Stability and bifurcations in an epidemic model with varying immunity period

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.Comment: 16 pages, 5 figure

Modelling of deep wells thermal modes

Purpose. Investigation of various heat-exchange conditions influence of the tower liquid on the deep wells thermal conditions. Methods. Methods of heat-exchange processes mathematical modeling are used. On the basis of the developed scheme for calculation, the thermal condition in a vertical well with a concentric arrangement of the drill-string was investigated. It was assumed that the walls of the well are properly insulated, and there is no flow or loss of fluid. The temperature distribution in the Newtonian (water) and non-Newtonian (clay mud) liquid along the borehole was simulated taking into account changes in the temperature regime of rocks with depth. To verify the calculation method and determine the reliability of the results, a comparative analysis of the calculated and experimental data to determine the temperature of the drilling liquid in the well was performed. Findings. A mathematical model for the study of temperature fields along the well depth was proposed and verified. A steady-state temperature distribution along the borehole is obtained for various types (Newtonian or non-Newtonian) tower liquid, with a linear law of change in rocks temperature with depth. It has been established that the temperature of the liquid flow at the face of hole and at the exit to the surface depends on the type of liquid used and the flow regime. It has been established that due to thermal insulation of drill pipe columns, heat-exchange between the downward and upward flow is reduced, which leads to a decrease in the temperature of the downward flow at the face of hole, providing a more favorable temperature at the face, which contributes to better destruction of the rock and cooling the tool during drilling. Originality. The nature of temperature distribution and changes along the borehole under the steady-state mode of heat-exchange in a turbulent and structural flow regime for both Newtonian and non-Newtonian circulating liquid are revealed. Practical implications. The proposed mathematical model and obtained results can be used to conduct estimates of the thermal conditions of wells and the development of recommendations for controlling the intensity of heat-exchange processes in the well, in accordance with the requirements of a specific technology.Мета. Дослідження впливу різних умов теплообміну циркулюючої рідини на тепловий режим глибоких свердловин. Методика. Використано методи математичного моделювання процесів теплообміну. На основі розробленої схеми до розрахунку досліджувався тепловий режим у вертикальній свердловині з концентричним розташуванням бурильної колони. Передбачалося, що стінки свердловини належним чином ізольовані, приплив і втрати рідини відсутні. Моделювався розподіл температур у потоках ньютонівської (води) та неньютонівської (глинистого розчину) рідин уздовж стовбура свердловини з урахуванням зміни температурного режиму гірських порід з глибиною. Для верифікації методики розрахунку і визначення достовірності результатів був виконаний порівняльний аналіз розрахункових та експериментальних даних з визначення температури промивної рідини у свердловині. Результати. Запропонована і верифіційована математична модель для дослідження температурних полів з глибиною свердловини. Отримано стаціонарний розподіл температур уздовж стовбура свердловини для різних типів (ньютонівських або неньютонівських) циркулюючих рідин при лінійному законі зміни температури гірських порід з глибиною. Виявлено, що температура потоку рідини на вибої свердловини і на виході на денну поверхню залежить від типу використовуваної рідини і режиму течії. Встановлено, що за рахунок термоізоляції колони бурильних труб знижується теплообмін між низхідним і висхідним потоками, що призводить до зниження температури низхідного потоку на вибої свердловини, забезпечуючи більш сприятливий температурний режим на вибої, який сприяє кращому руйнування породи та охолодженню інструменту при бурінні. Наукова новизна. Виявлено характер розподілу та зміни температури вздовж стовбура свердловин при стаціонарному режимі теплообміну в турбулентному і структурному режимах течії як для ньютонівських, так і неньютонівських циркулюючих рідин. Практична значимість. Запропонована математична модель і отримані результати можуть використовуватися для проведення оціночних розрахунків теплових режимів свердловин та розробки рекомендацій з управління інтенсивністю теплообмінних процесів у свердловині відповідно до вимог конкретної технології.Цель. Исследование влияния различных условий теплообмена циркулирующей жидкости на тепловой режим глубоких скважин. Методика. Использованы методы математического моделирования процессов теплообмена. На основе разработанной схемы к расчету исследовался тепловой режим в вертикальной скважине с концентрическим расположением бурильной колоны. Предполагалось, что стенки скважины надлежащим образом изолированы, приток и потери жидкости отсутствуют. Моделировалось распределение температур в потоках ньютоновской (воды) и неньютоновской (глинистого раствора) жидкостей вдоль ствола скважины с учетом изменения температурного режима горных пород с глубиной. Для верификации методики расчета и определения достоверности результатов был выполнен сравнительный анализ расчетных и экспериментальных данных по определению температуры промывочной жидкости в скважине. Результаты. Предложена и верифицирована математическая модель для исследования температурных полей по глубине скважины. Получено стационарное распределение температур вдоль ствола скважины для различных типов (ньютоновских или неньютоновских) циркулирующих жидкостей при линейном законе изменения температуры горных пород с глубиной. Выявлено, что температура потока жидкости на забое скважины и на выходе на дневную поверхность зависит от типа используемой жидкости и режима течения. Установлено, что за счет термоизоляции колоны бурильных труб снижается теплообмен между нисходящим и восходящим потоками, что приводит к снижению температуры нисходящего потока на забое скважины, обеспечивая более благоприятный температурный режим на забое, который способствует лучшему разрушению породы и охлаждению инструмента при бурении. Научная новизна. Выявлен характер распределения и изменения температуры вдоль ствола скважин при стационарном режиме теплообмена в турбулентном и структурном режимах течения как для ньютоновских, так и неньютоновских циркулирующих жидкостей. Практическая значимость. Предложенная математическая модель и полученные результаты могут использоваться для проведения оценочных расчетов тепловых режимов скважин и разработки рекомендаций по управлению интенсивностью теплообменных процессов в скважине в соответствии с требованиями конкретной технологии.The authors thank the Institute of Geotechnical Mechanics named by N. Poljakov of National Academy of Sciences of Ukraine (Dnipro, Ukraine) for providing technical and informational support in this work

Dynamics of neural systems with discrete and distributed time delays

In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings

Control of unstable steady states in neutral time-delayed systems

We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between the control strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.Comment: 11 pages, 8 figure

Asymptotic properties of the spectrum of neutral delay differential equations

Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.Comment: 14 pages, 6 figure

Time-delayed models of gene regulatory networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems

A class of pairwise models for epidemic dynamics on weighted networks

In this paper, we study the $SIS$ (susceptible-infected-susceptible) and $SIR$ (susceptible-infected-removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both models start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with network simulation to evaluate the impact of different weight distributions on epidemic threshold and dynamics in general. For the $SIR$ dynamics, the basic reproductive ratio $R_0$ is computed, and we show that (i) for both network models $R_{0}$ is maximised if all weights are equal, and (ii) when the two models are equally matched, the networks with a random weight distribution give rise to a higher $R_0$ value. The models are also used to explore the agreement between the pairwise and simulation models for different parameter combinations

Time-delayed model of autoimmune dynamics

Among various environmental factors associated with triggering or exacerbating autoimmune response, an important role is played by infections. A breakdown of immune tolerance as a byproduct of immune response against these infections is one of the major causes of autoimmune disease. In this paper we analyse the dynamics of immune response with particular emphasis on the role of time delays characterising the infection and the immune response, as well as on interactions between different types of T cells and cytokines that mediate their behaviour. Stability analysis of the model provides insights into how different model parameters affect the dynamics. Numerical stability analysis and simulations are performed to identify basins of attraction of different dynamical states, and to illustrate the behaviour of the model in different regime

Modelling the effects of awareness-based interventions to control the mosaic disease of Jatropha curcas

Plant diseases are responsible for substantial and sometimes devastating economic and societal costs and thus are a major limiting factor for stable and sustainable agricultural production. Diseases of crops are particular crippling in developing countries that are heavily dependent on agriculture for food security and income. Various techniques have been developed to reduce the negative impact of plant diseases and eliminate the associated parasites, but the success of these approaches strongly depends on population awareness and the degree of engagement with disease control and prevention programs. In this paper we derive and analyse a mathematical model of mosaic disease of Jatropha curcas, an important biofuel plant, with particular emphasis on the effects of interventions in the form of nutrients and insecticides, whose use depends on the level of population awareness. Two contributions to disease awareness are considered in the model: global awareness campaigns, and awareness from observing infected plants. All steady states of the model are found, and their stability is analysed in terms of system parameters. We identify parameter regions associated with eradication of disease, stable endemic infection, and periodic oscillations in the level of infection. Analytical results are supported by numerical simulations that illustrate the behaviour of the model in different dynamical regimes. Implications of theoretical results for practical implementation of disease control are discussed