Maria Vittoria Barbarossa

Transmission Dynamics and Final Epidemic Size of Ebola Virus Disease Outbreaks with Varying Interventions

The 2014 Ebola Virus Disease (EVD) outbreak in West Africa was the largest and longest ever reported since the first identification of this disease. We propose a compartmental model for EVD dynamics, including virus transmission in the community, at hospitals, and at funerals. Using time-dependent parameters, we incorporate the increasing intensity of intervention efforts. Fitting the system to the early phase of the 2014 West Africa Ebola outbreak, we estimate the basic reproduction number as 1.44. We derive a final size relation which allows us to forecast the total number of cases during the outbreak when effective interventions are in place. Our model predictions show that, as long as cases are reported in any country, intervention strategies cannot be dismissed. Since the main driver in the current slowdown of the epidemic is not the depletion of susceptibles, future waves of infection might be possible, if control measures or population behavior are relaxed.

State-dependent neutral delay equations from population dynamics

A novel class of state-dependent delay equations is derived from the balance laws of age-structured population dynamics, assuming that birth rates and death rates, as functions of age, are piece-wise constant and that the length of the juvenile phase depends on the total adult population size. The resulting class of equations includes also neutral delay equations. All these equations are very different from the standard delay equations with state-dependent delay since the balance laws require non-linear correction factors. These equations can be written as systems for two variables consisting of an ordinary differential equation (ODE) and a generalized shift, a form suitable for numerical calculations. It is shown that the neutral equation (and the corresponding ODE—shift system) is a limiting case of a system of two standard delay equations.

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 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 ordinary differential equations (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.

Analysis of N-acylhomoserine lactone dynamics in continuous cultures of Pseudomonas putida IsoF by use of ELISA and UHPLC/qTOF-MS-derived measurements and mathematical models.

In this interdisciplinary approach, the dynamics of production and degradation of the quorum sensing signal 3-oxo-decanoylhomoserine lactone were studied for continuous cultures of Pseudomonas putida IsoF. The signal concentrations were quantified over time by use of monoclonal antibodies and ELISA. The results were verified by use of ultra-high-performance liquid chromatography. By use of a mathematical model we derived quantitative values for non-induced and induced signal production rate per cell. It is worthy of note that we found rather constant values for different rates of dilution in the chemostat, and the values seemed close to those reported for batch cultures. Thus, the quorum-sensing system in P. putida IsoF is remarkably stable under different environmental conditions. In all chemostat experiments, the signal concentration decreased strongly after a peak, because emerging lactonase activity led to a lower concentration under steady-state conditions. This lactonase activity probably is quorum sensing-regulated. The potential ecological implication of such unique regulation is discussed.

A delay model for quorum sensing of Pseudomonas putida

The bacterial strain Pseudomonas putida IsoF, isolated from a tomato rhizosphere, possesses a quorum sensing regulation system, which allows the bacteria to recognise aspects of their environment or to communicate with each other by the so-called autoinducer molecules. In an experimental study, the time series of the autoinducer production did not show the expected behaviour, as it was observed for other bacterial species by indirect measurements. The modelling approach introduced here allows an explanation of the behaviour, supporting the hypothesis of the existence of a further (not yet detected) enzyme, which degrades the autoinducer into an inactive form. Especially the properties of the considered delay differential system allow for the description of the time series. For example the appearance of a first small maximum in the initial phase can be explained by a delay differential equation.

Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells.

In this work we present a mathematical model for tumor growth based on the biology of the cell cycle. For an appropriate description of the effects of phase-specific drugs, it is necessary to look at the cell cycle and its phases. Our model reproduces the dynamics of three different tumor cell populations: quiescent cells, cells during the interphase and mitotic cells. Starting from a partial differential equations (PDEs) setting, a delay differential equations (DDE) model is derived for an easier and more realistic approach. Our equations also include interactions of tumor cells with immune system effectors. We investigate the model both from the analytical and the numerical point of view, give conditions for positivity of solutions and focus on the stability of the cancer-free equilibrium. Different immunotherapeutic strategies and their effects on the tumor growth are considered, as well.

Predator-Prey Interactions, Age Structures and Delay Equations

A general framework for age-structured predator-prey systems is introduced. Individuals are distinguished into two classes, juveniles and adults, and several possible interactions are considered. The initial system of partial differential equations is reduced to a system of (neutral) delay differential equations with one or two delays. Thanks to this approach, physically correct models for predator-prey with delay are provided. Previous models are considered and analysed in view of the above results. A Rosenzweig-MacArthur model with delay is presented as an example.

Stability Switches Induced by Immune System Boosting in an SIRS Model with Discrete and Distributed Delays

We consider an epidemiological model that includes waning and boosting of immunity. Assuming that repeated exposure to the pathogen fully restores immunity, we derive an SIRS-type model with discrete and distributed delays. First we prove usual results, namely that if the basic reproduction number,

MATHEMATICAL MODELS FOR VACCINATION, WANING IMMUNITY AND IMMUNE SYSTEM BOOSTING: A GENERAL FRAMEWORK

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Temporal Evolution of Immunity Distributions in a Population with Waning and Boosting

, is less or equal than