from a population. A rigorous mathematical characterization can inform public health decision making.

A key result of this paper is an antibody response modeling framework for an arbitrary number of multiclass immune events-the first of its kind to the best of our knowledge. Our model is ideal for characterizing immune event sequences, referred to as personal trajectories. To illustrate our ideas, we apply our mathematical framework to longitudinal severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) data from individuals with multiple documented infection and vaccination events. This approach is fully generalizable to other diseases that exhibit waning immunity, such as influenza, respiratory syncytial virus (RSV), and pertussis. Our work is an important step towards a comprehensive understanding of antibody kinetics for infectious diseases that could lead to an effective way to analyze the protective power of natural immunity or vaccination, predict missed immune events at an individual level, and inform booster timing recommendations.

We design a rigorous mathematical characterization in terms of a time-inhomogeneous Markov chain model for event-to-event transitions coupled with a probabilistic framework for the post-event antibody kinetics of multiple immune events. Probabilistic models appropriately describe these measurements as they capture the natural variability in a population's antibody response. We build probability density models for population response since the emergence of a disease via a discrete convolution of immune state transmission probabilities and personal response models, repeatedly invoking the definition of conditional probability and the law of total probability. Importantly, our coupled framework simultaneously tracks immune state and antibody response. This novel modeling approach surpasses the susceptible-infected-recovered (SIR) characterizations by rigorously tracing the probability distribution of population antibody response across time.