This work aims at presenting different approaches to epidemics modelling. It consists of two main topics, which cover both theoretical and computational approaches to the development and analysis of mathematical models of infectious diseases. The first half regards the formulation and the analysis of SAIRS (Susceptible - Asyptomatics infected - Infected symptomatic - Recovered - Susceptible) epidemic models, including the possibility of vaccination. The model is formulated as a system of ordinary differential equations (ODEs), for which we provided a complete global stability analysis, combining two different approaches: the classical Lyapunov stability theorem, and a geometric approach, which generalises the Poincaré-Bendixon theorem. Afterwards, the model has been generalised using heterogeneous networks, which may describe different groups of individuals or different cities. For this model, the global stability analysis has been developed using the graph-theoretic approach to find Lyapunov functions. The second part of the thesis covers simulations based approaches to modelling heterogeneous humans interactions in epidemics. The first example we provide is an application with synthetic data. We investigate a stochastic SIR (Susceptible - Infected symptomatic - Recovered) dynamics on a network, by using a specialised version of the Gillespie algorithm. The other two examples we show consist of real data applications. Both regard the cost-benefit analysis of the introduction of new influenza vaccines. Both analyses have been performed using a multi-group SEIR (Susceptible - Exposed - Infected - Recovered) epidemiological model divided by age classes.

Mathematical models represent a powerful tool for investigating the dynamics of human infection diseases, providing useful predictions about the spread of a disease and the effectiveness of possible control measures. One of the central aspects to understand the dynamics of human infection is the heterogeneity in behavioral patters adopted by the host population. Beyond control measures imposed by public authorities, human behavioral changes can be triggered by uncoordinated responses driven by the diffusion of fear in the general population or by the risk perception. In order to assess how and when behavioral changes can affect the spread of an epidemic, spontaneous social distancing - e.g. produced by avoiding crowded environments, using face masks or limiting travels - is investigated. Moreover, in order to assess whether vaccine preventable diseases can be eliminated through not compulsory vaccination programs, vaccination choices are investigated as well. The proposed models are based on an evolutionary game theory framework. Considering dynamical games allows explicitly modeling the coupled dynamics of disease transmission and human behavioral changes. Specifically, the information diffusion is modeled through an imitation process in which the convenience of different behaviors depends on the perceived risk of infection and vaccine side effects. The proposed models allow the investigation of the effects of misperception of risks induced by partial, delayed or incorrect information (either concerning the state of the epidemic or vaccine side effects) as well. The performed investigation highlights that a small reduction in the number of potentially infectious contacts in response to an epidemic and an initial misperception of the risk of infection can remarkably affect the spread of infection. On the other hand, the analysis of vaccination choices showed that concerns about proclaimed risks of vaccine side effects can result in widespread refusal of vaccination which in turn leads to drops in vaccine uptake and suboptimal vaccination coverage.

The main object of our research is the notion of "intrinsic regular surfaces" introduced and studied by Franchi, Serapioni, Serra Cassano in a Carnot group G. More precisely, an intrinsic regular hypersurface (i.e. a topological codimension 1 surface) S is a subset of G which is locally defined as a non critical level set of a C^1 intrinsic function. In a similar way, a k-codimensional intrinsic regular surface is locally defined as a non critical level set of a C^1 intrinsic vector function. Through Implicit Function Theorem, S can be locally represented as an intrinsic graph by a function phi. Here the intrinsic graph is defined as follows: let V and W be complementary subgroups of G, then the intrinsic graph of phi defined from W to V is the set { A \cdot phi(A) | A belongs to W}, where \cdot indicates the group operation in G. A fine characterization of intrinsic regular surfaces in Heisenberg groups (examples of Carnot groups) as suitable 1-codimensional intrinsic graphs has been established in [1]. We extend this result in a general Carnot group introducing an appropriate notion of differentiability, denoted uniformly intrinsic differentiability, for maps acting between complementary subgroups of G. Finally we provide a characterization of intrinsic regular surfaces in terms of existence and continuity of suitable "derivatives" of phi introduced by Serra Cassano et al. in the context of Heisenberg groups. All the results have been obtained in collaboration with Serapioni. [1] L.Ambrosio, F. Serra Cassano, D. Vittone, \emph{Intrinsic regular hypersurfaces in Heisenberg groups}, J. Geom. Anal. 16, (2006), 187-232.

In this thesis we study intrinsic Lipschitz functions. In particular we provide a regular approximation result and a Poincarè type inequality for this class of functions. Moreover we study the obstacle problem in the Heisenberg group and we prove a geometric Poincarè inequality for a class of semilinear equations in the Engel group.

This dissertation consists of an introduction and four papers. The papers deal with several problems of non-Riemannian metric spaces, such as sub-Riemannian Carnot groups and homogeneous metric spaces. The research has been carried out between the University of Trento (Italy) and the University of Jyväskylä (Finland) under the supervision of prof. F. Serra Cassano and E. Le Donne, respectively. In the following we present the abstracts of the four papers. 1) REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS E. Le Donne AND S. Nicolussi Golo We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps. 2) ASYMPTOTIC BEHAVIOR OF THE RIEMANNIAN HEISENBERG GROUP AND ITS HOROBOUNDARY E. Le Donne, S. Nicolussi Golo, AND A. Sambusetti The paper is devoted to the large scale geometry of the Heisenberg group H equipped with left-invariant Riemannian metrics. We prove that two such metrics have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian metric d on H there is a unique sub-Riemanniann metric d' for which d − d' goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group. 3) FROM HOMOGENEOUS METRIC SPACES TO LIE GROUPS M. G. Cowling, V. Kivioja, E. Le Donne, S. Nicolussi Golo, AND A. Ottazzi We study connected, locally compact metric spaces with transitive isometry groups. For all $\epsilon\in R_+$, each such space is $(1,\epsilon)$- quasi-isometric to a Lie group equipped with a left-invariant metric. Further, every metric Lie group is $(1,C)$-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is $(1,C)$-quasi-isometrically homeomorphic to a solvable-by-compact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation. 4) SOME REMARKS ON CONTACT VARIATIONS IN THE FIRST HEISENBERG GROUP S. Nicolussi Golo We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f : R^2 \rightarrow R^2$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff = 0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.

The thesis surveys some recent results obtained in the field of nonlinear partial differential equations and calculus of variations about eigenvalues of nonlinear operators.

In the classical theory, the fractional diffusion is ruled by two different types of fractional Laplacians. Formerly known since 60s, the spectral fractional Laplacian had an important development in the recent mathematical study with the initial contributes of L. Caffarelli, L. Silvestre and X. Cabré, X.Tan. The integral version of the fractional Laplacian, recently discussed by M. Fukushima, Y. Oshima, M Takeda, and Song, Vondracek, is considered in a semilinear elliptic problem in presence of a general logistic function and an indefinite weight. In particular we look for a multiplicity result for the associated Dirichlet problem. In the second part, starting from the classical works of T.Hillen and G. Othmer and taking the Generalized velocity jump processes presented in a recent work of J.T.King, we obtain the fractional diffusion as limit of this last processes using the technique used in another recent work of Mellet, without the classical Hilbert or Cattaneo approximation's methods.

We present a mathematical model describing the dynamics occurring between two interacting populations, one of amoebae and one of virulent bacteria; it is meant to describe laboratory experiments with these two species in a mathematical framework and help understanding the role of the different mechanisms involved. In particular we aim to focus on how bacterial virulence may affect the dynamics of the system. The model is a modified reaction-diffusion-chemotaxis predator-prey one with a mechanism of redistribution of ingested biomass between amoeboid cells. The spatially homogeneous case is analyzed in detail; conditions for pattern formation are established; numerical simulations for the complete model are performed.

The interplay between human behavior and the spreading of an epidemics represents a challenge in modeling the dynamics of infectious diseases. The technological revolution that we are experiencing nowadays gives access to new sources of digital data, capable of capturing behavioral patterns and social dynamics of our society and opening, in fact, the path to new opportunities for mathematical modelers. Provided by such tools, we discuss two different aspects of the dynamics of infectious diseases associated with human behavior. In the first part of the thesis, we focus on the mechanism driving the awareness of individuals during public health emergencies and describe epidemiological models especially tailored to better understand the underline features of the risk perception. The proposed framework is able to disentangle and characterize the contribution of media drivers and social contagion mechanisms in the building of awareness of individuals about infectious diseases. In the second part of the thesis, we present a data driven computational model aiming to assess the potential risk of experiencing measles re-emergence in Turkey. This study takes into consideration the recent massive migration of Syrian refugees in Turkey, which changed the social structure and focuses on the possible outbreak of an infectious disease, such as measles, as a consequence of the great concentration of Syrian refugees not adequately immunized against it. The model proposed is informed with mobility patterns inferred from mobile phone data and accounts for the different hypothetical policies adopted to integrate the refugees with the Turkish population.

The thesis deals with the study of a natural extension of classical finite-dimensional time-optimal control problem to the space of positive Borel measures. This approach has two main motivations: to model real-life situations in which the knowledge of the initial state is only probabilistic, and to model the statistical distribution of a huge number of agents for applications in multi-agent systems. We deal with a deterministic dynamics and treat the problem first in a mass-preserving setting: we give a definition of generalized target, its properties, admissible trajectories and generalized minimum time function, we prove a Dynamic Programming Principle, attainability results, regularity results and an Hamilton-Jacobi-Bellman equation solved in a suitable viscosity sense by the generalized minimum time function, and finally we study the definition of an object intended to reflect the classical Lie bracket but in a measure-theoretic setting. We also treat a case with mass loss thought for modelling the situation in which we are interested in the study of an averaged cost functional and a strongly invariant target set. Also more general cost functionals are analysed which takes into account microscopical and macroscopical effects, and we prove sufficient conditions ensuring their lower semicontinuity and a dynamic programming principle in a general formulation.