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, R 0 , is less or equal than 1, then the disease free equilibrium is globally asymptotically stable, whereas for R 0 > 1 the disease persists in the population. The interesting features of boosting appear with respect to the endemic equilibrium, which can go through multiple stability switches by changing the key model parameters. We construct two-parameter stability charts, showing that increasing the delay can stabilize the positive equilibrium. Increasing R 0 , the endemic equilibrium can cross two distinct regions of instability, separated by Hopf-bifurcations. Our results show that the dynamics of infectious diseases with boosting of immunity can be more complex than most epidemiological models, and calls for careful mathematical analysis.
We investigate the temporal evolution of the distribution of immunities in a population, which is determined by various epidemiological, immunological, and demographical phenomena: after a disease outbreak, recovered individuals constitute a large immune population; however, their immunity is waning in the long term and they may become susceptible again. Meanwhile, their immunity can be boosted by repeated exposure to the pathogen, which is linked to the density of infected individuals present in the population. This prolongs the length of their immunity. We consider a mathematical model formulated as a coupled system of ordinary and partial differential equations that connects all these processes and systematically compare a number of boosting assumptions proposed in the literature, showing that different boosting mechanisms lead to very different stationary distributions of the immunity at the endemic steady state. In the situation of periodic disease outbreaks, the waveforms of immunity distributions are studied and visualized. Our results show that there is a possibility to infer the boosting mechanism from the population level immune dynamics.
Using the Hodge decomposition on bounded domains the compressible Euler equations of gas dynamics are reformulated using a density weighted vorticity and dilatation as primary variables, together with the entropy and density. This formulation is an extension to compressible flows of the well-known vorticity-stream function formulation of the incompressible Euler equations. The Hamiltonian and associated Poisson bracket for this new formulation of the compressible Euler equations are derived and extensive use is made of differential forms to highlight the mathematical structure of the equations. In order to deal with domains with boundaries also the Stokes-Dirac structure and the port-Hamiltonian formulation of the Euler equations in density weighted vorticity and dilatation variables are obtained.
We present and analyze a new space-time finite element method for the solution of neural field equations with transmission delays. The numerical treatment of these systems is rare in the literature and currently has several restrictions on the spatial domain and the functions involved, such as connectivity and delay functions. The use of a space-time discretization, with basis functions that are discontinuous in time and continuous in space (dGcG-FEM), is a natural way to deal with space-dependent delays, which is important for many neural field applications. In this article we provide a detailed description of a space-time dGcG-FEM algorithm for neural delay equations, including an a-priori error analysis. We demonstrate the application of the dGcG-FEM algorithm on several neural field models, including problems with an inhomogeneous kernel.
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