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Synchronization is one of the most striking instances of collective behavior, occurring in many natural phenomena. For example, in some ant species, ants are inactive within the nest most of the time, but their bursts of activity are highly synchronized and involve the entire nest population. Here we revisit a simulation model that generates this synchronized rhythmic activity through autocatalytic behavior, i.e., active ants can activate inactive ants, followed by a period of rest. We derive a set of delay differential equations that provide an accurate description of the simulations for large ant colonies. Analysis of the fixed-point solutions, complemented by numerical integration of the equations, indicates the existence of stable limit-cycle solutions when the rest period is greater than a threshold and the event of spontaneous activation of inactive ants is very unlikely, so that most of the arousal of ants is done by active ants. Furthermore, we argue that the persistent oscillations observed in the simulations for colonies of finite size are due to resonant amplification of demographic noise.

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Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S's). The model takes the form of a coupled system of nonlinear reaction-diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression.

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The fractional stochastic delay differential equation (FSDDE) is a powerful mathematical tool for modeling complex systems that exhibit both fractional order dynamics and stochasticity with time delays. The purpose of this study is to explore the stability analysis of a system of FSDDEs. Our study emphasizes the interaction between fractional calculus, stochasticity, and time delays in understanding the stability of such systems. Analyzing the moments of the system's solutions, we investigate stochasticity's influence on FSDDS. The article provides practical insight into solving FSDDS efficiently using various numerical techniques. Additionally, this research focuses both on asymptotic as well as Lyapunov stability of FSDDS. The local stability conditions are clearly presented and also the effects of a fractional orders with delay on the stability properties are examine. Through a comprehensive test of a stability criteria, practical examples and numerical simulations we demonstrate the complexity and challenges concern with the analyzing FSDDEs.

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The present study aimed to narrow such gaps by applying nonlinear differential equations to biostability in drinking water. Biostability results from the integrated dynamics of nutrients and disinfectants. The linear dynamics of biostability have been well studied, while there remain knowledge gaps concerning nonlinear effects. The nonlinear effects are explained by phase plots for specific scenarios in a drinking water system, including continuous nutrient release, flush exchange with the adjacent environment, periodic pulse disinfection, and periodic biofilm development. The main conclusions are, (1) The correlations between the microbial community and nutrients go through phases of linear, nonlinear, and chaotic dynamics. Disinfection breaks the chaotic phase and returns the system to the linear phase, increasing the microbial growth potential. (2) Post-disinfection after multiple microbial peaks produced via metabolism can increase disinfection efficiency and decrease the risks associated with disinfectant byproduct risks. This can provide guidelines for optimizing the disinfection strategy, according to the long-term water safety target or a short management. Limited disinfection and ultimate disinfection may be more effective and have low chemical risk, facing longer stagnant conditions. (3) Periodic biofilm formation and biofilm detachment increase the possibility of uncertainty in the chaotic phase. For future study, nonlinear differential equation models can accordingly be applied at the molecular and ecological levels to further explore more nonlinear regulation mechanisms.

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Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

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A four-dimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number R 0. Specifically, it shows that the disease-free equilibrium E 0 is globally asymptotically stable (GAS) for R 0 < 1, and globally attractive (GA) for R 0 = 1, while the endemic equilibrium E* is GAS and E 0 is unstable for R 0 > 1. Especially, to obtain the global stability of the equilibrium E* for R 0 > 1, the weak persistence of the model is proved by some analysis techniques.

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In this work, we briefly describe our technology developed for computing periodic solutions of time-delay systems and discuss the results of computing periodic solutions for the Marchuk-Petrov model with parameter values, corresponding to hepatitis B infection. We identified the regions in the model parameter space in which an oscillatory dynamics in the form of periodic solutions exists. The respective solutions can be interpreted as active forms of chronic hepatitis B. The period and amplitude of oscillatory solutions were traced along the parameter determining the efficacy of antigen presentation by macrophages for T- and B-lymphocytes in the model.. The oscillatory regimes are characterized by enhanced destruction of hepatocytes as a consequence of immunopathology and temporal reduction of viral load to values which can be a prerequisite of spontaneous recovery observed in chronic HBV infection. Our study presents a first step in a systematic analysis of the chronic HBV infection using Marchuk-Petrov model of antiviral immune response.

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In this paper we deal with two aspects of the Covid epidemic. The first is a phase change during the epidemic. The empirical observation is that once a certain threshold of active infections is reached, the rate of infection is increasing significantly. This threshold depends, among others, also on the season. We model this phenomenon as a jump in the coefficient of the virus exposition, giving the force of infection. In a chemical mass action law this coefficient corresponds to the reaction rate. We get a free boundary problem in time, which exhibits deterministic 'metastability'. In a population which is in a state of herd immunity, still, if the number of imported infections is large enough, an epidemic wave can start. The second aspect is the two scale nature of the infection network. On one hand side, there is always a finite number of reoccuring-deterministic-contacts, and on the other hand there is a large number of possible random contacts. We present a simple example, where the group size of deterministic contacts is two, and the graph of random contacts is complete.

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In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.

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Most plant viral infections are vector-borne. There is a latent period of disease inside the vector after obtaining the virus from the infected plant. Thus, after interacting with an infected vector, the plant demonstrates an incubation time before becoming diseased. This paper analyzes a mathematical model for persistent vector-borne viral plant disease dynamics. The backpropagated neural network based on the Levenberg-Marquardt algorithm (NN-BLMA) is used to study approximate solutions for fluctuations in natural plant mortality and vector mortality rates. A state-of-the-art numerical technique is utilized to generate reference data for obtaining surrogate solutions for multiple cases through NN-BLMA. Curve fitting, regression analysis, error histograms, and convergence analysis are used to assess accuracy of the calculated solutions. It is evident from our simulations that NN-BLMA is accurate and reliable.

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Fractional order systems of delay differential equations are very advantageous in analyzing the dynamics of various fields such as population dynamics, neural networking, ecology, and physiology. The aim of this paper is to present an implicit numerical scheme along with its error analysis to solve a fractional-order system of delay differential equations. The proposed method is an extension of the L1 numerical scheme and has the error estimate of O ( h 2 ) , where h denotes the step size. Further, we solve various non-trivial examples using the proposed method and compare the results with those obtained by some other established methods such as the fractional Adams method and the three-term new predictor-corrector method. We observe that the proposed method is more accurate as compared to the fractional Adams method and the new predictor-corrector method. Moreover, it converges for very small values of the order of fractional derivative.

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A deterministic model is proposed to describe the interaction between an immune system and an invading virus whose target cells circulate in the blood. The model is a system of two ordinary first order quadratic delay-differential equations with stipulated initial conditions, whose coefficients are eventually constant, so that the system becomes autonomous. The long-term behavior of the solution is investigated with some success. In particular, we find two simple functions of the parameters of the model, whose signs often, but not always, determine whether the virus persists above a nonzero threshold in the circulation or heads toward extinction.

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In the wake of the 2020 COVID-19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic and of disease models generally. Most works follow the susceptible-infected-removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID-19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult-to-estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model's ability to reproduce measured data on realistic problems.

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Stability in distribution for uncertain delay differential equations based on the strong Lipschitz condition only involving the current state has been successfully investigated. In reality, the uncertain delay differential equation is not only relate to the current state, but also relate to the past state, so it is very hard to obtain the strong Lipschitz condition. In this paper, the new Lipschitz condition concerning the current state and the past state is provided, if the uncertain delay differential equation satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz condition, conversely, it may not be established. By means of the new Lipschitz condition, a sufficient theorem for the uncertain delay differential equation being stable in distribution is proved. Meanwhile, a class of uncertain delay differential equation is certified to be stable in distribution without any limited condition. Besides, the effectiveness of the above sufficient theorem is verified by two numerical examples.

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We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of Arino et al. (J Theor Biol 241(1):109-119, 2006) by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition between two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, [Formula: see text], such that the evolutionary trend is for the delay to approach [Formula: see text].

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Tick infestation and tick-borne disease spread in a region of multiple adjacent patches with different environmental conditions depend heavily on the host mobility and patch-specific suitability for tick growth. Here we introduce a two-patch model where environmental conditions differ in patches and yield different tick developmental delays, and where feeding adult ticks can be dispersed by the movement of larger mammal hosts. We obtain a coupled system of four delay differential equations with two delays, and we examine how the dynamical behaviours depend on patch-specific basic reproduction numbers and host mobility by using singular perturbation analyses and monotone dynamical systems theory. Our theoretical results and numerical simulations provide useful insights for tick population control strategies.

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Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.

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Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown. Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model. We study the phase diagram and the bifurcations of the non-linear model, based on our conclusions on the linear one. Such dynamic analysis of the delay visco-elastic model in the presence of friction is absent in the literature for both linear and non-linear rheologies. Our work also shows how increasing values of some parameters such as delay and friction decrease its stability, while other parameters such as stiffness stabilise the oscillatory response.

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Many studies have shown that periodic erythrocytic (red blood cell linked) diseases are extremely rare in humans. To explain this observation, we develop here a simple model of erythropoiesis in mammals and investigate its stability in the parameter space. A bifurcation analysis enables us to sketch stability diagrams in the plane of key parameters. Contrary to some other mammal species such as rabbits, mice or dogs, we show that human-specific parameter values prevent periodic oscillations of red blood cells levels. In other words, human erythropoiesis seems to lie in a region of parameter space where oscillations exclusively concerning red blood cells cannot appear. Further mathematical analysis show that periodic oscillations of red blood cells levels are highly unusual and if exist, might only be due to an abnormally high erythrocytes destruction rate or to an abnormal hematopoietic stem cell commitment into the erythrocytic lineage. We also propose numerical results only for an improved version of our approach in order to give a more realistic but more complex approach of our problem.

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Starting from recent experimental observations of starlings and jackdaws, we propose a minimal agent-based mathematical model for bird flocks based on a system of second-order delayed stochastic differential equations with discontinuous (both in space and time) right-hand side. The model is specifically designed to reproduce self-organized spontaneous sudden changes of direction, not caused by external stimuli like predator's attacks. The main novelty of the model is that every bird is a potential turn initiator, thus leadership is formed in a group of indistinguishable agents. We investigate some theoretical properties of the model and we show the numerical results. Biological insights are also discussed.

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