Showing papers in "Ricerche Di Matematica in 2018"

Seasonality in epidemic models: a literature review

Abstract: We provide a review of some key literature results on the influence of seasonality and other time heterogeneities of contact rates, and other parameters, such as vaccination rates, on the spread of infectious diseases. This is a classical topic where highly theoretical methodologies have provided new insight on the seemingly random behavior observed in epidemic time-series. We follow the line of providing a highly personal non-systematic review of this topic, mainly based on the history of mathematical epidemiology and on the impact of reviewed articles. Our aim is to stress some issues of increasing interest, such as the public health implications of the biomathematical literature and the impact of seasonality on epidemic extinction or elimination.

The fractional Dodson diffusion equation: a new approach

Abstract: In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the $$(1+1)$$ -dimensional Dodson’s diffusion equation. For the latter we then compute the fundamental solution, which turns out to be expressed in terms of an M-Wright function of two variables. Then, we conclude the paper providing a few interesting results for some nonlinear fractional Dodson-like equations.

Modelling and optimization of least-cost water distribution networks with multiple supply sources and users

Abstract: The proper allocation of water resources is a very important practical problem in the field of water network planning. Optimization models that are expeditious and easy to use for all stakeholders of the sector play an important role for water resource management. The present work resumes and reviews a least-cost optimization model proposed by our group (Maiolo and Pantusa in Water Sci Tech-W Sup. https://doi.org/10.2166/ws.2015.114 , 2016), able to design a water distribution network with multiple supply sources and multiple users. This approach requires of solving an optimization problem based on a nonlinear objective function which is proportional to the cost of the water distribution network. The cost of pre-existing pipelines is considered null. A more realistic scenario, able to consider the maximum flow rate allowed for existing sources-users connections, is considered here. In order to illustrate the usefulness and flexibility of the proposed approach, an application of the model to the real case of the province of Croton, Southern Italy, is presented.

Seasonal forcing in stochastic epidemiology models

Abstract: The goal of this paper is to motivate the need and lay the foundation for the analysis of stochastic epidemiological models with seasonal forcing. We consider stochastic SIS and SIR epidemic models, where the internal noise is due to the random interactions of individuals in the population. We provide an overview of the general theoretic framework that allows one to understand noise-induced rare events, such as spontaneous disease extinction. Although there are many paths to extinction, there is one path termed the optimal path that is probabilistically most likely to occur. By extending the theory, we have identified the quasi-stationary solutions and the optimal path to extinction when seasonality in the contact rate is included in the models. Knowledge of the optimal extinction path enables one to compute the mean time to extinction, which in turn allows one to compare the effect of various control schemes, including vaccination and treatment, on the eradication of an infectious disease.

Some new results on integration for multifunction

Abstract: It has been proven in Di Piazza and Musial (Set Valued Anal 13:167–179, 2005, Vector measures, integration and related topics, Birkhauser Verlag, Basel, vol 201, pp 171–182, 2010) that each Henstock–Kurzweil–Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable (Theorem 34) Moreover, in case of strongly measurable (multi)functions, a characterization of the Birkhoff integrability is given using a kind of Birkhoff strong property

Characterized subgroups of the circle group

Abstract: A subgroup H of the circle group $$\mathbb T$$ is said to be characterized by a sequence $$\mathbf {u}= (u_n)_{n\in \mathbb N}$$ of integers if $$H=\{x\in \mathbb T: u_nx\rightarrow 0\}$$ . The characterized subgroups of $$\mathbb T$$ are known also under the name topologically $$\mathbf {u}$$ -torsion subgroups. This survey paper is dedicated to the characterized subgroups of $$\mathbb T$$ : we recall their main properties and collect most of the basic results from the wide bibliography, following, when possible, the historical line, and trying to show the deep roots of this topic in several areas of Mathematics. Due to this universality of the topic, many notions and results were found independently by various authors working unaware of each other, so our effort is also directed towards giving credit to all of them to the best of our knowledge. We provide also some background on the notions of characterized subgroup and topologically $$\mathbf {u}$$ -torsion subgroup in the general case of topological abelian groups, where they differ very substantially.

Uniqueness theorems about high-order time differential thermoelastic models

Abstract: The purpose of the present manuscript is to investigate the well-posedness question for three different stand-alone and self-consistent thermoelastic models derived from the time differential formulation of the dual-phase-lag heat conduction law and characterized by Taylor expansion orders higher than those most commonly considered in literature up to now. The main motivation at the basis of this study is that the interaction among multiple energy carriers progressively gains significance as the observation scales reduce and has, as a direct consequence, the involvement of high-order terms in the time differential dual-phase-lag heat conduction constitutive equation. Considering inhomogeneous and anisotropic linear thermoelastic materials, we are able to prove three uniqueness results through the use of appropriate integral operators and Lagrange identities; the results are proved without any restriction imposed on the delay times other than their positivity.

Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

Abstract: In this paper, we shall be concerned with the existence result of the following problem, 0.1 $$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\left( a(x,u, abla u)\right) -\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ with the second term f belongs to $$L^1(\Omega )$$ . The growth and the coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M. We assume any restriction on M, therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term $$\Phi $$ is a Caratheodory function satisfying only a growth condition.

Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination

Abstract: We study the newborn, non compulsory, vaccination in a SIR model with vital dynamics. The evolution of each individual is modeled as a Markov chain. His/Her vaccination decision optimizes a criterion depending on the time-dependent aggregate (societal) vaccination rate and the future epidemic dynamics. We prove the existence of a Nash-Mean Field Games equilibrium among all individuals in the population. Then we propose a novel numerical approach to find the equilibrium and test it numerically.

Discrete-time models for releases of sterile mosquitoes with Beverton–Holt-type of survivability

Abstract: In this paper, we formulate discrete-time mathematical models for the interactive wild and sterile mosquitoes. Instead of the Ricker-type of nonlinearity for the survival functions, we assume the Beverton–Holt-type in these models. We consider three different strategies for the releases of sterile mosquitoes and investigate the model dynamics. Threshold values for the releases of sterile mosquitoes are derived for all of the models that determine whether the wild mosquitoes are wiped out or coexist with the sterile mosquitoes. Numerical examples are given to demonstrate the dynamics of the models.

An isoperimetric inequality in the plane with a log-convex density

Abstract: Given a positive lower semi-continuous density f on $$\mathbb {R}^2$$ the weighted volume $$V_f:=f\mathscr {L}^2$$ is defined on the $$\mathscr {L}^2$$ -measurable sets in $$\mathbb {R}^2$$ . The f-weighted perimeter of a set of finite perimeter E in $$\mathbb {R}^2$$ is written $$P_f(E)$$ . We study minimisers for the weighted isoperimetric problem $$\begin{aligned} I_f(v):=\inf \Big \{ P_f(E):E\text { is a set of finite perimeter in }\mathbb {R}^2\text { and }V_f(E)=v\Big \} \end{aligned}$$ for $$v>0$$ . Suppose f takes the form $$f:\mathbb {R}^2\rightarrow (0,+\infty );x\mapsto e^{h(|x|)}$$ where $$h:[0,+\infty )\rightarrow \mathbb {R}$$ is a non-decreasing convex function. Let $$v>0$$ and B a centred ball in $$\mathbb {R}^2$$ with $$V_f(B)=v$$ . We show that B is a minimiser for the above variational problem and obtain a uniqueness result.

Comparative Analysis of Dengue versus Chikungunya Outbreaks in Costa Rica

Abstract: For decades, dengue virus has been a cause of major public health concern in Costa Rica, due to its landscape and climatic conditions that favor the circumstances in which the vector, Aedes aegypti, thrives. The emergence and introduction throughout tropical and subtropical countries of the chikungunya virus, as of 2014, challenged Costa Rican health authorities to provide a correct diagnosis since it is also transmitted by the same vector and infected hosts may share similar symptoms. We study the 2015–2016 dengue and chikungunya outbreaks in Costa Rica while establishing how point estimates of epidemic parameters for both diseases compare to one another. Longitudinal weekly incidence reports of these outbreaks signal likely misdiagnosis of infected individuals: underreporting of chikungunya cases, while overreporting cases of dengue. Our comparative analysis is formulated with a single-outbreak deterministic model that features an undiagnosed class. Additionally, we also used a genetic algorithm in the context of weighted least squares to calculate point estimates of key model parameters and initial conditions, while formally quantifying misdiagnosis.

On Lagrange duality theory for dynamics vaccination games

Abstract: The authors study an infinite dimensional duality theory finalized to obtain the existence of a strong duality between a convex optimization problem connected with the management of vaccinations and its Lagrange dual. Specifically, the authors show the solvability of a dual problem using as basic tool an hypothesis known as Assumption S. Roughly speaking, it requires to show that a particular limit is nonnegative. This technique improves the previous strong duality results that need the nonemptyness of the interior of the convex ordering cone. The authors use the duality theory to analyze the dynamic vaccination game in order to obtain the existence of the Lagrange multipliers related to the problem and to better comprehend the meaning of the problem.

Thermal convection in an inclined porous layer with Brinkman law

Abstract: A model for thermal convection of a fluid saturating an inclined layer of porous medium with a Brinkman law and stress-free boundary conditions is studied. When the Darcy number $$\tilde{D}a$$ is zero, this problem has been studied by Rees and Bassom (Acta Mech 144(1–2):103–118, 2000). When the Brinkman term is present in the model ( $$\tilde{D}a ot =0$$ ) the basic motion is a combination of hyperbolic and polynomial functions. With the Chebyshev collocation method we study the linear instability of the basic motion for three-dimensional perturbations. We also give nonlinear stability conditions and, for longitudinal perturbations, we prove the coincidence of linear and nonlinear critical Rayleigh numbers.

The effect of immigrant communities coming from higher incidence tuberculosis regions to a host country

Abstract: We introduce a new tuberculosis (TB) mathematical model, with 25 state-space variables where 15 are evolution disease states (EDSs), which generalises previous models and takes into account the (seasonal) flux of populations between a high incidence TB country (A) and a host country (B) with low TB incidence, where (B) is divided into a community (G) with high percentage of people from (A) plus the rest of the population (C). Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of semi-closed communities are beneficial for the TB control from a global viewpoint. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness. Simulations show that variations of the transmission coefficient on the origin country has a big influence on the number of infected (and infectious) individuals on the community and the host country. Moreover, there is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number $$R_0$$ . Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on $$R_0$$ and on EDSs are pursuit in order to understand the TB effects in the global statistics, by measuring the variability of the relevant parameters. We also show that the TB transmission rate $$\beta $$ does not act linearly on $$R_0$$ , as it is common in compartment models where system feedback or group interaction do not occur. Further, we find the most important parameters for the increase of each EDS.

Set-valued Brownian motion

Abstract: Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach spaces X. The present paper is an application of the paper (Labuschagne et al. in Quaest Math 30(3):285–308, 2007) in which an embedding result is obtained which considers also the ordered structure of the family of compact convex subsets of a Banach space X and of Grobler and Labuschagne (J Math Anal Appl 423(1):797–819, 2015; J Math Anal Appl 423(1):820–833, 2015) in which these processes are considered in f-algebras. Moreover, in the space of continuous functions defined on a Stonian space, a direct Levy’s result follows.

Multi-component Ermakov and non-autonomous many-body system connections

Abstract: It is established that classes of autonomous 3-body and 4-body systems previously set down in the literature may be embedded in solvable, non-autonomous, multi-component Ermakov systems as originally derived in a multi-layer hydrodynamic context The Ermakov connection is exploited, in particular, to construct integrable non-autonomous many-body type systems parametrised in terms of an arbitrary function J(y / x) where x, y are Jacobi variables

Some Calabi–Yau fourfolds verifying Voisin’s conjecture

Abstract: Motivated by the Bloch–Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi–Yau varieties. This note contains some examples of Calabi–Yau fourfolds verifying Voisin’s conjecture.

Modeling magnetic domain-wall evolution in trilayers with structural inversion asymmetry

Abstract: The one-dimensional motion of magnetic domain walls in a thin ferromagnetic nanostrip sandwiched between a heavy metal and a metal oxide is investigated analytically in the framework of the extended Landau–Lifshitz–Gilbert equation. The trilayer system under investigation exhibits structural inversion asymmetry and exploits the combined effects of spin-transfer-torque and spin-orbit-torque to optimize the domain-wall propagation along the nanostrip. Through the traveling-wave formalism, an explicit expression for the key features involved in both steady and precessional regimes is provided, with a particular emphasis on the role played by the two spin-orbit-torque contributions, Rashba and Spin-Hall. In particular, it is shown how the domain-wall velocity and mobility, the direction of propagation, the depinning threshold and the Walker breakdown can be controlled via a suitable combination of Rashba and Spin-Hall coefficients. A comparison between analytical results and numerical data extracted from literature is also addressed revealing a qualitative agreement between them. Additional information on spin-orbit-torque-driven DW dynamics is extracted from such an analysis and, in particular, a linear dependence between the spin-Hall angle and the azimuthal angle is outlined as a possible mechanism responsible for the reversal of propagation direction.

High order variational numerical schemes with application to Nash–MFG vaccination games

Abstract: This paper introduces high-order explicit Runge–Kutta numerical schemes in metric spaces. We show that our approach reduces to the corresponding Runge–Kutta schemes if the ambient space is Hilbert. We apply these schemes to compute the Nash equilibrium in a mean field vaccination game. Numerical simulations show improvement in the speed of convergence towards the Nash equilibrium; the numerical scheme has high order (2–4) in time.

Some combinatorial properties of the Hurwitz series ring

Abstract: We study some properties and perspectives of the Hurwitz series ring $$H_R[[t]]$$ , for an integral domain R, with multiplicative identity and zero characteristic. Specifically, we provide a closed form for the invertible elements by means of the complete ordinary Bell polynomials, we highlight some connections with well–known transforms of sequences, and we see that the Stirling transforms are automorphisms of $$H_R[[t]]$$ . Moreover, we focus the attention on some special subgroups studying their properties. Finally, we introduce a new transform of sequences that allows to see one of this subgroup as an ultrametric dynamic space.

Measures and submeasures on $$\hbox {d}_{0}$$ d 0 -algebras

Abstract: The structure of $$\hbox {d}_{0}$$ -algebra is a generalization of a D-lattice. We extend to this structure the definitions of measure and submeasure, and investigate their properties. We also investigate the relationships between uniformities and measures (or submeasures).

Modeling basic aspects of bacterial resistance of Mycobacterium tuberculosis to antibiotics

Abstract: In this work we study the effect of antibiotics treatment and the immune system in the development of Mycobacterium tuberculosis infection through the formulation and analysis of a nonlinear system of ordinary differential equations. The results are given in terms of forward and backward bifurcation and they reveal the importance of combining a suitable treatment with a good stimulation of the immune system.

Existence of entropy solutions to nonlinear parabolic problems with variable exponent and $$L^1$$ L 1 -data

Abstract: In the present paper we prove existence results of entropy solutions to a class of nonlinear parabolic p(.)-Laplacian problem with Neumann-type boundary conditions and $$L^1$$ data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

Elliptical flows perturbed by shear waves

Abstract: We consider the superimposition of two shear waves on a pseudo-plane motion of the first kind with elliptical streamlines. If the shear waves satisfy some special assumptions it is possible to establish a recurrence relation among the Rivlin–Ericksen tensors associated with the flow at hand. This remarkable kinematical result allows to determine new exact solutions for a large class of materials and to generalize some well known solutions modelling special flows (such as the celebrated Berker’s solution for a Navier–Stokes fluid in an orthogonal rheometer).

Final attack ratio in SIR epidemic models for multigroup populations

Abstract: We analyse an SIR epidemic model in a closed population subdivided in n groups. Population mixing occurs at two levels: within each group, and uniformly in the population. We prove that, if within-group transmission rates are large enough and not all identical to each other, then the final attack ratio is lower than what would occur in a population mixing homogeneously with the average transmission rate. We also show that the opposite may hold for certain parameter values and explore numerically the parameter regions in which the final attack ratio is higher or lower than in the corresponding homogeneous model. Finally, we analyse simulations of the corresponding stochastic model with finite group size, studying how well final attack ratio is approximated by the deterministic outcome and its relations with exponential growth rate.

On the operations of sequences in rings and binomial type sequences

Abstract: Given a commutative ring with identity R, many different and interesting operations can be defined over the set $$H_R$$ of sequences of elements in R. These operations can also give $$H_R$$ the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between $$H_R$$ equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

Approximation theorems for a family of multivariate neural network operators in Orlicz-type spaces

Abstract: In this paper, we study the theory of a Kantorovich version of the multivariate neural network operators. Such operators, are activated by suitable kernels generated by sigmoidal functions. In particular, the main result here proved is a modular convergence theorem in Orlicz spaces. As special cases, convergence theorem in $$L^p$$ -spaces, interpolation spaces, and exponential-type spaces can be deduced. In general, multivariate approximations by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of activation functions of sigmoidal-type for which the above theory holds have been described.

Dirichlet and Neumann eigenvalue problems on CR manifolds

Abstract: We study the properties of Carnot–Caratheodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point $$x \in M$$ , versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form $$\theta $$ on M. The weak Dirichlet problem for the sublaplacian $$\Delta _b$$ on $$(M, \theta )$$ is solved on domains $$\Omega \subset M$$ supporting the Poincare inequality. The solution to Neumann problem for the sublaplacian $$\Delta _b$$ on a $$C^{1,1}$$ connected $$(\epsilon , \delta )$$ -domain $$\Omega \subset {{\mathbb {G}}}$$ in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of $$\Delta _b$$ on $$\Omega \subset M$$ in a Carnot–Cartheodory complete pseudohermitian manifold $$(M, \theta )$$ .

Hyperbolic balance laws with dissipative source relaxing to adiabatic conservation laws

Abstract: The paper discusses the long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws with partial dissipation, when the relaxed system is adiabatic.