Mathematical models for vaccination, waning immunity and immune system boosting: a general framework

When the body gets infected by a pathogen or receives a vaccine dose, the immune system develops pathogen-specific immunity. Induced immunity decays in time and years after recovery/vaccination the host might become susceptible again. Exposure to the pathogen in the environment boosts the immune system thus prolonging the duration of the protection. Such an interplay of within host and population level dynamics poses significant challenges in rigorous mathematical modeling of immuno-epidemiology. The aim of this paper is twofold. First, we provide an overview of existing models for waning of disease/vaccine-induced immunity and immune system boosting. Then a new modeling approach is proposed for SIRVS dynamics, monitoring the immune status of individuals and including both waning immunity and immune system boosting. We show that some previous models can be considered as special cases or approximations of our framework.Comment: 18 pages, 1 figure keywords: Immuno-epidemiology, Waning immunity, Immune status, Boosting, Physiological structure, Reinfection, Delay equations, Vaccination. arXiv admin note: substantial text overlap with arXiv:1411.319

Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting

When the body gets infected by a pathogen the immune system develops pathogen-specific immunity. Induced immunity decays in time and years after recovery the host might become susceptible again. Exposure to the pathogen in the environment boosts the immune system thus prolonging the time in which a recovered individual is immune. Such an interplay of within host processes and population dynamics poses significant challenges in rigorous mathematical modeling of immuno-epidemiology. We propose a new framework to model SIRS dynamics, monitoring the immune status of individuals and including both waning immunity and immune system boosting. Our model is formulated as a system of two ODEs coupled with a PDE. After showing existence and uniqueness of a classical solution, we investigate the local and the global asymptotic stability of the unique disease-free stationary solution. Under particular assumptions on the general model, we can recover known examples such as large systems of ODEs for SIRWS dynamics, as well as SIRS with constant delay. Moreover, a new class of SIS models with delay can be obtained in this framework.Comment: 33 page

The significance of case detection ratios for predictions on the outcome of an epidemic - a message from mathematical modelers

In attempting to predict the further course of the novel coronavirus disease (COVID-19) pandemic caused by SARS-CoV-2, mathematical models of different types are frequently employed and calibrated to reported case numbers. Among the major challenges in interpreting these data is the uncertainty about the amount of undetected infections, or conversely: the detection ratio. As a result, some models make assumptions about the percentage of detected cases among total infections while others completely neglect undetected cases. Here, we illustrate how model projections about case and fatality numbers vary significantly under varying assumptions on the detection ratio. Uncertainties in model predictions can be significantly reduced by representative testing, both for antibodies and active virus RNA, to uncover past and current infections that have gone undetected thus far

Compliance with NPIs and possible deleterious effects on mitigation of an epidemic outbreak

The first attempt to control and mitigate an epidemic outbreak caused by a previously unknown virus occurs primarily via non-pharmaceutical interventions (NPIs). In case of the SARS-CoV-2 virus, which since the early days of 2020 caused the COVID-19 pandemic, NPIs aimed at reducing transmission-enabling contacts between individuals. The effectiveness of contact reduction measures directly correlates with the number of individuals adhering to such measures. Here, we illustrate by means of a very simple compartmental model how partial noncompliance with NPIs can prevent these from stopping the spread of an epidemic

Mathematical Modeling of Bacteria Communication in Continuous Cultures

Quorum sensing is a bacterial cell-to-cell communication mechanism and is based on gene regulatory networks, which control and regulate the production of signaling molecules in the environment. In the past years, mathematical modeling of quorum sensing has provided an understanding of key components of such networks, including several feedback loops involved. This paper presents a simple system of delay differential equations (DDEs) for quorum sensing of Pseudomonas putida with one positive feedback plus one (delayed) negative feedback mechanism. Results are shown concerning fundamental properties of solutions, such as existence, uniqueness, and non-negativity; the last feature is crucial for mathematical models in biology and is often violated when working with DDEs. The qualitative behavior of solutions is investigated, especially the stationary states and their stability. It is shown that for a certain choice of parameter values, the system presents stability switches with respect to the delay. On the other hand, when the delay is set to zero, a Hopf bifurcation might occur with respect to one of the negative feedback parameters. Model parameters are fitted to experimental data, indicating that the delay system is sufficient to explain and predict the biological observations

Germany’s next shutdown—Possible scenarios and outcomes

In attempting to predict the further course of the novel coronavirus disease (COVID-19) pandemic caused by SARS-CoV-2, mathematical models of different types are frequently employed and calibrated to reported case numbers. Among the major challenges in interpreting these data is the uncertainty about the amount of undetected infections, or conversely: the detection ratio. As a result, some models make assumptions about the percentage of detected cases among total infections while others completely neglect undetected cases. Here, we illustrate how model projections about case and fatality numbers vary significantly under varying assumptions on the detection ratio. Uncertainties in model predictions can be significantly reduced by representative testing, both for antibodies and active virus RNA, to uncover past and current infections that have gone undetected thus far