System Of Partial Integro-differential Equations Research Articles

An immunity-structured SEIRS epidemic model with variable infectivity and spatial heterogeneity

A mathematical model is proposed for the spread of an epidemic disease of agedependent infectivity through an asexual population with spatial heterogeneity, assuming that some individuals recover from the disease with temporary immunity, another part recover with permanent immunity, and the last part recover with no immunity. The demographic changes such as births and deaths due to natural causes and the chronological age of individuals are not taken into account. The model is based on a system of partial integro-differential equations including a differential equation to describe the evolution of individuals who have recovered with temporary immunity. The existence and uniqueness of the globally defined solution is proved, and its long-time behaviour is studied.

A stochastic-local volatility model with [formula omitted] jumps for pricing derivatives

We introduce a new mixed model unifying local volatility, pure stochastic volatility, and Le´vy type of jumps in this paper. Our model framework allows the pure stochastic volatility and the jump intensity to be functions of fast (and slowly) varying stochastic processes and the local volatility to be given in a constant elasticity of variance type of parametric form. We use asymptotic analysis to derive a system of partial integro-differential equations for the prices of European derivatives and use Fourier analysis to obtain an explicit formula for the prices. Our result is an extension of a stochastic-local volatility model from no jumps to Le´vy jumps and an extension of a multiscale stochastic volatility model with Le´vy jumps from the zero elasticity of variance to the non-zero elasticity of variance. We find that our model outperforms two benchmark models in view of fitting performance when the time-to-maturities of European options are relatively short and outperforms one benchmark model in terms of pricing time cost.

A de-singularized meshfree approach to default probability estimation under a regime-switching synchronous-jump tempered stable Lévy model

The default probability of a publicly-traded company is modeled by a synchronous-jump regime-switching model in the paper (Hainaut and Colwell, 2016). In this investigation, we first generalize the proposed Lévy model to a more general setting of tempered stable processes recently introduced into the finance literature. However, the resulting integro-partial differential operator suffers from a singularity, thus a general framework based on strictly positive-definite functions is proposed as a remedy to de-singularize the operator. We then analyze an efficient meshfree collocation method based on radial basis functions to approximate the solution of the corresponding system of partial integro-differential equations arising from the structural credit risk model. We prove that considering some regularity assumptions, the proposed method automatically de-singularizes the problem in the tempered stable case. Finally, we perform some numerical experiments using the proposed method on some standard examples from the literature which confirm the validity of our theoretical results and the stability of the numerical algorithm.

Thermoelasticity with temperature and microtemperatures with fading memory

In this paper, we investigate a model of poro-thermoelasticity with microtemperatures, where the behavior of the body is influenced by the history of both temperature and microtemperatures. Mathematically, this translates into a system of partial integro-differential equations. Under suitable condition on the tensors appearing in the model, we prove that the resulting system is well posed. In the one-dimensional case, the exponential decay of the energy is proved.

Second-order IMEX scheme for a system of partial integro-differential equations from Asian option pricing under regime-switching jump-diffusion models

Second-order IMEX scheme for a system of partial integro-differential equations from Asian option pricing under regime-switching jump-diffusion models

A Mathematical Study of the Influence of Hypoxia and Acidity on the Evolutionary Dynamics of Cancer

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells.

Time-consistent mean-variance investment with unit linked life insurance contracts in a jump-diffusion setting

We consider a time-consistent mean–variance portfolio selection problem of an insurer and allow for the incorporation of basis (mortality) risk. The optimal solution is identified with a Nash subgame perfect equilibrium. We characterize an optimal strategy as solution of a system of partial integro-differential equations (PIDEs), a so called extended Hamilton–Jacobi–Bellman (HJB) system. We prove that the equilibrium is necessarily a solution of the extended HJB system. Under certain conditions we obtain an explicit solution to the extended HJB system and provide the optimal trading strategies in closed-form. A simulation shows that the previously found strategies yield payoffs whose expectations and variances are robust regarding the distribution of jump sizes of the stock. The same phenomenon is observed when the variance is correctly estimated, but erroneously ascribed to the diffusion components solely. Further, we show that differences in the insurance horizon and the time to maturity of a longevity asset do not add to the variance of the terminal wealth.

Viscosity Solution of System of Integro-Partial Differential Equations with Interconnected Obstacles of Non-local Type Without Monotonicity Conditions

In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial growth. In our study, the main tool is the associated of reflected backward stochastic differential equations with jumps with interconnected obstacles for which we show the existence of a unique Markovian solution.

Semi-linear Poisson-mediated Flocking in a Cucker-Smale Model

We propose a family of compactly supported parametric interaction functions in the general Cucker-Smale flocking dynamics such that the mean-field macroscopic system of mass and momentum balance equations with non-local damping terms can be converted from a system of partial integro-differential equations to an augmented system of partial differential equations in a compact set. We treat the interaction functions as Green’s functions for an operator corresponding to a semi-linear Poisson equation and compute the density and momentum in a translating reference frame, i.e. one that is taken in reference to the flock’s centroid. This allows us to consider the dynamics in a fixed, flock-centered compact set without loss of generality. We approach the computation of the non-local damping using the standard finite difference treatment of the chosen differential operator, resulting in a tridiagonal system which can be solved quickly.

Population Balance Modeling with Coupled Agglomeration and Disintegration Processes for TiO2 Nanoparticles Formation and Experimental Validation

Particle size distribution of nanoparticles plays an important role in modelling many scientific and engineering problems. In this article, we proposed a Finite Volume Method (FVM) to model TiO2 nanoparticles formation using population balance equations (PBEs) by incorporating the simultaneous agglomeration and disintegration processes. The superposition of the PBEs for agglomeration and disintegration with different kernels leads to a system of partial-integro differential equations, which are numerically solved by using FVM. The precipitation of TiO2 nanoparticles in the batch reactor is studied experimentally as well as by numerical simulations based on Austin and Diemer disintegration kernels and Shear agglomeration kernel. Finally, the capability of the precipitation model is evaluated and the experimental results on particle sizes are compared with the numerical results.

Stability for a competing system of hierarchical age-structured populations

In this paper, we are concerned with the stability for a model in the form of system of integro-partial differential equations, which governs the evolution of two competing age-structured populations. The age-specified environment is incorporated in the vital rates, which displays the hierarchy of ages. By a non-zero fixed-point result, we show the existence of positive equilibria. Some conditions for the stability of steady states are derived by means of semigroup method. Furthermore, numerical experiments are also presented.

A hybrid thermostatted kinetic framework for the modeling of a hybrid multisource system with storage

The planning and development of a hybrid energy source network is based on an optimal analysis which takes into account the connection costs and the production facilities. This paper aims at modeling a hybrid multisource system by employing the methods of the thermostatted kinetic theory. Specifically a network of energy sources is allocated into a geographical domain, which is divided into different regions each of them containing different energy sources. A hybrid kinetic theory framework is proposed where the energy source is identified by a quality parameter which can attain discrete values, whereas the energy supplied by the source is modeled by defining a continuous variable. The mathematical framework consists into a system of partial integro-differential equations with quadratic type nonlinearities whose parameters are interaction rates and probability density functions. The storage system is also modeled by introducing an external force field coupled with a mathematical thermostat for ensuring the conservation of the activation energy of the energy network. The existence and uniqueness of the solution is investigated by employing fixed point arguments and integration along the characteristic curves. A critical analysis and research perspectives conclude the paper showing that the proposed thermostatted kinetic theory frameworks can be considered as general paradigms for the derivation of specific models.

A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps

In this paper, we propose the spectral element method to price European, digital, butterfly, American, discrete and continuous barrier options in a Markovian jump-diffusion regime-switching economy. The spectral element method discretisation is considered for the approximation of the spatial derivatives in a system of partial integro-differential equations and is chosen because it possesses spectral accuracy such that highly accurate option prices can be obtained using a small number of grid discretisation nodes. Essentially, the spectral element method consists of splitting the computational domain into as many elements as needed and approximating the basis functions by high-order orthogonal polynomials within each element. In order to sustain the high-order convergence in time, we also use an exponential time integration scheme to solve the semi-discrete system. Our numerical examples support our error analysis and indicate that the spectral element method converges exponentially for the values and the hedging parameters of the regime-dependent options. Therefore, the proposed scheme provides a viable alternative to the finite difference or finite element methods which usually converge only algebraically.

On the Cauchy Problem of Vectorial Thermostatted Kinetic Frameworks

This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper.

Bounded and multiperiodic solutions of the system of partial integro-differential equations

Bounded and multiperiodic solutions of the system of partial integro-differential equations

Asymptotics of Solutions in the Problem about Small Motions of a Compressible Maxwell Fluid

A model of a viscoelastic compressible Maxwell fluid is studied. This model is described by a system of partial integro-differential equations with appropriate boundary and initial conditions. An abstract analog of the problem under study is also considered. It is proved that a uniformly exponentially stable C0-semigroup emerges in this problem. Based on this fact, an estimate for the solution of the evolution problem is derived in the case where the external load is close to being almost periodic.

Space dependent models for studying the spread of some diseases

In this paper we present some extensions of the classical SIR model with non-symmetric spatial dependence. SIR-type models usually describe the epidemic in a population, which is split into three categories, namely the susceptibles (S), the infected (I) and the recovered (R). The proposed model yields a system of partial integro-differential equations. Two methods handling the integrals of the equations are presented. We give numerical examples which show that the discrete models preserve the basic qualitative properties of the original biological process, and these results are supported by the oretical theorems.

A lipid-structured model for macrophage populations in atherosclerotic plaques

Atherosclerosis is a chronic inflammatory disease driven by the accumulation of pro-inflammatory, lipid-loaded macrophages at sites inside artery walls. These accumulations lead to the development of atherosclerotic plaques. The rupture of plaques that contain lipid-rich necrotic cores can trigger heart attacks and strokes via occlusion of blood vessels. We construct and analyse a system of partial integro-differential equations that model lipid accumulation by macrophages, the generation of apoptotic cells and the formation of the necrotic core. The model accounts for the following cell behaviours: monocyte recruitment into the plaque and differentiation into macrophages; macrophage ingestion of low density lipoproteins (LDL) and of apoptotic cells and necrotic material; lipid offloading to high density lipoproteins (HDL); macrophage emigration; and apoptosis of macrophages and necrosis of apoptotic cells. With this model, we study how changes in parameters predict the characteristic features of plaque pathology. In particular, we find the qualitative form of lipid distribution across the macrophage population and show that high lipid loads can occur in the absence of LDL ingestion. We also demonstrate the importance of macrophage emigration in mitigating and resolving inflammation and plaque lipid accumulation.

Localized kernel-based approximation for pricing financial options under regime switching jump diffusion model

In this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.

Regularized asymptotic solutions of the initial problem for the system of integro-partial differential equations

The Lomov regularization method [1] is generalized to integro-partial differential equations. It turns out that the regularization procedure essentially depends on the type of integral operator. The case in which the upper limit of the integral is not the differentiation variable is the most difficult one. It is not considered in the present paper. Only the case in which the upper limit of the integral operator coincides with the differentiation variable is studied. For such problems, an algorithm for constructing regularized asymptotics is developed.