Category Archives: Epidemiology

Derivation of macroscopic epidemic models from multi-agent systems

by Mattia Zanella

M. Zanella

Preprint arXiv, 2024

We present a systematic review of some basic results on the derivation of classical epidemiological models from simple agent-based dynamics. The evolution of large populations is coupled with the dynamics of the contact distribution, providing insights into how individual behaviors impact macroscopic epidemiological trends.

The resulting set of equations incorporates local characteristics of the operator governing the emergence of a family of contact distributions. To validate the proposed approach, we provide several numerical results based on asymptotic preserving methods, demonstrating their effectiveness in capturing the multi-scale nature of the problem and ensuring robust performance across different regimes.

Impact of opinion formation phenomena in epidemic dynamics: kinetic modeling on networks

by Mattia Zanella

G. Albi, E. Calzola, G. Dimarco, M. Zanella.

Preprint arXiv, 2024

After the recent COVID-19 outbreaks, it became increasingly evident that individuals’
thoughts and beliefs can have a strong impact the disease transmission. It becomes therefore important to understand how information and opinions on protective measures evolve during epidemics. To this end, incorporating the impact of social media is essential to take into account the hierarchical structure of these platforms. In this context, we present a novel approach to take into account the interplay between infectious disease dynamics and socially-structured opinion dynamics. Our work extends a conventional compartmental framework including behavioral attitudes in shaping public opinion and promoting the adoption of protective measures under the influence of different degrees of connectivity. The proposed approach is capable to reproduce the emergence of epidemic waves. Specifically, it provides a clear link between the social influence of highly connected individuals and the epidemic dynamics. Through a heterogeneity of numerical tests we show how this comprehensive framework offers a more nuanced understanding of epidemic dynamics in the context of modern information dissemination and social behavior.

Predictability of viral load kinetics in the early phases of SARS-CoV-2 through a model-based approach

by Mattia Zanella

A. Bondesan, A. Piralla, E. Ballante, A. M. G. Pitrolo, S. Figini, F. Baldanti, M. Zanella

Preprint arXiv, 2024.

A pipeline to evaluate the evolution of viral dynamics based on a new model-driven approach has been developed in the present study. The proposed methods exploit real data and the multiscale structure of the infection dynamics to provide robust predictions of the epidemic dynamics. We focus on viral load kinetics whose dynamical features are typically available in the symptomatic stage of the infection. Hence, the epidemiological evolution is obtained by relying on a compartmental approach characterized by a varying infection rate to estimate early-stage viral load dynamics, of which few data are available. We test the proposed approach with real data of SARS-CoV-2 viral load kinetics collected from patients living in an Italian province. The considered database refers to early-phase infections, whose viral load kinetics are not affected by mass vaccination policies in Italy. Our contribution is devoted to provide an effective computational pipeline to evaluate in real time the evolution of infectivity. Comprehending the factors influencing the in-host viral dynamics represents a fundamental tool to provide robust public health strategies. This pilot study could be implemented in further investigations involving other respiratory viruses, to better clarify the process of viral dynamics as a preparatory action for future pandemics.

Effects of heterogeneous opinion interactions in many-agent systems for epidemic dynamics

by Mattia Zanella

S. Bonandin, M. Zanella

Netw. Heterog. Media, 19(1): 235-261, 2024.. (Preprint arXiv)

In this work we define a kinetic model for understanding the impact of heterogeneous opinion formation dynamics on epidemics. The considered many-agent system is characterized by nonsymmetric interactions which define a coupled system of kinetic equations for the evolution of the opinion density in each compartment.

In the quasi-invariant limit we may show positivity and uniqueness of the solution of the problem together with its convergence towards an equilibrium distribution exhibiting bimodal shape. The tendency of the system towards opinion clusters is further analyzed by means of numerical methods, which confirm the consistency of the kinetic model with its moment system whose evolution is approximated in several regimes of parameters.

Kinetic compartmental models driven by opinion dynamics: vaccine hesitancy and social influence

by Mattia Zanella

A. Bondesan, G. Toscani, M. Zanella

Math. Mod. Meth. Appl. Sci., 34(06): 1043-1076, 2024. (Preprint arXiv)

We propose a kinetic model for understanding the link between opinion formation phenomena and epidemic dynamics. The recent pandemic has brought to light that vaccine hesitancy can present different phases and temporal and spatial variations, presumably due to the different social features of individuals.

The emergence of patterns in societal reactions permits to design and predict the trends of a pandemic. This suggests that the problem of vaccine hesitancy can be described in mathematical terms, by suitably coupling a kinetic compartmental model for the spreading of an infectious disease with the evolution of the personal opinion of individuals, in the presence of leaders. The resulting model makes it possible to predict the collective compliance with vaccination campaigns as the pandemic evolves and to highlight the best strategy to set up for maximizing the vaccination coverage. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.

Kinetic models for epidemic dynamics in the presence of opinion polarization

by Mattia Zanella

M. Zanella

Bulletin of Mathematical Biology, 85(5):36, 2023. (Preprint arXiv)

Understanding the impact of collective social phenomena in epidemic dynamics is a crucial task to effectively contain the disease spread. In this work, we build a mathematical description for assessing the interplay between opinion polarization and the evolution of a disease.

The proposed kinetic approach describes the evolution of aggregate quantities characterizing the agents belonging to epidemiologically relevant states and will show that the spread of the disease is closely related to consensus dynamics distribution in which opinion polarization may emerge. In the present modelling framework, microscopic consensus formation dynamics can be linked to macroscopic epidemic trends to trigger the collective adherence to protective measures. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic.

On the optimal control of kinetic epidemic models with uncertain social features

by Mattia Zanella

J. Franceschi, A. Medaglia, M. Zanella.

Optim. Contr. Appl. Meth., 45(2): 494-522, 2024. (Preprint arXiv)

It is recognized that social heterogeneities in terms of the contact distribution have a strong influence on the spread of infectious diseases. Nevertheless, few data are available and their statistical description does not possess universal patterns and may vary spatially and temporally. It is therefore essential to design optimal control strategies, mimicking the effects of non-pharmaceutical interventions, to limit efficiently the number of infected cases.

In this work, starting from a recently introduced kinetic model for epidemiological dynamics that takes into account the impact of social contacts of individuals, we consider an uncertain contact formation dynamics leading to slim-tailed as well as fat-tailed distributions of contacts. Hence, we analyse the effects of an optimal control strategy of the system of agents. Thanks to classical methods of kinetic theory, we couple uncertainty quantification methods with the introduced mathematical model to assess the effects of social limitations. Finally, using the proposed modelling approach and starting from available data, we show the effectiveness of the proposed selective measures to dampen uncertainties together with the epidemic trends.

Effects of vaccination efficacy on wealth distribution in kinetic epidemic models

by Mattia Zanella

E. Bernardi, L. Pareschi, G. Toscani, M. Zanella

Entropy, 22:216, 2022. (Preprint arXiv)

The spreading of Covid-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modelling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming on the one hand that individuals in different compartments act differently in the economic process and on the other hand that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society

Kinetic and macroscopic epidemic models in presence of multiple heterogeneous populations

by Mattia Zanella

A. Medaglia, M. Zanella

Preprint arXiv, 2021.

We study the impact of contact heterogeneity on epidemic dynamics. A system characterized by multiple susceptible populations is considered. The description of the spread of an infectious disease is obtained through the study of a system of Boltzmann-type equations for the number densities of social contacts of the introduced compartments. A macroscopic system of equations characterizing observable effects of the epidemic is then derived to assess the impact of contact heterogeneity.

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

by Mattia Zanella

G. Albi, G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani, M. Zanella.

Predicting Pandemics in a Globally Connected World Vol.1, 2022. (Preprint arXiv)

In this survey we report some recent results in the mathematical modeling of epidemic phenomena through the use of kinetic equations.

We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker’s perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modeling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings.