scholarly journals An Investigation on Analytical Properties of Delayed Fractional Order HIV Model: A Case Study

2021 ◽  
Vol 16 (1) ◽  
pp. 57-85
Author(s):  
M. Pitchaimani ◽  
A. Saranya Devi
Keyword(s):  
Fractional Order ◽  
Hiv Transmission ◽  
Equation Model ◽  
Model Parameters ◽  
Hiv Model ◽  
Proposed Model ◽  
Positivity Of Solutions ◽  
The Stability ◽  
Delay Differential ◽  
Spread Of Infection

In this manuscript, we design a fractional order delay differential equation model for HIV transmission with the implementation of three distinct therapies for three different infectious stages. We investigate the positivity of solutions, analyze the stability properties, followed by Hopf bifurcation analysis. To probe the parameters that expedite the spread of infection, uncertainty and sensitivity analysis were performed. The numerical review was carried out to substantiate our theoretical results. Our proposed model parameters have been calibrated to fit yearly data from Afghanistan, Australia, France, Italy, Netherlands and New Zealand.

scholarly journals Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2069
Author(s):  
Anton V. Kovalev ◽  
Evgeny A. Viktorov ◽  
Thomas Erneux
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Optical Feedback ◽  
Laser Pulses ◽  
Equation Model ◽  
Rate Equations ◽  
Differential Equation Model ◽  
The Stability ◽  
Delay Differential ◽  
Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

scholarly journals Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Haiping Ye ◽  
Yongsheng Ding
Keyword(s):  
Fractional Order ◽  
Equation Model ◽  
Viral Diversity ◽  
Human Immune System ◽  
Order Model ◽  
Hiv Model ◽  
Interior Equilibrium ◽  
Infected Cells ◽  
The One ◽  
Predictor Corrector

We introduce fractional order into an HIV model. We consider the effect of viral diversity on the human immune system with frequency dependent rate of proliferation of cytotoxic T-lymphocytes (CTLs) and rate of elimination of infected cells by CTLs, based on a fractional-order differential equation model. For the one-virus model, our analysis shows that the interior equilibrium which is unstable in the classical integer-order model can become asymptotically stable in our fractional-order model and numerical simulations confirm this. We also present simulation results of the chaotic behaviors produced from the fractional-order HIV model with viral diversity by using an Adams-type predictor-corrector method.

Differential-Delay Equation Model of Gene Expression: A Continuum Approach

2008 ◽  
Author(s):  
Anael Verdugo ◽  
Richard H. Rand
Keyword(s):  
Analytical Study ◽  
Equation Model ◽  
Differential Delay Equation ◽  
Differential Delay ◽  
Differential Integral Equation ◽  
Gene Regulatory ◽  
The Stability ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Nonlinear Delay

This paper presents an analytical study of the stability of the steady state solutions of a gene regulatory network with time delay. The system is modeled as a continuous network and takes the form of a nonlinear delay differential-integral equation coupled to an ordinary differential equation. Two examples are given in which the critical delay causing instability is computed.

scholarly journals Effect of time delay on a detritus-based ecosystem

2006 ◽  
Vol 2006 ◽  
pp. 1-28 ◽  
Author(s):  
Nurul Huda Gazi ◽  
Malay Bandyopadhyay
Keyword(s):  
Time Delay ◽  
Equilibrium Points ◽  
Equation Model ◽  
Model System ◽  
Trophic Levels ◽  
Dynamical Analysis ◽  
Discrete Time Delay ◽  
Growth Equations ◽  
The Stability ◽  
Delay Differential

Models of detritus-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with the dynamical analysis of a nonlinear model of a detritus-based ecosystem involving detritivores and predator of detritivores. We have obtained the criteria for local stability of various equilibrium points and persistence of the model system. Next, we have introduced discrete time delay due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. With delay differential equation model system we have studied the effect of time delay on the stability behaviour. Next, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Finally, the existence of Hopf-bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter.

scholarly journals Mathematical Modeling for Prediction Dynamics of the Coronavirus Disease 2019 (COVID-19) Pandemic, Quarantine Control Measures

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1404
Author(s):  
Din Prathumwan ◽  
Kamonchat Trachoo ◽  
Inthira Chaiya
Keyword(s):  
Endemic Equilibrium ◽  
Control Measures ◽  
Single Infection ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Expected Number ◽  
Susceptible Population ◽  
Proposed Model ◽  
The Stability ◽  
Disease Free

A mathematical model for forecasting the transmission of the COVID-19 outbreak is proposed to investigate the effects of quarantined and hospitalized individuals. We analyze the proposed model by considering the existence and the positivity of the solution. Then, the basic reproduction number (R0)—the expected number of secondary cases produced by a single infection in a completely susceptible population—is computed by using the next-generation matrix to carry out the stability of disease-free equilibrium and endemic equilibrium. The results show that the disease-free equilibrium is locally asymptotically stable if R0<1, and the endemic equilibrium is locally asymptotically stable if R0>1. Numerical simulations of the proposed model are illustrated. The sensitivity of the model parameters is considered in order to control the spread by intervention strategies. Numerical results confirm that the model is suitable for the outbreak that occurred in Thailand.

Fractional Modeling Approach of Course Evaluation

2011 ◽  
Vol 219-220 ◽  
pp. 130-134
Author(s):  
Chun Na Zhao ◽  
Min Hua Wu ◽  
Yu Zhao ◽  
Ying Shun Li ◽  
Li Ming Luo
Keyword(s):  
Fractional Order ◽  
Evaluation Model ◽  
Course Evaluation ◽  
Model Parameters ◽  
Fractional Order Systems ◽  
Proposed Model ◽  
Fractional Modeling ◽  
Information Engineering ◽  
Normal University ◽  
Evaluation Work

Course evaluation is necessary means to ensure the improvement of course construction level. A new course evaluation model is proposed based on the fractional order systems in this paper. An algorithm for linear fractional order systems founded on Grunwald–Letnikov’s fractional calculus definition is described. The fractional evaluation model is composed of fractional order and common coefficient. Model parameters can be determined by a large number of data and mathematical statistics method. The proposed model was applied to actual course evaluation work of Capital Normal University Information Engineering College. The practicability and effectiveness of the method have been validated.

scholarly journals A nonlinear fractional-order damage model of stress relaxation of net-like red soil

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mingwu Wang ◽  
Xinyu Xu ◽  
Qiuyan Liu ◽  
Yingxun Ding ◽  
Fengqiang Shen
Keyword(s):  
Stress Relaxation ◽  
Fractional Order ◽  
Damage Model ◽  
Red Soil ◽  
Analytical Models ◽  
Model Parameters ◽  
Nonlinear Characteristics ◽  
Loading Mode ◽  
Proposed Model ◽  
Analytical Result

AbstractIt is essential to precisely describe the nonlinear characteristics of the stress relaxation behavior to ensure the long-term stability of geotechnical structures in the net-like red soil. A novel damage model of variable fractional-order was discussed here to accurately analyze the progress of stress relaxation for the net-like red soil. Moreover, unsaturated triaxial experiments on stress relaxation under a step-loading mode were performed to identify model parameters and investigate the nonlinear relaxation characteristics of the net-like red soil. The feasibility and validity of the proposed model were furthermore verified by comparisons with the experimental results and fitting curves obtained from the Nishihara model and the generalized Kelvin model. Results show that the analytical result by the proposed model is consistent with the measured data, and the proposed model can better depict the nonlinear characteristics of stress relaxation relative to other analytical models. It can better exhibit the relaxation evolution of soil compared with the conventional models.

scholarly journals Modeling, analysis and numerical solution to malaria fractional model with temporary immunity and relapse

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Attiq ul Rehman ◽  
Ram Singh ◽  
Thabet Abdeljawad ◽  
Eric Okyere ◽  
Liliana Guran
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Modeling Analysis ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Temporary Immunity

AbstractThe present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number $\mathcal{R}_{0}$ R 0 is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

Effect of Regenerative Process on the Sample Stability of a Multiple Delay Differential Equation

2000 ◽  
Author(s):  
M. S. Fofana
Keyword(s):  
Chip Thickness ◽  
Fixed Time ◽  
Regenerative Process ◽  
Model Parameters ◽  
Combined Use ◽  
Linearized Stability ◽  
Sample Stability ◽  
The Stability ◽  
Delay Differential ◽  
Analytical Expressions

Abstract In this paper, explicit analytical expressions for the stability behaviour of a single degree of freedom turning model have been derived. The regenerative chatter due to chip thickness and feed rate variations, a delay in the spring stiffness coefficient and the probabilistic property of the displacement process of the chatter induced by the earlier tool cuts in the undeformed chip thickness have been taken into account. A characteristic equation for the linearized stability at equilibrium machining is presented, and regions of stable and unstable machining for multiple fixed time delays are captured in the parameter plane of two model parameters. By a combined use of the classical Hopf bifurcation theorem and the centre manifold, equations governing stochastic chattering, which are infinite in character, are reduced to two-dimensional ordinary differential equations. The integral averaging method and the Lyapunov exponent have been employed to explicitly derive the required analytical expressions.

scholarly journals Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Nasser Hassan Sweilam ◽  
Seham Mahyoub Al-Mekhlafi ◽  
Taghreed Abdul Rahman Assiri
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Numerical Study ◽  
Equilibrium Points ◽  
Fractional Delay ◽  
The Stability ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

A novel mathematical fractional model of multistrain tuberculosis with time delay memory is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the sense of the Grünwald–Letinkov definition. Modified parameters are introduced to account for the fractional order. The stability of the equilibrium points is investigated for any time delay. Nonstandard finite deference method is proposed to solve the resulting system of fractional-order delay differential equations. Numerical simulations show that nonstandard finite difference method can be applied to solve such fractional delay differential equations simply and effectively.

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