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Discrete & Continuous Dynamical Systems - S

September 2020 , Volume 13 , Issue 9

Issue on delay differential equations: Theory, applications and new trends

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From the guest editors: "Delay Differential Equations: Theory, Applications and New Trends"
Fathalla A. Rihan, Yang Kuang and Gennady Bocharov
2020, 13(9): i-iv doi: 10.3934/dcdss.2020404 +[Abstract](107) +[HTML](50) +[PDF](379.29KB)
Dynamics of a model of tumor-immune interaction with time delay and noise
Lifeng Han, Changhan He and Yang Kuang
2020, 13(9): 2347-2363 doi: 10.3934/dcdss.2020140 +[Abstract](711) +[HTML](275) +[PDF](756.67KB)

We propose a model of tumor-immune interaction with time delay in immune reaction and noise in tumor cell reproduction. Immune response is modeled as a non-monotonic function of tumor burden, for which the tumor is immunogenic at nascent stage but starts inhibiting immune system as it grows large. Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and uncertainty in the tumor innate proliferation rate are studied by including delay and noise in the appropriate model terms. Stability, persistence and extinction of the tumor are analyzed. We find that delay and noise can both induce the transition from low tumor burden equilibrium to high tumor equilibrium. Moreover, our result suggests that the elimination of cancer depends on the basal level of the immune system rather than on its response speed to tumor growth.

Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays
Nikolay Pertsev, Konstantin Loginov and Gennady Bocharov
2020, 13(9): 2365-2384 doi: 10.3934/dcdss.2020141 +[Abstract](500) +[HTML](282) +[PDF](552.58KB)

We formulate a novel mathematical model to describe the development of acute HIV-1 infection and the antiviral immune response. The model is formulated using a system of delay differential- and integral equations. The model is applied to study the possibility of eradication of HIV infection during the primary acute phase of the disease. To this end a combination of analytical examination and computational treatment is used. The model belongs to the Wazewski differential equation systems with delay. The conditions of asymptotic stability of a trivial steady state solution are expressed in terms of the algebraic Sevastyanov–Kotelyanskii criterion. The results of the computational experiments with the model calibrated according to the available estimates of parameters suggest that a complete elimination of HIV-1 infection after acute phase of infection is feasible.

Bistability analysis of virus infection models with time delays
Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov and Gennady Bocharov
2020, 13(9): 2385-2401 doi: 10.3934/dcdss.2020166 +[Abstract](529) +[HTML](257) +[PDF](404.46KB)

Mathematical models with time delays are widely used to analyze the mechanisms of the immune response to virus infections and predict various therapeutic effects. Using the lymphocytic choriomeningitis virus infection model as an example, this work describes an original computational technology for searching the bistable regimes of such models. This technology includes numerical methods for finding all possible steady states at fixed values of parameters, for tracing these states along the parameters and for analyzing their stability.

A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies
Nasser Sweilam, Fathalla Rihan and Seham AL-Mekhlafi
2020, 13(9): 2403-2424 doi: 10.3934/dcdss.2020120 +[Abstract](692) +[HTML](358) +[PDF](651.21KB)

In this paper, we present an optimal control problem of fractional-order delay-differential model for cancer treatment based on the synergy between anti-angiogenic and immune cells therapies. The governed model consists of eighteen differential equations. A discrete time-delay is incorporated to represent the time required for the immune system to interact with the cancer cells, and fractional-order derivative is considered to reflect the memory and hereditary properties in the process. Two control variables for immunotherapy and anti-angiogenic therapy are considered to reduce the load of cancer cells. Necessary conditions that guarantee the existence and the uniqueness of the solution for the control problem have been considered. We approximate numerically the solution of the optimal control problem by solving the state system forward and adjoint system backward in time. Some numerical simulations are provided to validate the theoretical results.

Dynamics of solutions of a reaction-diffusion equation with delayed inhibition
Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov and Vitaly Volpert
2020, 13(9): 2425-2442 doi: 10.3934/dcdss.2020193 +[Abstract](492) +[HTML](245) +[PDF](1085.14KB)

Reaction-diffusion equation with a logistic production term and a delayed inhibition term is studied. Global stability of the homogeneous in space equilibrium is proved under some conditions on the delay term. In the case where these conditions are not satisfied, this solution can become unstable resulting in the emergence of spatiotemporal pattern formation studied in numerical simulations.

Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes
Ahmed Aghriche, Radouane Yafia, M. A. Aziz Alaoui and Abdessamad Tridane
2020, 13(9): 2443-2463 doi: 10.3934/dcdss.2020194 +[Abstract](412) +[HTML](285) +[PDF](1413.44KB)

Aedes aegypti (Ae. aegypti: mosquito) is a known vector of several viruses including yellow fever, dengue, chikungunya and zika. In the current paper, we present a delayed mathematical model describing the dynamics of Ae. aegypti. Our model is governed by a system of three delay differential equations modeling the interactions between three compartments of the Ae. aegypti life cycle (females, eggs and pupae). By using time delay as a parameter of bifurcation, we prove stability/switch stability of the possible equilibrium points and the existence of bifurcating branch of small amplitude periodic solutions when time delay crosses some critical value. We establish an algorithm determining the direction of bifurcation and stability of bifurcating periodic solutions. In the end, some numerical simulations are carried out to support theoretical results..

Oscillation criteria for second-order quasi-linear neutral functional differential equation
Osama Moaaz and Omar Bazighifan
2020, 13(9): 2465-2473 doi: 10.3934/dcdss.2020136 +[Abstract](552) +[HTML](344) +[PDF](284.17KB)

In this work, new sufficient conditions for oscillation of solution of second order neutral delay differential equation are established. One objective of our paper is to further simplify and complement some results which were published lately in the literature. In order to support our results, we introduce illustrating examples.

Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations
Hasib Khan, Cemil Tunc and Aziz Khan
2020, 13(9): 2475-2487 doi: 10.3934/dcdss.2020139 +[Abstract](616) +[HTML](310) +[PDF](406.09KB)

In this paper, we are dealing with singular fractional differential equations (DEs) having delay and \begin{document}$ \mho_p $\end{document} (\begin{document}$ p $\end{document}-Laplacian operator). In our problem, we Contemplate two fractional order differential operators that is Riemann–Liouville and Caputo's with fractional integral and fractional differential initial boundary conditions.The SFDE is given by

\begin{document}$ \zeta_1 $\end{document} is a continuous function and singular at \begin{document}$ t $\end{document} and \begin{document}$ x(t) $\end{document} for some values of \begin{document}$ t\in [0, 1] $\end{document}. The operator \begin{document}$ \mathcal{D}^{\gamma}, \, $\end{document} is Riemann–Liouville fractional derivative while \begin{document}$ \mathcal{D}^{\delta^*}, \mathcal{D}^{\kappa} $\end{document} stand for Caputo fractional derivatives and \begin{document}$ \delta^*, \, \gamma\in(1, 2] $\end{document}, \begin{document}$ n-1<\kappa\leq n, $\end{document} where \begin{document}$ n\geq3 $\end{document}. For the study of the EUS, fixed point approach is followed in this paper and an application is given to explain the findings.

Minimum energy compensation for discrete delayed systems with disturbances
Salma Souhaile and Larbi Afifi
2020, 13(9): 2489-2508 doi: 10.3934/dcdss.2020119 +[Abstract](464) +[HTML](266) +[PDF](410.27KB)

This work is devoted to the remediability problem for a class of discrete delayed systems. We investigate the possibility of reducing the disturbance effect with a convenient choice of the control operator. We give the main properties and characterization results of this concept, according to the delay and the observation. Then, under an appropriate hypothesis, we demonstrate how to find the optimal control which ensures the compensation of a disturbance measured through the observation (measurements, signals, ...). The discrete version of the wave equation, as well as the usual actuators and sensors, are examined. Numerical results are also presented.

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model
Majid Bani-Yaghoub, Chunhua Ou and Guangming Yao
2020, 13(9): 2509-2535 doi: 10.3934/dcdss.2020195 +[Abstract](453) +[HTML](269) +[PDF](1019.02KB)

The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays
Udhayakumar Kandasamy and Rakkiyappan Rajan
2020, 13(9): 2537-2559 doi: 10.3934/dcdss.2020137 +[Abstract](521) +[HTML](256) +[PDF](1792.7KB)

In this paper, the hopf bifurcation of a fractional-order octonion-valued neural networks with time delays is investigated. With this constructed model all the parameters would belong to the normed division algebra of octonians. Because of the non-commutativity of the octonians, the fractional-order octonian-valued neural networks can be decomposed into four-dimensional real-valued neural networks. Furthermore, the conditions for the occurrence of Hopf bifurcation for the considered model are firstly given by taking time delay as a bifurcation parameter. Also we investigate their bifurcation when the system loses its stability. Finally, we give one numerical simulation to verify the effectiveness of the our proposed method.

Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects
Baskar Sundaravadivoo
2020, 13(9): 2561-2573 doi: 10.3934/dcdss.2020138 +[Abstract](531) +[HTML](364) +[PDF](346.99KB)

This manuscript prospects the controllability analysis of non-instantaneous impulsive Volterra type fractional differential systems with state delay. By enroling an appropriate Grammian matrix with the assistance of Laplace transform, the conditions to obtain the necessary and sufficiency for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived using algebraic approach and Cayley-Hamilton theorem. A distinctive approach presents in the manuscript, i have taken non-instantaneous impulses into the fractional order dynamical system with state delay and studied the controllability analysis, since this not exists in the available source of literature. Inclusively, i have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures
Odo Diekmann, Francesca Scarabel and Rossana Vermiglio
2020, 13(9): 2575-2602 doi: 10.3934/dcdss.2020196 +[Abstract](472) +[HTML](270) +[PDF](902.22KB)

In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the \begin{document}$ \odot* $\end{document}-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the \begin{document}$ \odot* $\end{document}-shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system "mimics" the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the \begin{document}$ \odot* $\end{document}-shift operator and of the \begin{document}$ \odot* $\end{document}-generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research.

Functionally-fitted block $ \theta $-methods for ordinary differential equations
Jingwen Wu, Jintao Hu and Hongjiong Tian
2020, 13(9): 2603-2617 doi: 10.3934/dcdss.2020164 +[Abstract](427) +[HTML](244) +[PDF](346.26KB)

We propose a new family of functionally-fitted block \begin{document}$ \theta $\end{document}-methods for numerically solving ordinary differential equations which integrates a chosen set of linearly independent functions exactly. The advantage of such variable coefficient methods is that the basis functions can be chosen to exploit specific properties of the problem that may be known in advance. The basic theory for the proposed methods is established. First, we derive a sufficient condition to ensure the existence of the functionally-fitted block \begin{document}$ \theta $\end{document}-methods, and discuss the independence on integration time for a set of separable basis functions. We then obtain some basic characteristics of the methods by Taylor series expansions, and show that the order of accuracy of \begin{document}$ r $\end{document}-dimensional functionally-fitted block \begin{document}$ \theta $\end{document}-method is at least \begin{document}$ r $\end{document} for ordinary differential equations. Numerical experiments are conducted to illustrate the efficiency of the functionally-fitted block \begin{document}$ \theta $\end{document}-methods.

Numerical continuation and delay equations: A novel approach for complex models of structured populations
Alessia Andò, Dimitri Breda and Francesca Scarabel
2020, 13(9): 2619-2640 doi: 10.3934/dcdss.2020165 +[Abstract](619) +[HTML](292) +[PDF](477.2KB)

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.

2019  Impact Factor: 1.233

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