Vincenzo Capasso

A generalization of the Kermack-McKendrick deterministic epidemic model☆

Abstract In this paper the Kermack-McKendrick deterministic model is generalized, introducing an interaction term in which the dependence upon the number of infectives occurs via a nonlinear bounded function which may take into account saturation phenomena for large numbers of infectives. An extension of the well-known threshold theorem is obtained, after a stability analysis of the equilibrium points of the system. A numerical example is carried out in detail.

Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases

A reaction-diffusion system which describes the spatial spread of bacterial diseases is studied. It consists of two nonlinear parabolic equations which concern the evolution of the bacteria population and of the human infective population in an urban community, respectively. Different boundary conditions of the third type are considered, for the two variables. This model is suitable to study oro-faecal transmitted diseases in the European Mediterranean regions. A threshold parameter is introduced such that for suitable values of it the epidemic eventually tends to extinction, otherwise a globally asymptotically stable spatially inhomogeneous stationary endemic state appears. The case in which the bacteria diffuse but the human population does not, has also been considered.

An introduction to optimal control problems in life sciences and economics

An Introduction to MATLAB. Elementary Models with Applications.- Optimal Control of Ordinary Differential Systems. Optimality Conditions.- Optimal Control of Ordinary Differential Systems. Gradient Methods.- Optimal Harvesting for Age-Structured Population.- Optimal Control of Diffusive Models.- Appendices.- References.- Index.

Global Solution for a Diffusive Nonlinear Deterministic Epidemic Model

A diffusive epidemic model with removal of infectives is proposed here in which the interaction term depends on the density of infectives via a smooth bounded map. This function allows either “local” or “distant” interactions between susceptibles and infectives to be taken into account.We study the nonlinear initial-value problem arising from the proposed model and show that a unique positive and bounded strong solution exists at any time

On the general structure of epidemic systems. Global asymptotic stability

t \geqq 0

Sensitive Observables of Quantum Mechanics

.An upper bound is also given for the number of susceptibles and the number of infectives respectively.The results can be easily extended to ecological systems with the same kind of interaction.

Stochastic modelling of tumour-induced angiogenesis

Abstract In this paper a general ODE model is proposed to describe epidemic systems. The mathematical structure of such a model is so general that it includes many epidemic systems already analyzed via different methods by various authors. The asymptotic analysis of the general system is carried out with applications to several models.

On the Geometric Densities of Random Closed Sets

We prove that there are quantum mechanical observables which are sensitive to the type of state-vector (first type or second type) describing two correlated physical systems, in the sense that the expectation value of these ‘sensitive observables’ is measurably different in the two cases. The proof centers around Bells inequality since we show that in quantum mechanics forall state-vectors of the second type (and only for them) sensitive observables exist in the absence of super-selection rules. Experimental verification of the existence of sensitive observables rules out local hidden variables.

Mesoscale averaging of nucleation and growth models

A major source of complexity in the mathematical modelling of an angiogenic process derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network with a family of interacting underlying fields. The aim of this paper is to propose a novel mathematical approach for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. This method leads to models which are known as hybrid models. In this paper, as a working example, we apply our method to a simplified stochastic geometric model, inspired by the relevant literature, for a spatially distributed angiogenic process. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips, according to a system of stochastic differential equations à la Langevin.

MATHEMATICAL MODELLING AND SIMULATION OF NON-ISOTHERMAL CRYSTALLIZATION OF POLYMERS

Abstract In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper we suggest to cope with these difficulties by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. In this last one we analyze mean generalized densities, and relate them to densities of the expected values of the relevant measures. Many models of interest in material science and in biomedicine are based on time dependent random closed sets, as the ones describing the evolution of (possibly space and time inhomogeneous) growth processes; in such a situation, the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the local relevant kinetic parameters of birth and growth. In this context connections with the concepts of hazard function, and spherical contact distribution function are offered.