arXiv subjects

Mattia Sensi

Publications and source records attributed to Mattia Sensi.

30 records · Page 2Linked to original sources

arXiv2023

A survey on Lyapunov functions for epidemic compartmental models

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, together with a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.

math.DS↗

arXiv2022

A minimal model for adaptive SIS epidemics

The interplay between disease spreading and personal risk perception is of key importance for modelling the spread of infectious diseases. We propose a planar system of ordinary differential equations (ODEs) to describe the co-evolution of a spreading phenomenon and the average link density in the personal contact network. Contrary to standard epidemic models,we assume that the contact network changes based on the current prevalence of the disease in the population, i.e.\ it adapts to the current state of the epidemic. We assume that personal risk perception is described using two functional responses: one for link-breaking and one for link-creation. The focus is on applying the model to epidemics, but we highlight other possible fields of application. We derive an explicit form for the basic reproduction number and guarantee the existence of at least one endemic equilibrium, for all possible functional responses. Moreover, we show that for all functional responses, limit cycles do not exist.

math.DS↗

arXiv2022

Entry-exit functions in fast-slow systems with intersecting eigenvalues

We study delayed loss of stability in a class of fast-slow systems with two fast variables and one slow one, where the linearisation of the fast vector field along a one-dimensional critical manifold has two real eigenvalues which intersect before the accumulated contraction and expansion are balanced along any individual eigendirection. That interplay between eigenvalues and eigendirections renders the use of known entry-exit relations unsuitable for calculating the point at which trajectories exit neighbourhoods of the given manifold. We illustrate the various qualitative scenarios that are possible in the class of systems considered here, and we propose novel formulae for the entry-exit functions that underlie the phenomenon of delayed loss of stability therein.

math.DS↗

arXiv2022

A Generalization of Lanchester's Model of Warfare

The classical Lanchester's model is shortly reviewed and analysed, with particular attention to the critical issues that intrinsically arise from the mathematical formalization of the problem. We then generalize a particular version of such a model describing the dynamics of warfare when three or more armies are involved in the conflict. Several numerical simulations are provided.

physics.soc-ph↗

arXiv2021

Evolutionary dynamics in an SI epidemic model with phenotype-structured susceptible compartment

We present an SI epidemic model whereby a continuous variable captures variability in proliferative potential and resistance to infection among susceptibles. The occurrence of heritable, spontaneous changes in these phenotype and the presence of a fitness trade-off between resistance to infection and proliferative potential are incorporated into the model. The model comprises an ODE for the number of infected individuals that is coupled with a partial integrodifferential equation for the population density of susceptibles through an integral term. The expression for the basic reproduction number $\mathcal{R}_0$ is derived, the disease-free and endemic equilibrium of the model are characterised and a threshold theorem is proved. Analytical results are integrated with numerical simulations of a calibrated version of the model based on the results of artificial selection experiments in a host-parasite system. The results of our mathematical study disentangle the impact of different evolutionary parameters on the spread of infectious diseases and the consequent phenotypic adaption of susceptible individuals. In particular, these results provide a theoretical basis for the observation that infectious diseases exerting stronger selective pressures on susceptibles and being characterised by higher infection rates are more likely to spread. Moreover, our results indicate that spontaneous phenotypic changes in proliferative potential and resistance to infection can either promote or prevent the spread of diseases depending on the strength of selection acting on susceptible individuals prior to infection. Finally, we demonstrate that, when an endemic equilibrium is established, higher levels of resistance to infection and lower degrees of phenotypic heterogeneity are to be expected in the presence of infections which are characterised by lower rates of death and exert stronger selective pressures.

q-bio.PE↗

arXiv2021

Global stability of SAIRS epidemic models

We study an SAIRS-type epidemic model with vaccination, where the role of asymptomatic and symptomatic infectious individuals are explicitly considered in the transmission patterns of the disease. We provide a global stability analysis for the model. We determine the value of the basic reproduction number $\mathcal{R}_0$ and prove that the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$, condition under which a positive endemic equilibrium exists. We investigate the global stability of the endemic equilibrium for some variations of the original model under study and answer to an open problem proposed in Ansumali et al. \cite{ansumali2020modelling}. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when $\mathcal{R}_0=1$. We provide a thorough numerical exploration of our model, to validate our analytical results.

math.DS↗

arXiv2020

How network properties and epidemic parameters influence stochastic SIR dynamics on scale-free random networks

With the premise that social interactions are described by power-law distributions, we study a SIR stochastic dynamic on a static scale-free random network generated via configuration model. We verify our model with respect to deterministic considerations and provide a theoretical result on the probability of the extinction of the disease. Based on this calibration, we explore the variability in disease spread by stochastic simulations. In particular, we demonstrate how important epidemic indices change as a function of the contagiousness of the disease and the connectivity of the network. Our results quantify the role of starting node degree in determining these indices, commonly used to describe epidemic spread.

physics.soc-ph↗

arXiv2020

A geometric analysis of the SIRS epidemiological model on a homogeneous network

We study a fast-slow version of an SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. We use GSPT to study the model, taking into account that the infection period is much shorter than the average duration of immunity. We show that the dynamics occurs through a sequence of fast and slow flows, that can be described through 2-dimensional maps that, under some assumptions, can be approximated as 1-dimensional maps. Using this method, together with numerical bifurcation tools, we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.

math.DS↗

arXiv2020

Exact solutions for a Solow-Swan model with non-constant returns to scale

The Solow-Swan model is shortly reviewed from a mathematical point of view. By considering non-constant returns to scale, we obtain a general solution strategy. We then compute the exact solution for the Cobb-Douglas production function, for both the classical model and the von Bertalanffy model. Numerical simulations are provided.

econ.TH↗

arXiv2020

Notes on a conformal characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature

We present a characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature. It is well known that every $2$-dimensional Lorentzian manifolds is conformally flat, so we rewrite the Ricci scalar curvature in terms of the conformal factor and we study the solutions of the corresponding differential equations. Several remarkable examples are provided.

math-ph↗

arXiv2020

A geometric characterization of VES and Kadiyala-type production functions

The basic concepts of the differential geometry are shortly reviewed and applied to the study of VES production function in the spirit of the works of Vîlcu and collaborators. A similar characterization is given for a more general production function, namely the Kadiyala production function, in the case of developable surfaces.

econ.TH↗

arXiv2020

A geometric analysis of the SIR, SIRS and SIRWS epidemiological models

We study fast-slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behavior is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast-slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov's method, we are able to not only analyze the stability of the endemic equilibria but also to show that in some of the models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model, for which we provide a detailed description of its dynamics by combining analytic and numerical techniques.

math.DS↗