A mathematical model of dengue transmission with memory

In this paper, a novel hybrid compartmental model of the dengue transmission process is proposed and studied with memory and relapse between host-to-vector and vice versa. The memory and correlated learning system in the dengue models by using the fractional differential operators such as Riemann–Liouville and Caputo has been a fascinating area of research. A threshold parameter which is called basic reproduction number R 0 is investigated and calculated by next-generation technique. It’s also shows that if basic reproduction number R 0 1 , the disease-free equilibrium(DFE) is locally asymptotically stable(LAS) and if R 0 > 1 then, the DFE is unstable. It’s also found that the fractional-order α also depends upon R 0 . Therefore, if fractional-order α = 1 and R 0 > 1 , then dengue fever model doesn’t show Hopf-type bifurcation. Further, it’s also worth mentioning that although R 0 1 , the DFE E 0 may not be always stable but it’s necessary and the model shows a Hopf-type bifurcation. We employed the scheme of Adams–Bashforth–Moulton predictor-corrector to find an approximate the solution of the dengue model. The numerical simulation is carried out to validate the analytic solution.

Numerical solution of hybrid mathematical model of dengue transmission with relapse and memory via Adam–Bashforth–Moulton predictor-corrector scheme

The 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 $\mathcal{R}_{0}$
 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.

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

A Novel Decentralized Consensus-Based Tracking Control for Exploration of Saturn's Rings

We consider the asymptotic consensus problem for 2nd-order nonlinear multi-agent systems subject to predefined constraints for the system response, such as maximum overshoot or minimum convergence rate. We design a distributed discontinuous adaptive control protocol that guarantees that the inter-agent consensus errors evolve in a prescribed funnel and converge to zero. The multi-agent dynamics contain parametric and structural uncertainties, without boundedness or approximation/parametric factorization assumptions. The response of the closed-loop system is solely determined by the predefined funnel and is independent from the control gain selection. Finally, simulation results verify the theoretical findings.

Asymptotic Consensus of Unknown Nonlinear Multi-Agent Systems with Prescribed Transient Response

Funnel asymptotic tracking of nonlinear multi-agent systems with unmatched uncertainties

In this paper, the dynamical behaviors of an incommensurate fractional order Van der Pol oscillator are studied. First, based on the stability theory of fractional order systems, some sufficient conditions for the stability and Hpof-type bifurcation are given for such fractional order system. Then, the frequency and amplitude of periodic oscillations are determined by numerical simulations. It is shown that the frequency of oscillations gets smaller as the fractional order is increased, while the amplitude of oscillations increases with the fractional order. Finally, the synchronization of two coupled fractional order Van der Pol oscillators is analyzed with the help of the linear feedback control method. Numerical simulations are performed to verify the theoretical results.

http://ieeexplore.ieee.org/iel5/6373891/6389889/06389926.pdf

Hopf-type bifurcation and synchronization of a fractional order Van der Pol oscillator

This article investigates the finite-time distributed adaptive consensus for nonlinear uncertain multiagent systems (MASs). The performance of the consensus errors can be prespecified in a funnel sense. Three achievements make this work depart from available results on prespecified performance and funnel control: first, a novel error transformation is constructed to prespecify the performance, which avoids the singularity problem pointed out in the literature in the differentiation of the control law; second, the funnel control method is extended in a MAS setting by suitably modifying the backstepping technique in a power integrator sense; and third, it is shown that finite-time stability notions can be attained without extra complexity being involved in the design. Stability analysis and comparative simulations demonstrate the method.

Finite-Time Distributed Control of Nonlinear Multiagent Systems via Funnel Technique

Low-complexity control (i.e. control avoiding complex function approximation and dynamic adaptive laws) is important in small mobile robots, which cannot be equipped with high-end communication and control hardware. However, the literature has shown that attaining low-complexity control is challenging in the presence of formation constraints such as connectivity maintenance, safety and performance constraints. This work proposes a low-complexity control design (in the framework of funnel control) which is able to handle a large set of formation constraints, while merely relying on static nonlinear feedback, without any function approximation nor dynamic adaptation mechanism. A simulation study further illustrates the method as compared to the state of the art.

Low-Complexity Control of Nonholonomic Mobile Robots With Formation Constraints

The optimal control of the dynamical behavior of reaction-diffusion systems has been rarely studied. In this paper, we put forward a novel one-dimensional reaction-diffusion marine planktonic biological system depicted by partial differential equations, and propose a class of hybrid control optimization algorithms for this system. In the analysis, the local stability of the system is determined firstly. Then, the condition of appearances of Turing patterns induced by diffusion and the criterion of Hopf bifurcation induced by time delay are obtained, respectively. Next, select the appropriate controller parameters to optimize the dynamic behavior of the system. Finally, several numerical simulations are utilized to verify the correctness of the theoretical results.

Xiao Min

Papers

Funnel asymptotic tracking of nonlinear multi-agent systems with unmatched uncertainties

Hopf-type bifurcation and synchronization of a fractional order Van der Pol oscillator

Finite-Time Distributed Control of Nonlinear Multiagent Systems via Funnel Technique

Low-Complexity Control of Nonholonomic Mobile Robots With Formation Constraints

Turing Instability and Hopf Bifurcation Analysis for A Hybrid Control Biological System with One-Dimensional Diffusion