Search Results (107)

A co-infection model for onchocerciasis and Lassa fever (OLF) with periodic variational vectors and optimal control is studied and analyzed to assess the impact of controls against incidence infections. The model is qualitatively examined in order to evaluate its asymptotic behavior in relation to the equilibria. Employing a Lyapunov function, we demonstrated that the disease-free equilibrium (DFE) is globally asymptotically stable; that is, the related basic reproduction number is less than unity. When it is bigger than one, we use a suitable nonlinear Lyapunov function to demonstrate the existence of a globally asymptotically stable endemic equilibrium (EE). Furthermore, the necessary conditions for the presence of optimum control and the optimality system for the co-infection model are established using Pontryagin’s maximum principle. The model is quantitatively analyzed by studying how sensitive the basic reproduction number is to the model parameters and the model simulation using Runge–Kutta technique of order 4 is also presented to study the effects of the treatments. We deduced from the quantitative analysis that, if there is an effective treatment and diagnosis of those exposed to and infected with the disease, the spread of the viral disease can be effectively managed. The results presented in this work will be useful for the proper mitigation of the disease. Full article

It is common in financial markets for market makers to offer prices on derivative instruments even though they are uncertain about the underlying asset’s value. This paper studies the mathematical problem that arises as a result. Derivatives are priced in the risk-neutral framework, so as the market maker acquires more information about the underlying asset, the change of measure for transition to the risk-neutral framework (the pricing kernel) evolves. This evolution takes a precise form when the market maker is Bayesian. It is shown that Bayesian updates can be characterized as additional informational drift in the underlying asset’s stochastic process. With Bayesian updates, the change of measure needed for pricing derivatives is two-fold: the first change is from the prior probability measure to the posterior probability measure, and the second change is from the posterior probability measure to the risk-neutral measure. The relation between the regular pricing kernel and the pricing kernel under this two-fold change of measure is characterized. Full article

The following differential quadratic polynomial differential system  $\frac{dx}{dt}=y-x, \frac{dy}{dt}=2y-\frac{y}{\gamma -1}\left(2-\gamma y-\frac{5\gamma -4}{\gamma -1}x\right),$ when the parameter $\gamma \in (1,2]$ models the structure equations of an isotropic star having a linear barotropic equation of state, being $x=m\left(r\right)/r$ where $m\left(r\right)\ge 0$ is the mass inside the sphere of radius r of the star, $y=4\pi r^2\rho$ where $\rho$ is the density of the star, and $t=ln(r/R)$ where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter $\gamma \in \mathbb{R}\setminus \left\{1\right\}$. Second, using the information of the different phase portraits obtained we classify the possible limit values of $m\left(r\right)/r$ and $4\pi r^2\rho$ of an isotropic star when r decreases. Full article

If a quantum field theory has a Landau pole, the theory is usually called ‘sick’ and dismissed as a candidate for an interacting UV-complete theory. In a recent study on the interacting 4d O(N) model at large N, it was shown that at the Landau pole, observables remain well-defined and finite. In this work, I investigate both relevant and irrelevant deformations of the said model at the Landau pole, finding that physical observables remain unaffected. Apparently, the Landau pole in this theory is benign. As a phenomenological application, I compare the O(N) model to QCD by identifying ${\Lambda }_{\overline{\mathrm{MS}}}$ with the Landau pole in the O(N) model. Full article

Modeling the vulnerabilities lifecycle and exploitation frequency are at the core of security of networks evaluation. Pareto, Weibull, and log-normal models have been widely used to model the exploit and patch availability dates, the time to compromise a system, the time between compromises, and the exploitation volumes. Random samples (systematic and simple random sampling) of the time from publication to update of cybervulnerabilities disclosed in 2021 and in 2022 are analyzed to evaluate the goodness-of-fit of the traditional Pareto and log-normal laws. As censoring and thinning almost surely occur, other heavy-tailed distributions in the domain of attraction of extreme value or geo-extreme value laws are investigated as suitable alternatives. Goodness-of-fit tests, the Akaike information criterion (AIC), and the Vuong test, support the statistical choice of log-logistic, a geo-max stable law in the domain of attraction of the Fréchet model of maxima, with hyperexponential and general extreme value fittings as runners-up. Evidence that the data come from a mixture of differently stretched populations affects vulnerabilities scoring systems, specifically the common vulnerabilities scoring system (CVSS). Full article

Our aim is to provide an insight into the procedures and the dynamics that lead the spread of contagious diseases through populations. Our simulation tool can increase our understanding of the spatial parameters that affect the diffusion of a virus. SIR models are based on the hypothesis that populations are “well mixed”. Our model constitutes an attempt to focus on the effects of the specific distribution of the initially infected individuals through the population and provide insights, considering the stochasticity of the transmission process. For this purpose, we represent the population using a square lattice of nodes. Each node represents an individual that may or may not carry the virus. Nodes that carry the virus can only transfer it to susceptible neighboring nodes. This important revision of the common SIR model provides a very realistic property: the same number of initially infected individuals can lead to multiple paths, depending on their initial distribution in the lattice. This property creates better predictions and probable scenarios to construct a probability function and appropriate confidence intervals. Finally, this structure permits realistic visualizations of the results to understand the procedure of contagion and spread of a disease and the effects of any measures applied, especially mobility restrictions, among countries and regions. Full article

The innate immune system is the first line of defense against pathogens. Its composition includes barriers, mucus, and other substances as well as phagocytic and other cells. The purpose of the present paper is to compare tissues with regard to their immune response to infections and to cancer. Simple ideas and the qualitative theory of differential equations are used along with general principles such as the minimization of the pathogen load and economy of resources. In the simplest linear model, the annihilation rate of pathogens in any tissue should be greater than the pathogen’s average replication rate. When nonlinearities are added, a stability condition emerges, which relates the strength of regular threats, barrier height, and annihilation rate. The stability condition allows for a comparison of immunity in different tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response. The way the tissue responds to an infection shows a correlation with the way it responds to cancer. The results of this paper are formulated in the form of precise statements in such a way that they could be checked by present-day quantitative immunology. Full article

New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating parameters computed using the Hermite interpolating polynomial. The corresponding R-order of convergence of the proposed uni- and bi-parametric with-memory methods is increased from 8 to 9 and 10, respectively. This increase in convergence order is accomplished without requiring additional function evaluations, making the with-memory method computationally efficient. The efficiency of our with-memory methods NWM9 and NWM10 increases from 1.6818 to 1.7320 and 1.7783, respectively. Numeric testing confirms the theoretical findings and emphasizes the superior efficacy of suggested methods when compared to some well-known methods in the existing literature. Full article

Max-C and min-D projection auto-associative fuzzy morphological memories (max-C and min-D PAFMMs) are two-layer feedforward fuzzy morphological neural networks designed to store and retrieve finite fuzzy sets. This paper addresses the main features of these auto-associative memories: unlimited absolute storage capacity, fast retrieval of stored items, few spurious memories, and excellent tolerance to either dilative or erosive noise. Particular attention is given to the so-called Zadeh’ PAFMM, which exhibits the most significant noise tolerance among the max-C and min-D PAFMMs besides performing no floating-point arithmetic operations. Computational experiments reveal that Zadeh’s max-C PFAMM, combined with a noise masking strategy, yields a fast and robust classifier with a strong potential for face recognition tasks. Full article

Computing Shapley values for large cooperative games is an NP-hard problem. For practical applications, stochastic approximation via permutation sampling is widely used. In the context of machine learning applications of the Shapley value, the concept of antithetic sampling has become popular. The idea is to employ the reverse permutation of a sample in order to reduce variance and accelerate convergence of the algorithm. We study this approach for the Shapley and Banzhaf values, as well as for the Owen value which is a solution concept for games with precoalitions. We combine antithetic samples with established stratified sampling algorithms. Finally, we evaluate the performance of these algorithms on four different types of cooperative games. Full article

In this study, we tackle the subject of interval quadratic equations and we aim to accurately determine the root enclosures of quadratic equations, whose coefficients constitute interval variables. This study focuses on interval quadratic equations that contain only one coefficient considered as an interval variable. The four methods reviewed here in order to solve this problem are: (i) the method of classic interval analysis used by Elishakoff and Daphnis, (ii) the direct method based on minimizations and maximizations also used by the same authors, (iii) the method of quantifier elimination used by Ioakimidis, and (iv) the interval parametrization method suggested by Elishakoff and Miglis and again based on minimizations and maximizations. We will also compare the results yielded by all these methods by using the computer algebra system Mathematica for computer evaluations (including quantifier eliminations) in order to conclude which method would be the most efficient way to solve problems relevant to interval quadratic equations. Full article

We review some general aspects about the Black–Scholes equation, which is used for predicting the fair price of an option inside the stock market. Our analysis includes the symmetry properties of the equation and its solutions. We use the Hamiltonian formulation for this purpose. Taking into account that the volatility inside the Black–Scholes equation is a parameter, we then introduce the Merton–Garman equation, where the volatility is stochastic, and then it can be perceived as a field. We then show how the Black–Scholes equation and the Merton–Garman one are locally equivalent by imposing a gauge symmetry under changes in the prices over the Black–Scholes equation. This demonstrates that the stochastic volatility emerges naturally from symmetry arguments. Finally, we analyze the role of the volatility on the decisions taken by the holders of the options when they use the solution of the Black–Scholes equation as a tool for making investment decisions. Full article

In this paper, we provide a first systematic treatment of binomial sum relations involving (generalized) Fibonacci and Lucas numbers. The paper introduces various classes of relations involving (generalized) Fibonacci and Lucas numbers and different kinds of binomial coefficients. We also present some novel relations between sums with two and three binomial coefficients. In the course of exploration, we rediscover a few isolated results existing in the literature, commonly presented as problem proposals. Full article

This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius of the associated adjacency matrices, which expresses the network topology. The epidemic threshold dynamics are analyzed, depending on the global dynamics of the network structure. The main contribution of this work is the relationship established between the epidemic threshold and the topological entropy of the Erdös-Rényi contact networks. In addition, a relationship between the basic reproduction number and the topological entropy is also stated. The trigger of the infectious state is studied, where the probability value of the stability of the infected state after the first instant, depending on the degree of the node in the seed set, is proven. Some numerical studies are included and illustrate the implementation of the probabilistic procedures introduced, complementing the discussion on the choice of the seed set. Full article

For linear optimization problems with a parametric objective, so-called parametric linear programs (PLP), we show that the optimal decision values are, under few technical restrictions, unimodal functions of the parameter, at least in the two-degrees-of-freedom case. Assuming that the parameter is random and follows a known probability distribution, this allows for an efficient algorithm to determe the quantiles of linear combinations of the optimal decisions. The novel results are demonstrated with probabilistic economic dispatch. For an example setup with uncertain fuel costs, quantiles of the resulting inter-regional power flows are computed. The approach is compared against Monte Carlo and piecewise computation techniques, proving significantly reduced computation times for the novel procedure. This holds especially when the feasible set is complex and/or extreme quantiles are desired. This work is limited to problems with two effective degrees of freedom and a one-dimensional uncertainty. Future extensions to higher dimensions could yield a key tool for the analysis of probabilistic PLPs and, specifically, risk management in energy systems. Full article