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Using delay differential equations to study mathematical models of Parkinson's disease and Huntington's disease is important to show how important it is for synchronization between basal ganglia loops to work together. We used the delay circuit RLC (resistor, inductor, capacitor) model to show how the direct pathway and the indirect pathway in the basal ganglia excite and inhibit the motor cortex, respectively. A term has been added to the mathematical model without time delay in the case of the hyperdirect pathway. It is proposed to add a non-linear term to adjust the synchronization. We studied Hopf bifurcation conditions for the proposed models. The desynchronization of response times between the direct pathway and the indirect pathway leads to different symptoms of Parkinson's disease. Tremor appears when the response time in the indirect pathway increases at rest. The simulation confirmed that tremor occurs and the motor cortex is in an inhibited state. The direct pathway can increase the time delay in the dopaminergic pathway, which significantly increases the activity of the motor cortex. The hyperdirect pathway regulates the activity of the motor cortex. The simulation showed bradykinesia occurs when we switch from one movement to another that is less exciting for the motor cortex. A decrease of GABA in the striatum or delayed excitation of the substantia nigra from the subthalamus may be a major cause of Parkinson's disease. An increase in the response time delay in one of the pathways results in the chaotic movement characteristic of Huntington's disease. [ABSTRACT FROM AUTHOR]

The relaxation self-oscillations of the Mackey–Glass equation are studied under the assumption that the exponent in the nonlinearity denominator is a large parameter. We consider the case where the limit relay equation, which arises as the large parameter tends to infinity, has a periodic solution with the smallest number of breaking points on the period. In this case, we prove the existence of a periodic solution of the Mackey–Glass equation that is asymptotically close to the periodic solution of the limit equation. [ABSTRACT FROM AUTHOR]

We introduce an innovative postprocessing technique aimed at refining the accuracy of the discontinuous Galerkin method for solving linear delay differential equations (DDEs) with vanishing delays. The fundamental idea behind this postprocessing technique is to enhance the discontinuous Galerkin solution of degree k by incorporating a generalized Jacobi polynomial of degree k + 1 . We demonstrate that this postprocessing step enhances convergence by one order under the L ∞ -norm. Moreover, we apply this technique to both nonlinear DDEs with vanishing delays and linear DDEs with non-vanishing delays. We further validated the theoretical results through a series of numerical examples. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic came with many setbacks, be it to a country's economy or the global missions of organizations like WHO, UNICEF or GTFCC. One of the setbacks is the rise in cholera cases in developing countries due to the lack of cholera vaccination. This model suggested a solution by introducing another public intervention, such as adding Chlorine to water bodies and vaccination. A novel delay differential model of fractional order was recommended, with two different delays, one representing the latent period of the disease and the other being the delay in adding a disinfectant to the aquatic environment. This model also takes into account the population that will receive a vaccination. This study utilized sensitivity analysis of reproduction number to analytically prove the effectiveness of control measures in preventing the spread of the disease. This analysis provided the mathematical evidence for adding disinfectants in water bodies and inoculating susceptible individuals. The stability of the equilibrium points has been discussed. The existence of stability switching curves is determined. Numerical simulation showed the effect of delay, resulting in fluctuations in some compartments. It also depicted the impact of the order of derivative on the oscillations. [ABSTRACT FROM AUTHOR]

In this paper, a delay - induced pest control model is proposed. We have introduced a time delay in healthy trees and whitefly population in the infected tree density of the proposed system of equations to reduce the probability of healthy trees becoming infected, as well as the level of infection. We have analyzed the impact of time delay on the stability of the equilibrium and establish requirements to verify its asymptotic stability over all delays. The solutions of this system of non-linear ordinary differential equations(ODEs) and delay differential equations(DDEs) are presented by using homotopy perturbation method(HPM). Numerical simulation is also obtained for the same model of both ODE and DDE by using MATLAB software. The proposed method works very well and is easy to use, as shown by these findings. Our aim is to control the spread of whitefly such that harvest is not affected. [ABSTRACT FROM AUTHOR]

This study is concerned with finding numerical solutions of nonlinear delay differential equations involving extended Mittag-Leffler fractional derivatives of the Caputo-type. The main benefit of the used extension is to address the complexity resulting from the limitations of using fractional derivatives with non-singular Mittag-Leffler kernels. We discussed the existence and uniqueness of solutions for the studied delay models. Next, we modified an Adams-type method to numerically solve fractional delay differential equations combined with Mittag-Leffler kernels. A new type of solution belonging to the L 1 space is presented for the studied models using the proposed scheme. [ABSTRACT FROM AUTHOR]

We study the oscillation of a first-order delay equation with negative feedback at the critical threshold . We apply a novel center manifold method, proving that the oscillation of the delay equation is equivalent to the oscillation of a -dimensional system of ordinary differential equations (ODEs) on the center manifold. It is well known that the delay equation oscillation is equivalent to the oscillation of a certain second-order ODE, and we furthermore show that the center manifold system is asymptotically equivalent to this same second-order ODE. In addition, the center manifold method has the advantage of being applicable to the case where the parameters oscillate around the critical value , thereby extending and refining previous results in this case. [ABSTRACT FROM AUTHOR]

We generalize the model proposed in [Adimy, Babin, Pujo-Menjouet, SIAM Journal on Applied Dynamical Systems (2022)] for prion infection to a network of neurons. We do so by applying a so-called multigroup approach to the system of Delay Differential Equations (DDEs) proposed in the aforementioned paper. We derive the classical threshold quantity $ \mathcal{R}_0 $, i.e. the basic reproduction number, exploiting the fact that the DDEs of our model qualitatively behave like Ordinary Differential Equations (ODEs) when evaluated at the Disease Free Equilibrium. We prove analytically that the disease naturally goes extinct when $ \mathcal{R}_0<1 $, whereas it persists when $ \mathcal{R}_0>1 $. We conclude with some selected numerical simulations of the system, to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convectiondiffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem's asymptotic analysis. The Euler's method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation parameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of µ (free parameter, when the free parameter µ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature. [ABSTRACT FROM AUTHOR]

This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem’s asymp-totic analysis. The Euler’s method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation pa-rameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of μ (free param-eter, when the free parameter μ tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature.

Abstract The COVID-19 pandemic came with many setbacks, be it to a country’s economy or the global missions of organizations like WHO, UNICEF or GTFCC. One of the setbacks is the rise in cholera cases in developing countries due to the lack of cholera vaccination. This model suggested a solution by introducing another public intervention, such as adding Chlorine to water bodies and vaccination. A novel delay differential model of fractional order was recommended, with two different delays, one representing the latent period of the disease and the other being the delay in adding a disinfectant to the aquatic environment. This model also takes into account the population that will receive a vaccination. This study utilized sensitivity analysis of reproduction number to analytically prove the effectiveness of control measures in preventing the spread of the disease. This analysis provided the mathematical evidence for adding disinfectants in water bodies and inoculating susceptible individuals. The stability of the equilibrium points has been discussed. The existence of stability switching curves is determined. Numerical simulation showed the effect of delay, resulting in fluctuations in some compartments. It also depicted the impact of the order of derivative on the oscillations.

In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit. [ABSTRACT FROM AUTHOR]

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation. [ABSTRACT FROM AUTHOR]

Insect growth regulators (IGRs) have been developed as effective control measures against harmful insect pests to disrupt their normal development. This study is to propose a mathematical model to evaluate the cost-effectiveness of IGRs for pest management. The key features of the model include the temperature-dependent growth of insects and realistic impulsive IGRs releasing strategies. The impulsive releases are carefully modeled by counting the number of implements during an insect’s temperature-dependent development duration, which introduces a surviving probability determined by a product of terms corresponding to each release. Dynamical behavior of the model is illustrated through dynamical system analysis and a threshold-type result is established in terms of the net reproduction number. Further numerical simulations are performed to quantitatively evaluate the effectiveness of IGRs to control populations of harmful insect pests. It is interesting to observe that the time-changing environment plays an important role in determining an optimal pest control scheme with appropriate release frequencies and time instants. [ABSTRACT FROM AUTHOR]

This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall's inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp. [ABSTRACT FROM AUTHOR]

The menstrual cycle of fertile female is regulated by multiple organs and hormones, controlling the reproductive health and fertility of females. This paper is dedicated to exploring how two negative feedback loops regulate and influence the dynamic behavior of the menstrual cycle. We simplified the regulation circuit of the menstrual cycle into a nonlinear differential equation with two time delays. This simplified mathematical model integrates the hypothalamus–pituitary–ovary primary negative feedback loop and the negative feedback auxiliary loop between the ovary and the pituitary. We investigated the delay-dependent conditions for the stability of the model’s equilibrium points and also demonstrated the presence of Hopf bifurcations. Theoretical analysis suggests that the primary feedback loop is the major factor driving oscillation in the system. Further numerical simulations indicate that the negative feedback auxiliary loop allows the system to adjust its amplitude while maintaining the robustness of the cycle length. Consequently, the results obtained from this model provide new insights into the regulation of oscillations in menstrual cycle irregularities. [ABSTRACT FROM AUTHOR]

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. [ABSTRACT FROM AUTHOR]

Abstract.This paper chiefly takes into account the singularly perturbed delay Volterra-Fredholm integro-differential equations by numerically. In this context, firstly, priori estimates are given and a new discretization is constructed on piecewise-equidistant mesh by using interpolating quadrature rules [2] and composite integration formulas. Then, the convergence analysis and stability bounds of the presented method are discussed. Finally, numerical results are demonstrated with two test problems. [ABSTRACT FROM AUTHOR]

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically. [ABSTRACT FROM AUTHOR]

In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory. [ABSTRACT FROM AUTHOR]

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the $ L_\infty $ L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper. [ABSTRACT FROM AUTHOR]

Photovoltaic systems convert solar irradiance into electricity. Due to some factors, the amount of solar irradiance arriving at the solar photovoltaic collector at a specific location varies. The goal of this study was to develop a mathematical model for predicting the performance of a photovoltaic system, which depends on the amount of solar irradiance. A novel model for solar irradiance in the form of a delay differential equation is introduced by including the factor of delayed solar irradiance, hour angle and the sun’s motion. The simulation study is carried out for the three scenarios of weather conditions: a clear day, a slightly cloudy day, and a heavily overcast day. The numerical solution is obtained by adopting the 4th-order Runge Kutta method coupled with a parameter fitting technique, the Nelder Mead algorithm, which is implemented by using MATLAB software. The data from a solar plant in Pahang, Malaysia, was used for model validation and it is found that the prediction profile for solar irradiance aligns well with the intermediate and decay phases, but deviates slightly during the growth phase. The output current and power for the solar photovoltaic panel were treated as time-dependent functions. As the solar irradiance increases, the output current and power of the solar panel will increase. The result showed that the maximum output current and output power of STP250S-20/Wd crystalline solar module decreased by 42% and 76%, respectively, during slightly cloudy and heavily overcast conditions when compared to clear days. In other words, the performance of a photovoltaic module is better on clear days compared to cloudy days and heavily overcast. These findings highlight the relationship between delayed solar irradiance and the performance of the solar photovoltaic system. [ABSTRACT FROM AUTHOR]

Purpose: In this article, the transverse vibration motion of a self-excited beam which is subjected to a distributed and a singular load is analyzed using the differential transformation method (DTM). Methods: The Euler–Bernoulli beam model is employed. The beam is modeled to represent the cutting tool holder motion in machining. A delayed distributed load and a vibration velocity-dependent singular load are considered as forcing. Analysis is performed for different time delays, widths of distributed load, and beam lengths in the time domain. The Laplace transform method is deployed for the stability analysis. Multi-step DTM is applied for the mathematical solution. Matlab® ddesd solutions is used for mathematical comparison. Results: When the width of the distributed load increases and then the vibration amplitude increases. An increase in the beam length causes the amplitude to increase. The vibration amplitude increases as the delay time decreases. However, the reduction of some delay values reduces the amplitude. Conclusion: The equation that expresses the variation of the distributed load width with respect to the delay time is compatible results in accordance with the experimental studies in the literature. If beam length increases, the stability region of the width of the distributed load will decrease. The effect of beam length for stability can be adjusted by changing the time delay. [ABSTRACT FROM AUTHOR]

We construct and analyze a second-order parameter uniform numerical method for parabolic singularly perturbed space-delay problems with interior turning point. The considered problem's solution possesses an interior layer in addition to twin boundary layers due to the presence of delay. Some theoretical estimates on derivatives of the analytical solution, which are useful for conducting the error analysis, are given. The proposed technique employs an upwind scheme on a fitted Bakhvalov–Shishkin mesh in the spatial variable and implicit-Euler scheme on a uniform mesh in the time variable. This discretization of the problem is shown to be uniformly convergent of O (Δ τ + − 1) , where Δ τ is the step size in the temporal direction and K denotes the number of mesh-intervals in the spatial direction. Further, to improve the accuracy, we make use of Richardson extrapolation and establish parameter-uniform convergence of O ((Δ τ) 2 + − 2) for the resulting scheme. Numerical experiments are performed over two test problems for validation of the theoretical predictions. [ABSTRACT FROM AUTHOR]

Hepatitis B is a liver disease caused by the human hepatitis B virus (HBV). Mathematical models help further the understanding of the processes involved and help make predictions. The basic reproduction number, R0, is an index that predicts whether the disease will be chronic or not. This is the single most-important information that a mathematical model can give. Within-host virus processes involve delays. We study two within-host hepatitis B virus infection models without and with delay. One is a standard one, and the other considering additional processes and with two delays is new. We analyze the basic reproduction number and alternative threshold indices. The values of R0 and the alternative indices change depending on the model. All these indices predict whether the infection will persist or not, but they do not give the same rate of growth of the infection when it is starting. Therefore, the choice of the model is very important in establishing whether the infection is chronic or not and how fast it initially grows. We analyze these indices to see how to decrease their value. We study the effect of adding delays and how the threshold indices depend on how the delays are included. We do this by studying the local asymptotic stability of the disease-free equilibrium or by using an equivalent method. We show that, for some models, the indices do not change by introducing delays, but they change when the delays are introduced differently. Numerical simulations are presented to confirm the results. Finally, some conclusions are presented.

This paper presents the computational analysis of fractional differential equations of variable‐order delay systems. To the proposed problem, the existence of solutions is derived using Arzela‐Ascoli theorem, and the Banach fixed point theorem is used for uniqueness results. To investigate and address the computational solutions, Adams‐Bashforth‐Moulton technique is established. To demonstrate the method's efficiency, computational simulations of chaotic behaviors in several one‐dimensional delayed systems with distinct variable orders are employed. The numerical solution of the proposed problem gives high precision approximations. [ABSTRACT FROM AUTHOR]

In this work, we propose a numerical method for solving a singularly perturbed convection-diffusion problem that involves a time delay term. A priori bounds and properties of the continuous solution are discussed. Using the backward Euler method for the time derivative term, the problem is approximated by a set of singularly perturbed boundary value problems. Then, using a higher-order finite difference method, the boundary value problem is approximated on a piecewise uniform Shishkin mesh. The stability analysis of the method is studied using the comparison principle and discrete solution bounds. We proved that the proposed scheme is uniformly convergent, with an order of convergence of almost two in space and one in time. Two numerical examples are considered to validate the applicability of the proposed scheme. The proposed scheme has better accuracy than some schemes in the literature. [ABSTRACT FROM AUTHOR]

Hepatitis B is a liver disease caused by the human hepatitis B virus (HBV). Mathematical models help further the understanding of the processes involved and help make predictions. The basic reproduction number, R 0 , is an index that predicts whether the disease will be chronic or not. This is the single most-important information that a mathematical model can give. Within-host virus processes involve delays. We study two within-host hepatitis B virus infection models without and with delay. One is a standard one, and the other considering additional processes and with two delays is new. We analyze the basic reproduction number and alternative threshold indices. The values of R 0 and the alternative indices change depending on the model. All these indices predict whether the infection will persist or not, but they do not give the same rate of growth of the infection when it is starting. Therefore, the choice of the model is very important in establishing whether the infection is chronic or not and how fast it initially grows. We analyze these indices to see how to decrease their value. We study the effect of adding delays and how the threshold indices depend on how the delays are included. We do this by studying the local asymptotic stability of the disease-free equilibrium or by using an equivalent method. We show that, for some models, the indices do not change by introducing delays, but they change when the delays are introduced differently. Numerical simulations are presented to confirm the results. Finally, some conclusions are presented. [ABSTRACT FROM AUTHOR]

In the present paper, new oscillation criteria are established for fourth-order delay differential equations of the form (a 3 (t) (a 2 (t) a 1 (t) x ′ (t) ′ ) ′ ) ′ + b (t) x α (σ (t)) = 0 under the assumption (noncanonical) ∫ t 0 ∞ 1 a j (t) d t < ∞ , j = 1 , 2 , 3. We convert the equation into a canonical type, utilize the comparison method, and the Riccati transformation to find sufficient conditions for oscillation of all solutions to the aforementioned problem. This approach greatly simplifies the examination analysis, and provides a substantial improvement of the current results and this is documented by several evidences and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

We consider a periodic function p : R → R of minimal period 4 which satisfies a family of delay differential equations 0.1 x ′ (t) = g (x (t - d Δ (x t))) , Δ ∈ R , with a continuously differentiable function g : R → R and delay functionals d Δ : C ([ - 2 , 0 ] , R) → (0 , 2). The solution segment x t in Eq. (0.1) is given by x t (s) = x (t + s) . For every Δ ∈ R the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators S Δ , t : x 0 ↦ x t , t ≥ 0 , on a continuously differentiable submanifold X Δ of the space C 1 ([ - 2 , 0 ] , R) , with codim X Δ = 1 . At Δ = 0 the delay is constant, d 0 (ϕ) = 1 everywhere, and the orbit O = { p t : 0 ≤ t < 4 } ⊂ X 0 of the periodic solution is extremely stable in the sense that the spectrum of the monodromy operator M 0 = D S 0 , 4 (p 0) is σ 0 = { 0 , 1 } , with the eigenvalue 1 being simple. For | Δ | ↗ ∞ there is an increasing contribution of variable, state-dependent delay to the time lag d Δ (x t) = 1 + ⋯ in Eq. (0.1). We study how the spectrum σ Δ of M Δ = D S Δ , 4 (p 0) changes if | Δ | grows from 0 to ∞ . A main result is that at Δ = 0 an eigenvalue Λ (Δ) < 0 of M Δ bifurcates from 0 ∈ σ 0 and decreases to - ∞ as | Δ | ↗ ∞ . Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit O at the critical parameter Δ ∗ where Λ (Δ ∗) = - 1 . [ABSTRACT FROM AUTHOR]

Mosquito-borne diseases have become a serious threat to human health worldwide. As Wolbachia may induce cytoplasmic incompatibility (CI) and cause embryonic death when infected male mosquitoes mate with uninfected females, suppressing mosquito population by releasing infected male mosquitoes has become an effective strategy to prevent mosquito-borne diseases. In this paper, we build delay differential equations to study mosquito population suppression with constant releases of Wolbachia infected mosquitoes with incomplete CI. By discussing the existence of multiple equilibria and analyzing the global dynamics completely determined by the intensity threshold of CI and releasing thresholds, we confirm the possibility of bistable and semi-stable phenomena. Numerical simulations show that the intensity threshold $ s_h^* $ of CI combined with the releasing threshold $ r^* $ can help to select Wolbachia strains, and cut down the suppression duration by improving the intensity of CI. Sensitivity analysis shows that both the CI intensity and the time delay also have significant influence on the suppression duration. In addition, we illustrate how the releasing threshold $ r_1 $ and the suppression duration $ T $ change with different Wolbachia strains. [ABSTRACT FROM AUTHOR]

For autonomous delay differential equations x′(t) = f(xt) we construct a continuous semiflow of continuously differentiable solution operators x0 ↦ xt, t ≥ 0, on open subsets of the Fréchet space C((−∞, 0], ℝn). For nonautonomous equations this yields a continuous process of differentiable solution operators. As an application we obtain processes which incorporate all solutions of Volterra integro-differential equations x ′ t = ∫ t 0 k t , s h x s d s . [ABSTRACT FROM AUTHOR]

In the present paper, we mainly focus on an iterative erythropoiesis model with a nonlinear harvesting term involving a time-varying delay. Under certain conditions and by the Schauder fixed point theorem, we show that the existence of positive periodic solutions can be guaranteed and based on the Banach fixed point theorem, we establish the existence as well as the continuous dependence on parameters theorems of the unique solution. Moreover, two examples are provided to validate the correctness of our outcomes. Our theoretical findings extend and improve several related ones in the existing literature. [ABSTRACT FROM AUTHOR]

Involving intracellular delay into a mathematical model and investigating the delayed systems by incorporating optimal control is of great importance to study the cell‐to‐cell interactions of the disease leprosy. Keeping this in mind, we have proposed two different variants of delay‐induced mathematical models with time delay in the process of proliferation of Mycobacterium leprae bacteria from the infected cells and a similar delay to indicate the time‐lag both in the proliferation of M. leprae bacteria and the infection of healthy cells after getting attached with the bacterium. In this research article, we have performed a comparative study between these two delayed systems equipped with optimal control therapeutic approach to determine which one acts better to unravel the complexities of the transmission and dissemination of leprosy into a human body as far as scheduling a perfect drug dose regime depending on this analysis remains our main priority. Our investigations suggest that adopting optimal control strategy consisting of combined drug therapy eliminates the oscillatory behavior of the delayed systems completely. Existence of optimal control solutions are demonstrated in detail. To achieve the optimal control profiles of the drug therapies and to obtain the optimality systems, Pontryagin's Minimum principle with delay in state are employed for our controlled systems. Furthermore, the analytical as well as the numerical outcomes obtained in this research article indicate that the delayed bacterial proliferation and M. leprae‐induced infection model equipped with optimal control policy performs more realistically and accurately in the form of a safe and cost‐effective double‐drug therapeutic regimen. All the mathematical results are verified numerically and the numerical results are compared with some recent clinical data in our article as well. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the strong feedback limit of two negative feedback schemes which have proven to be efficient for many biological processes (protein synthesis, immune responses, breathing disorders). In this limit, the nonlinear delayed feedback function can be reduced to a function with a threshold nonlinearity. This will considerably help analytical and numerical studies of networks exhibiting different topologies. Mathematically, we compare the bifurcation diagrams for both the delayed and non-delayed feedback functions and show that Hopf classical theory needs to be revisited in the strong feedback limit.

The study of dynamics of diabetic population infected by COVID-19 is of pressing concern as people with diabetes are considered to be at higher risk of severe illness from COVID-19. A three-compartment mathematical model to describe the interactions of diabetic population and non-diabetic population both infected by COVID-19 with a susceptible population is considered. Time delays in incubation periods of COVID-19 in diabetic and non-diabetic populations are introduced. Besides the basic properties of such a dynamical system, both local and global stability of endemic equilibrium, are studied. The lengths of time delays are estimated for which the stability of the system is preserved locally, while sufficient conditions on system parameters are obtained for global stability. Numerical examples are provided to establish the theory, and simulations are provided to visualize the examples. It is noted that an increase in length of time delay in either of infected populations leads to oscillations in susceptible population but has no impact on infected populations.

Measles remains one of the leading causes of death among young children globally, even though a safe and cost-effective vaccine is available. Vaccine hesitancy and social response to vaccination continue to undermine efforts to eradicate measles. In this study, we consider data about measles vaccination and measles prevalence in Germany for the years 2008–2012 in 345 districts. In the first part of the paper, we show that the probability of a local outbreak does not significantly depend on the vaccination coverage, but—if an outbreak does take place—the scale of the outbreak depends significantly on the vaccination coverage. Additionally, we show that the willingness to be vaccinated is significantly increased by local outbreaks, with a delay of about one year. In the second part of the paper, we consider a deterministic delay model to investigate the consequences of the statistical findings on the dynamics of the infection. Here, we find that the delay might induce oscillations if the vaccination coverage is rather low and the social response to an outbreak is sufficiently strong. The relevance of our findings is discussed at the end of the paper.

In this article, we consider an epidemiological model in which we take into account the effects of direct and indirect transmissions. The first mode occurs through direct contact between infectious and susceptible individuals, and the second one will take place through the shedding of virus particles by infectious individuals and their acquisition by susceptible ones. We also study the effect of latency period and time needed for a susceptible person to become infected by indirect transmission mode. By considering the direct and indirect basic reproduction numbers, we define the basic reproduction number R0{R}_{0} of the model, which helps us to analyze the stability of equilibria and bifurcation and determine the most sensitive parameters. In conclusion, some numerical simulations are given to confirm the analytical analysis.

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations. [ABSTRACT FROM AUTHOR]

The asymptotical properties of a special dynamic two-person game are examined under best-response dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper. [ABSTRACT FROM AUTHOR]

Measles remains one of the leading causes of death among young children globally, even though a safe and cost-effective vaccine is available. Vaccine hesitancy and social response to vaccination continue to undermine efforts to eradicate measles. In this study, we consider data about measles vaccination and measles prevalence in Germany for the years 2008–2012 in 345 districts. In the first part of the paper, we show that the probability of a local outbreak does not significantly depend on the vaccination coverage, but—if an outbreak does take place—the scale of the outbreak depends significantly on the vaccination coverage. Additionally, we show that the willingness to be vaccinated is significantly increased by local outbreaks, with a delay of about one year. In the second part of the paper, we consider a deterministic delay model to investigate the consequences of the statistical findings on the dynamics of the infection. Here, we find that the delay might induce oscillations if the vaccination coverage is rather low and the social response to an outbreak is sufficiently strong. The relevance of our findings is discussed at the end of the paper. [ABSTRACT FROM AUTHOR]

A double Hopf bifurcation analysis is performed for the rolling of a low-freeboard ship model controlled with an active U-tube anti-roll tank (ART). We consider a single-degree-of-freedom system with nonlinear damping and restoring functions. The ART is modeled as a proportional-gain controller. A constant delay term is included in the controller since a finite amount of time is required to pump the fluid inside the ART from one container to the other. We perform a linear stability analysis to determine the critical control gain and delay corresponding to the double Hopf bifurcation point. We confirm the existence of the double Hopf bifurcation by finding slopes and roots of the characteristic equation of the linearized delay differential equation with the critical parameters. We use the method of multiple scales to obtain slow-flow equations at the double Hopf bifurcation, which are then used to identify six qualitatively distinct sets of fixed points. Our analysis reveals that one of these regions has a stable zero equilibrium, another has a stable limit cycle with amplitude smaller than the capsizing angle, and the remaining regions have no safe fixed points. These qualitative observations are validated numerically. Study of the control of ship roll motion is important to avoid dynamic instability and capsizing. [ABSTRACT FROM AUTHOR]

In this article, a time delay parabolic convection-reaction-diffusion singularly perturbed problem with two small parameters is considered. We investigate the layer behavior of the solution for both smooth and non-smooth data. A numerical method to solve the problems described is developed using the Crank-Nicolson scheme to discretize the time-variable on a uniform mesh while a hybrid finite difference is applied for the space-variable. The hybrid scheme is a combination of the central, upwind and mid-point differencing on a piecewise uniform mesh of Shishkin type. The convergence analysis shows that the proposed method is uniformly convergent of second order in both space and time. Numerical experiments conducted on some test examples confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The Wolbachia-induced incompatible insect technique is a promising strategy for controlling wild mosquito populations. However, recent experimental studies have shown that mosquito migration into target areas dilutes the strategy’s effectiveness. In this work, we formulate a delay differential equation model to assess the impact of migration on mosquito population suppression. We identify that mosquito migration into an idealized target area makes it impossible to eliminate the target population completely. Our analysis identifies a lower bound of the suppression rate γ ∗ for a given migration number, which reveals the possible maximum reduction of wild population size in the peak season. For a given suppression rate target γ 0 > γ ∗ , we identify the permitted maximum migration number D (γ 0) , above which is impossible to reduce the field mosquito density up to (1 - γ 0) × 100 % in peak season. To reduce more than 95 % of Aedes albopictus population during its peak season in Guangzhou within six weeks, the required minimum release number of Wolbachia-infected males climbs steeply as the migration number increases to D(0.05). [ABSTRACT FROM AUTHOR]

We present a new method for obtaining stability conditions for certain classes of delay differential equations. The method is based on the transition from an individual equation to a family of equations, and next the selection of a representative of this family, the test equation, asymptotic properties of which determine those of all equations in the family. This approach allows us to obtain the conditions that are the criteria for the stability of all equations of a given family. These conditions are formulated in terms of the parameters of the class of equations being studied, and are effectively verifiable. The main difference of the proposed method from the known general methods (using Lyapunov–Krasovsky functionals, Razumikhin functions, and Azbelev W-substitution) is the emphasis on the exactness of the result; the difference from the known exact methods is a significant expansion of the range of applicability. The method provides an algorithm for checking stability conditions, which is carried out in a finite number of operations and allows the use of numerical methods. [ABSTRACT FROM AUTHOR]

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings. [ABSTRACT FROM AUTHOR]