Theory Of Ordinary Differential Equations Research Articles

考虑媒体报道效应的谣言传播模型研究

Combined with the impacts of positive media coverage and negative publicity, a new rumor spreading model was provided. With the stability theory for ordinary differential equations, the existence and global stability of the equilibrium point between rumor spreading and no rumor were discussed for the model, respectively. The correctness of the theoretical analysis was verified through numerical simulation. The research results provide corresponding countermeasures and suggestions for the governmental management in public opinion monitoring and crisis decision making.

Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative

In this paper, we model an ecological system consisting of a predator and two preys with the newly derived two-step fractional Adams-Bashforth method via the Atangana-Baleanu derivative in the Caputo sense. We analyze the dynamical system for correct choice of parameter values that are biologically meaningful. The local analysis of the main model is based on the application of qualitative theory for ordinary differential equations. By using the fixed point theorem idea, we establish the existence and uniqueness of the solutions. Convergence results of the new scheme are verified in both space and time. Dynamical wave phenomena of solutions are verified via some numerical results obtained for different values of the fractional index, which have some interesting ecological implications.

Математическое моделирование экономической структуры общества на примере статистических данных по Пермскому краю

Monetary savings of the population are the subject of research of many specialists, since their detailed study in total with other various economic variables allows for a qualitative analysis of the current social and financial situation and make predictions. The definition of such an economic characteristic of society as the distribution of the population by money savings in practice is a very non-trivial task because of the lack of reliable information on the accumulation of citizens. Often, rather than a distributed value, a constant indicator is used – the average level of household savings. In the case when the savings are distributed according to the normal law in society (as in the developed countries), the replacement of the distributed quantity by a constant is possible to simplify the tasks. It is shown that Perm krai has bimodal structure of society savings, what essentially differs from the normal distribution law and therefore from the mathematical point of view it is unacceptable to replace one with a constant value, but it is necessary to take it into account as a distributed one. This ensures the relevance of this study, whose goal is to identify and analyze the numerical characteristics of the economic structure of Perm krai. The main idea of the work is to apply the mathematical model of the accumulation spectrum of the society of D.S. Chernavskii for Perm krai, the numerical calculation of the model and the economic analysis of the obtained characteristics. Similar numerical calculations of the distribution of the population of Perm krai on savings based on official statistics have not been conducted previously. Methods of economic analysis, mathematical and computer modeling, methods of theories of ordinary differential equations, partial differential equations and stochastic differential equations, probability theory and mathematical statistics are applied. A numerical study of the mathematical model of Perm krai economic society structure was carried out, the most probable stationary levels of population savings in Perm krai were calculated, which approximately amounted to 10 and 63 subsistence minimums for 2016 year. Families seem to accumulate in the vicinity of these savings values, like elementary particles under Brownian motion (this is not a coincidence, since both processes are described by the stochastic differential Fokker-Planck equation in partial derivatives of the parabolic type). Stationary savings levels numerically characterize the consumption standards and the standard of living of a society and are formed on the basis of the value of the consumer basket. In future, it is planned to conduct research on the optimal control of the economic structure of society. Keywords the economic society structure, population distribution density by savings, stationary levels of savings, spectrum of the savings, savings in the Perm Region, economic groups of the population, social stratification, Fokker – Planck equation, money balance equation, mathematical model of D.S. Chernavskii.

Dynamic behavior analysis of SIVS epidemic models with state-dependent pulse vaccination

To prevent and control the spread of infectious disease, in this paper, we formulate two susceptible–infected–vaccinated–susceptible (SIVS) epidemic models, where state-dependent pulse vaccination control strategies are introduced. The first model incorporates the density of infected individuals as the monitoring threshold value and some sufficient conditions for the existence and orbital stability of order-1 or order-2 periodic solutions are obtained by using the Poincaré map and qualitative theory of ordinary differential equations. Further, using the density of susceptible individuals as the monitoring threshold value, the second model with state-dependent pulse vaccination is proposed, and the existence and stability of the semi-trivial periodic solution and order-1 periodic solution can be obtained via utilizing the analog of the Poincaré criterion. Theoretical results imply that a state-dependent pulse vaccination strategy can eliminate the spread of infectious disease or keep the density of infected individuals at a desired low level for a long time. Finally, numerical simulations are given to verify the correctness of the theoretical results and to obtain the highest efficiency of state-dependent pulse control strategy.

The Degree of Compact Multivalued Perturbations of Fredholm Mappings of Positive Index and Its Application to a Certain Optimal Control Problem

Earlier a topological characteristic of the degree type for multivalued perturbations of Fredholm mappings with zero index was constructed and it was assumed that the multivalued perturbation permits a single-valued approximation. In this paper, a similar characteristic is constructed for multivalued perturbations of Fredholm mappings of positive index, and its application is given to the problem of existence of an optimal solution for the boundary-value problem in the theory of ordinary differential equations with feedback.

Modeling the effect of dust pollutants on plant biomass and their abatement from the near earth atmosphere

Scavenging of ground level dust pollution from the near earth atmosphere has been a challenging problem, causing various undesirable consequences to human beings and their ecosystem. Water sprays and plant biomass in a greenbelt can be very effective mechanisms for the mitigation of ground level dust pollution. In the present paper, a nonlinear mathematical model is proposed to investigate the impact of water sprays and plant biomass on the reduction of ground level dust pollution. Moreover, the effect of dust pollutants captured on plant biomass has also been discussed. In modeling process, we have considered three non-linearly dependent variables namely, the cumulative concentration of dust pollutants, the density of plant biomass and the number density of water sprays. Using stability theory of ordinary differential equations, local and global stability conditions of the model equilibria are obtained. Model analysis reveals that the equilibrium concentration of dust pollutants decreases as the rate of removal parameters increases. Further, it is also found that if the cumulative concentration of dust pollutants is less than its threshold concentration then there is no need to spray water droplets into the near earth atmosphere. Numerical simulation has also been performed to support the analytical findings.

Mathematical study of multispecies dynamics modeling predator–prey spatial interactions

Abstract In this work, we present analysis of a scaled time-dependent reaction–diffusion system modeling three competitive species dynamics that is of Lotka–Volterra type for coexistence, permanence and stability. The linear analysis is based on the application of qualitative theory of ordinary differential equations and dynamical systems. We consider two notable spatial discretization methods in conjunction with an adaptive time stepping method to verify the biological wave phenomena of the solutions and present the numerical results in one dimensional space. Adequate numerical resulting are provided in one and two dimensions to justify theoretical investigations. In addition, efficiency of the proposed numerical schemes are justified.

Existence of positive periodic solutions of some nonlinear fractional differential equations

The paper is devoted to study of existence and uniqueness of periodic solutions for a particular class of nonlinear fractional differential equations admitting its right-hand side with certain singularities. Our approach is based on Krasnosel’skiĭ and Schauder fixed point theorems and monotone iterative technique which enable us to extend some previously known results. The discussed problems are characterized by a Green’s function which has integrable singularities disallowing a direct use of classical techniques known from theory of ordinary differential equations, therefore proper modifications are proposed. Furher, the paper presents simple numerical algorithms directly built on the iterative technique used in theoretical proofs. Illustrative examples conclude the paper.

Partial mappings, Čech homology and ordinary differential equations

Consideration of partial mappings allows us to expose a noticeable part of the theory of ordinary differential equations at the axiomatic level. This gives us a possibility to cover equations with discontinuous right-hand sides and differential inclusions by standard well-known (but a little revised) theory. Homological properties of the corresponding solution sets allow us to create a new version of the shift method in the theory of boundary value problems as powerful as the application here of the Leray–Schauder theory.

Mathematical model and simulations of MERS outbreak: Predictions and implications for control measures

The Middle East Respiratory SyndromeВ (MERS) has been identified in 2012 and since thenВ outbreaks have been reported in various localities in theВ Middle East and in other parts of the world. To helpВ predict the possible dynamics of MERS, as well as waysВ to contain it, this paper develops a mathematical modelВ for the disease. It has a compartmental structure similarВ to SARS models and is in the form of a coupled systemВ of nonlinear ordinary differential equations (ODEs). TheВ model predictions are fitted to data from the outbreaksВ in Riyadh (Saudi Arabia) during 2013-2016. The resultsВ reveal that MERS will eventually be contained in the city.В However, the containment time and the severity of the outbreaks depend crucially on the contact coefficients andВ the isolation rate constant. When randomness is addedВ to the model coefficients, the simulations show that theВ model is sensitive to the scaled contact rate among peopleВ and to the isolation rate. The model is analyzed usingВ stability theory for ODEs and indicates that when usingВ only isolation, the endemic steady state is locally stableВ and attracting. Numerical simulations with parametersВ estimated from the city of Riyadh illustrate the analyticalВ results and the model behavior, which may have importantВ implications for the disease containment in the city. Indeed,В the model highlights the importance of isolation of infectedВ individuals and may be used to assess other controlВ measures. The model is general and may be used to analyzeВ outbreaks in other parts of the Middle East and other areas.

A Viral Product Diffusion Model to Forecast the Market Performance of Products

To investigate the diffusion of products in the market, this paper proposes a viral product diffusion model using an epidemiological approach. This model presents the process of product diffusion through the dynamics of human behaviors. Based on the stability theory of Ordinary Differential Equations, we demonstrate the conditions under which a product in the market persists or dies out eventually. Next, we use Google data to validate the model. Fitting results illustrate that the viral product diffusion model not only depicts the steady growth process of products, but also describes the whole diffusion process during which the products increase at the initial stage and then gradually decrease and sometimes even exhibit multiple peaks. This shows that the viral product diffusion model can be used to forecast the developing tendency of products in the market through early behavior of these products. Moreover, our model also provides useful insights on how to design effective marketing strategies via social contagions.

Qualitative Analysis of Critical Transitions in Complex Disease Propagation from a Dynamical Systems Perspective

Considerable evidence suggests that in complex diseases the deteriorations are often abrupt, and can be viewed as a bifurcation or a critical transition from one asymptotically stable equilibrium to another one at a tipping point. Here, we propose some new ideas to detect early-warning signals of such critical transitions from the perspective of qualitative theory of ordinary differential equations. Specifically, we theoretically derive three indicators that serve as a general early-warning signal indicating an imminent bifurcation or sudden deterioration before the critical transition occurs. Then, we verify our theoretical results by numerical simulations for three examples. Our work forms a starting point to motivate new mathematical insights into predictability for critical transitions of dynamical systems from a new perspective.

Elementary proof of convergence to the mean-field model for the SIR process.

The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (IMA J Appl Math, doi: 10.1093/imamat/hxw010 , 2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.

Dynamic responses of a beam with breathing cracks by precise integration method

The beam structure with breathing cracks subjected to harmonic excitations was modeled by FEM based on Euler-Bernoulli theory, and a piecewise dynamical system was deduced. The precise integration method (PIM) was employed to propose an algorithm for analyzing the dynamic responses of the deduced system. This system was first divided into linear sub-systems, between which there are switching points resulted from the breathing cracks. The inhomogeneous terms due to the external excitations were tackled by introducing auxiliary variables to express the harmonic functions, hence the sub-systems are homogeneous. The PIM was then applied to solve the homogeneous sub-systems one by one. During the procedures, a predictor-corrector algorithm was presented to determine the switching points accurately. The presented method can provide solutions with an accuracy to a magnitude of 10-12 compared with exact solutions obtained by the theories of ordinary differential equations. The PIM results are much more accurate than Newmark ones with the same time step. Moreover, it is found that the PIM can maintain a high level of accuracy even when the time step increases within a relatively wide range.

The osgood criterion and finite‐time cosmological singularities

In this paper, we apply Osgood's criterion from the theory of ordinary differential equations to detect finite‐time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti‐Chaplygin gas. In particular, we applied Osgood's criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We show that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finite‐time, Type 0 singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid such a finite‐time singularity. Osgood's criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models.

Geometric analysis of a pest management model with Holling’s type III functional response and nonlinear state feedback control

In this paper, based on a predator–prey model with Holling’s type III functional response, a pest management system with artificial interference is proposed. We assume that the artificial interference strategy will be taken to control pests when their number reaches a certain threshold. Based on this assumption, the artificial interference strategy of the system with nonlinear state feedback control is analyzed by using the geometric theory of ordinary differential equations. We first study the existence of periodic solutions of the model by successor functions and then the stability of periodic solutions. Finally, numerical simulations are given to illustrate our theoretic conclusions.

一类计算机病毒入侵的数学模型分析

本文研究一类具有标准发生率的SIR网络病毒传播模型，得到病毒入侵失败和成功的阈值及病毒存在情形下的平衡点的稳定性，在此基础上为控制病毒传播提出了一些策略。 This paper studies a network virus propagation SIR model with standard incidence rate. We get the threshold value which shows that when the virus will invade successfully and when it will fail. Further, using the balance theory of ordinary differential equations, we analyze the stability of the equilibrium and give some suggestions to control the spreading of the virus.

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h′(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1) h′(u) is constant k, (2) h′(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.

(Convex) Level Sets Integration

The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cyclically pseudoconvex multivalued map that sends vectors or “bundles” of a Euclidean space to convex sets in that space. That is, we are seeking a pseudoconvex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalizing the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations.

Maximum principle for optimal distributed control of viscous weakly dispersive Degasperis–Procesi equation

This paper is concerned with the optimal distributed control of the viscous weakly dispersive Degasperis–Procesi equation in nonlinear shallow water dynamics. It is well known that the Pontryagin maximum principle, which unifies calculus of variations and control theory of ordinary differential equations, sets up the theoretical basis of the modern optimal control theory along with the Bellman dynamic programming principle. In this paper, we commit ourselves to infinite dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. In contrast to the finite dimensional setting, the maximum principle for the infinite dimensional system does not generally hold as a necessary condition for optimal control. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled viscous weakly dispersive Degasperis–Procesi equation. The necessary optimality condition is established for the problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made. Copyright © 2015 John Wiley & Sons, Ltd.