arXiv subjects

Mattia Sensi

Publications and source records attributed to Mattia Sensi.

At least 19 recordsLinked to original sources

arXiv2026

Traveling fronts in a spatial epidemic model with slow loss of immunity

We investigate the emergence of traveling front solutions in a spatial SIRS epidemic model with diffusion acting on the infected population. The model exhibits a natural slow-fast structure due to the presence of a small parameter governing the loss of immunity, which induces a separation of scales in the dynamics. Using a traveling wave reduction, the PDE system is transformed into a singularly perturbed system of ODEs, which we analyze within the framework of Geometric Singular Perturbation Theory. In the singular limits, we study the fast excursions governed by the layer problem, and the slow evolution close to the critical manifold. In particular, we identify an entry-exit mechanism tracking the transitions between slow and fast regimes, and derive a quantitative characterization of the entry-exit dynamics. Numerical simulations of the full system confirm the validity of the proposed geometric picture. The traveling front is shown to consist of a concatenation of local, fast, and slow segments, in agreement with the theoretical analysis.

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arXiv2026

Slow-fast dynamics in a planar parasite--host model with an extinction singularity

We study a slow-fast parasite--host model featuring a singularity at the extinction state. Using techniques from Geometric Singular Perturbation Theory (GSPT), and in particular the so-called blow-up method, we desingularize that point and reconstruct the local and global dynamics. The system we consider is in non-standard GSPT form and is characterized by a rich dynamical behavior: families of slow-fast homoclinic orbits, canard-like transitions generated by trajectories that remain close to a repelling critical manifold, and topological changes produced by infinitesimal variations of the infection rate, including the creation and destruction of an endemic equilibrium. We also show that our model is able to reproduce dynamics observed in the spread of the Tasmanian devil facial tumor disease (DFTD), whose behavior resembles the one of a parasite. We conclude with a numerical exploration of the model, to illustrate our analytical results.

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arXiv2026

A model for mosquito-borne epidemic outbreaks with information-dependent protective behaviour

We investigate a model for a mosquito-borne epidemic in which human hosts may adopt protective behaviour against vector bites in response to information on both past and current disease prevalence. Assuming that mosquitoes can also feed on non-competent hosts (i.e.\ hosts that do not contribute to disease transmission), we first revisit existing results and show that behaviour-driven protection may either decrease or increase the basic reproduction number, depending on the interaction between behavioural response, host composition, and transmission parameters. Assuming that opinion dynamics evolves on a much faster time scale than disease transmission, we then apply Geometric Singular Perturbation Theory to effectively reduce the original two-group model to a model for a homogeneous host population. The reduced system enables a detailed investigation of the impact of information-induced behavioural changes on the transient dynamics of the epidemic, including scenarios in which protective measures lead to outbreaks with low attack rates. Our analysis shows that behavioural responses may either facilitate epidemic control or prolong disease persistence, potentially generating recurrent damped epidemic waves. Numerical simulations are provided to illustrate and support the analytical findings.

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arXiv2026

Non-Birkhoff periodic orbits in symmetric billiards

We study non-Birkhoff periodic orbits in symmetric convex planar billiards. Our main result provides a quantitative criterion for the existence of such orbits with prescribed minimal period, rotation number, and spatiotemporal symmetry. We exploit this criterion to find sufficient conditions for a symmetric billiard to possess infinitely many non-Birkhoff periodic orbits. It follows that arbitrarily small analytical perturbations of the circular billiard have non-Birkhoff periodic orbits of any rational rotation number and with arbitrarily long periods. We also generalize a known result for elliptical billiards to other $\mathbb{D}_2$-symmetric billiards. Lastly, we provide Matlab codes which can be used to numerically compute and visualize the non-Birkhoff periodic orbits whose existence we prove analytically.

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arXiv2026

Exploring Exponential Runge-Kutta Methods: A Survey

In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.

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arXiv2025

A geometric analysis of the Bazykin-Berezovskaya predator-prey model with Allee effect in an economic framework

We study a fast-slow version of the Bazykin-Berezovskaya predator-prey model with Allee effect evolving on two timescales, through the lenses of Geometric Singular Perturbation Theory (GSPT). The system we consider is in non-standard form. We completely characterize its dynamics, providing explicit threshold quantities to distinguish between a rich variety of possible asymptotic behaviors. Moreover, we propose numerical results to illustrate our findings. Lastly, we comment on the real-world interpretation of these results, in an economic framework and in the context of predator-prey models.

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arXiv2025

Heterogeneously structured compartmental models of epidemiological systems: from individual-level processes to population-scale dynamics

We develop a general modelling framework for compartmental epidemiological systems structured by continuous variables which are linked to the levels of expression of compartment-specific traits. We start by formulating an individual-based model that describes the dynamics of single individuals in terms of stochastic processes. Then we formally derive: (i) the mesoscopic counterpart of this model, which is formulated as a system of integro-differential equations for the distributions of individuals over the structuring-variable domains of the different compartments; (ii) the corresponding macroscopic model, which takes the form of a system of ordinary differential equations for the fractions of individuals in the different compartments and the mean levels of expression of the traits represented by the structuring variables. We employ a reduced version of the macroscopic model to obtain a general formula for the basic reproduction number, $\mathcal{R}_0$, in terms of key parameters and functions of the underlying microscopic model, so as to illustrate how such a modelling framework makes it possible to draw connections between fundamental individual-level processes and population-scale dynamics. Finally we apply the modelling framework to case studies based on classical compartmental epidemiological systems, for each of which we report on Monte Carlo simulations of the individual-based model as well as on analytical results and numerical solutions of the macroscopic model.

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arXiv2025

A minimal model for multigroup adaptive SIS epidemics

We propose a generalization of the adaptive N-Intertwined Mean-Field Approximation (aNIMFA) model studied in Achterberg and Sensi (2023) to a heterogeneous network of communities. In particular, the multigroup aNIMFA model describes the impact of both local and global disease awareness on the spread of a disease in a network. We obtain results on existence and stability of the equilibria of the system, in terms of the basic reproduction number $R_0$. Assuming individuals have no reason to decrease their contacts in the absence of disease, we show that the basic reproduction number $R_0$ is equivalent to the basic reproduction number of the NIMFA model on static networks. Based on numerical simulations, we demonstrate that with just two communities periodic behaviour can occur, which contrasts the case with only a single community, in which periodicity was ruled out analytically. We also find that breaking connections between communities is more fruitful compared to breaking connections within communities to reduce the disease outbreak on dense networks, but both strategies are viable to networks with fewer links. Finally, we emphasize that our method of modelling adaptivity is not limited to SIS models, but has huge potential to be applied in other compartmental models in epidemiology.

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arXiv2024

A refractory density approach to a multi-scale SEIRS epidemic model

We propose a novel multi-scale modeling framework for infectious disease spreading, borrowing ideas and modeling tools from the so-called Refractory Density (RD) approach. We introduce a microscopic model that describes the probability of infection for a single individual and the evolution of the disease within their body. From the individual-level description, we then present the corresponding population-level model of epidemic spreading on the mesoscopic and macroscopic scale. We conclude with numerical illustrations taking into account either a white Gaussian noise or an escape noise to showcase the potential of our approach in producing both transient and asymptotic complex dynamics as well as finite-size fluctuations consistently across multiple scales. A comparison with the epidemiology of coronaviruses is also given to corroborate the qualitative relevance of our new approach.

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arXiv2024

A geometric analysis of the SIRS compartmental model with fast information and misinformation spreading

We propose a novel slow-fast SIRS compartmental model with demography, by coupling a slow disease spreading model and a fast information and misinformation spreading model. Beside the classes of susceptible, infected and recovered individuals of a common SIRS model, here we define three new classes related to the information spreading model, e.g. unaware individuals, misinformed individuals and individuals who are skeptical to disease-related misinformation. Under our assumptions, the system evolves on two time scales. We completely characterize its asymptotic behaviour with techniques of Geometric Singular Perturbation Theory (GSPT). We exploit the time scale separation to analyse two lower dimensional subsystem separately. First, we focus on the analysis of the fast dynamics and we find three equilibrium point which are feasible and stable under specific conditions. We perform a theoretical bifurcation analysis of the fast system to understand the relations between these three equilibria when varying specific parameters of the fast system. Secondly, we focus on the evolution of the slow variables and we identify three branches of the critical manifold, which are described by the three equilibria of the fast system. We fully characterize the slow dynamics on each branch. Moreover, we show how the inclusion of (mis)information spread may negatively or positively affect the evolution of the epidemic, depending on whether the slow dynamics evolves on the second branch of the critical manifold, related to the skeptical-free equilibrium or on the third one, related to misinformed-free equilibrium, respectively. We conclude with numerical simulations which showcase our analytical results.

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arXiv2024

Transition from time-variant to static networks: timescale separation in NIMFA SIS epidemics

We extend the N-Intertwined Mean-Field Approximation (NIMFA) for the Susceptible-Infectious-Susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analysed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times $\mathrm{\underline{T}}(r)$ and $\mathrm{\overline{T}}(r)$ as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance $r$. By analysing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for $\mathrm{\overline{T}}(r)$. Our results provide insights/bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time $\mathrm{\overline{T}}(r)$ is almost entirely determined by the basic reproduction number $R_0$ of the network. The value of the upper-transition time $\mathrm{\overline{T}}(r)$ around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

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arXiv2023

Delayed loss of stability of periodic travelling waves: insights from the analysis of essential spectra

Periodic travelling waves (PTW) are a common solution type of partial differential equations. Such models exhibit multistability of PTWs, typically visualised through the Busse balloon, and parameter changes typically lead to a cascade of wavelength changes through the Busse balloon. In the past, the stability boundaries of the Busse balloon have been used to predict such wavelength changes. Here, motivated by anecdotal evidence from previous work, we provide compelling evidence that the Busse balloon provides insufficient information to predict wavelength changes due to a delayed loss of stability phenomenon. Using two different reaction-advection-diffusion systems, we relate the delay that occurs between the crossing of a stability boundary in the Busse balloon and the occurrence of a wavelength change to features of the essential spectrum of the destabilised PTW. This leads to a predictive framework that can estimate the order of magnitude of such a time delay, which provides a novel ``early warning sign'' for pattern destabilization. We illustrate the implementation of the predictive framework to predict under what conditions a wavelength change of a PTW occurs.

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arXiv2023

A multigroup approach to delayed prion production

We generalize the model proposed in [Adimy, Babin, Pujo-Menjouet, \emph{SIAM Journal on Applied Dynamical Systems} (2022)] for prion infection to a network of neurons. We do so by applying a so-called \emph{multigroup approach} to the system of Delay Differential Equations (DDEs) proposed in the aforementioned paper. We derive the classical threshold quantity $\mathcal{R}_0$, \textit{i.e.} the basic reproduction number, exploiting the fact that the DDEs of our model qualitatively behave like Ordinary Differential Equations (ODEs) when evaluated at the Disease Free Equilibrium. We prove analytically that the disease naturally goes extinct when $\mathcal{R}_0<1$, whereas it persists when $\mathcal{R}_0>1$. We conclude with some selected numerical simulations of the system, to illustrate our analytical results.

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arXiv2023

Global stability of multi-group SAIRS epidemic models

We study a multi-group SAIRS-type epidemic model with vaccination. The role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. This is a natural extension of the homogeneous mixing SAIRS model with vaccination studied in Ottaviano et. al (2021) to a network of communities. 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 $\mathcal{R}_0 < 1$. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when $\mathcal{R}_0=1$. Moreover, if $\mathcal{R}_0 > 1$, the disease-free equilibrium is unstable and a unique endemic equilibrium exists. First, we investigate the local asymptotic stability of the endemic equilibrium and subsequently its global stability, for two variations of the original model. Last, we provide numerical simulations to compare the epidemic spreading on different networks topologies.

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arXiv2023

A geometric analysis of the SIRS model with secondary infections

We propose a compartmental model for a disease with temporary immunity and secondary infections. From our assumptions on the parameters involved in the model, the system naturally evolves in three time scales. We characterize the equilibria of the system and analyze their stability. We find conditions for the existence of two endemic equilibria, for some cases in which $\mathcal{R}_0 < 1$. Then, we unravel the interplay of the three time scales, providing conditions to foresee whether the system evolves in all three scales, or only in the fast and the intermediate ones. We conclude with numerical simulations and bifurcation analysis, to complement our analytical results.

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arXiv2023

A geometric analysis of the impact of large but finite switching rates on vaccination evolutionary games

In contemporary society, social networks accelerate decision dynamics causing a rapid switch of opinions in a number of fields, including the prevention of infectious diseases by means of vaccines. This means that opinion dynamics can nowadays be much faster than the spread of epidemics. Hence, we propose a Susceptible-Infectious-Removed epidemic model coupled with an evolutionary vaccination game embedding the public health system efforts to increase vaccine uptake. This results in a global system ``epidemic model + evolutionary game''. The epidemiological novelty of this work is that we assume that the switching to the strategy ``pro vaccine'' depends on the incidence of the disease. As a consequence of the above-mentioned accelerated decisions, the dynamics of the system acts on two different scales: a fast scale for the vaccine decisions and a slower scale for the spread of the disease. Another, and more methodological, element of novelty is that we apply Geometrical Singular Perturbation Theory (GSPT) to such a two-scale model and we then compare the geometric analysis with the Quasi-Steady-State Approximation (QSSA) approach, showing a criticality in the latter. Later, we apply the GSPT approach to the disease prevalence-based model already studied in (Della Marca and d'Onofrio, Comm Nonl Sci Num Sim, 2021) via the QSSA approach by considering medium-large values of the strategy switching parameter.

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arXiv2023

Slow-fast dynamics in a neurotransmitter release model: delayed response to a time-dependent input signal

We propose a generalization of the neurotransmitter release model proposed in \emph{Rodrigues et al. (PNAS, 2016)}. We increase the complexity of the underlying slow-fast system by considering a degree-four polynomial as parametrization of the critical manifold. We focus on the possible transient and asymptotic dynamics, exploiting the so-called entry-exit function to describe slow parts of the dynamics. We provide extensive numerical simulations, complemented by numerical bifurcation analysis.

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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.

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