Search

Abstract In this paper, we establish sufficient conditions for various stability aspects of a nonlinear Volterra integro-dynamic matrix Sylvester system on time scales. We convert the nonlinear Volterra integro-dynamic matrix Sylvester system on time scale to an equivalent nonlinear Volterra integro-dynamic system on time scale using vectorization operator. Sufficient conditions are obtained to this system for stability, asymptotic stability, exponential stability, and strong stability. The obtained results include various stability aspects of the matrix Sylvester systems in continuous and discrete models.

Abstract A bacteriophage (in short, phage) is a virus that can infect and replicate within bacteria. Assuming that uninfected and infected bacteria are capable of reproducing with logistic law, we investigate a model of bacteriophage infection that resembles simple SI-models widely used in epidemiology. The dynamics of host-parasite co-extinctions may exhibit four scenarios: hosts and parasites go extinct, parasites go extinct, hosts go extinct, and hosts and parasites coexist. By using the Jacobian matrix and Bendixson–Dulac theory, local and global stability analysis of uninfected and infected steady states is provided; the basic reproduction number of the model is given; general results are supported by numerical simulations. We show that bacteriophages can reduce a host density. This provides a theoretical framework for studying the problem of whether phages can effectively prevent, control, and treat infectious diseases.

Abstract In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results.

Abstract Infection of hepatitis B virus (HBV) is a global health problem. We provide the study about hepatitis B virus dynamics that can be controlled by education campaign (awareness), vaccination, and treatment. Initially we bring constant controls in considerations for treatment, vaccination, and education campaign (awareness). In the case of constant controls, we study the stability and existence of the disease-free and endemic equilibria model’s solutions. Afterwards, we take time as a control and formulate the suitable optimal control problem, acquire optimal control strategy in order to reduce the number of humans that are infected and the costs associated. At the end, results of numerical simulations show that the optimal combination of education campaign (awareness), treatment, and vaccination is the most efficient way to control the infection of hepatitis B virus (HBV) infection.

Abstract In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.

Abstract In this paper, we study the uniqueness and existence of the solution of a non-autonomous and nonsingular delay difference equation using the well-known principle of contraction from fixed point theory. Furthermore, we study the Hyers–Ulam stability of the given system on a bounded discrete interval and then on an unbounded interval. An example is also given at the end to illustrate the theoretical work.

Abstract An explicit unconditionally stable scheme is proposed for solving time-dependent partial differential equations. The application of the proposed scheme is given to solve the COVID-19 epidemic model. This scheme is first-order accurate in time and second-order accurate in space and provides the conditions to get a positive solution for the considered type of epidemic model. Furthermore, the scheme’s stability for the general type of parabolic equation with source term is proved by employing von Neumann stability analysis. Furthermore, the consistency of the scheme is verified for the category of susceptible individuals. In addition to this, the convergence of the proposed scheme is discussed for the considered mathematical model.

Abstract We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using the finite volume element method to discretize the space-fractional derivative and Crank–Nicholson scheme for time discretization. We also analyze and prove the stability and convergence of the given scheme. Finally, we validate our theoretical analysis by data from three examples.

Abstract This paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number R 0 > 1 $R_{0} > 1$ ; a disease-free equilibrium E 0 $E_{0}$ and a disease endemic equilibrium E 1 $E_{1}$ . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number R 0 < 1 $R_{0} 1 $R_{0} > 1$ . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.

Abstract In this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

Abstract In this paper, an uncertain SIR (spreader, ignorant, stifler) rumor spreading model driven by one Liu process is formulated to investigate the influence of perturbation in the transmission mechanism of rumor spreading. The deduced process of the uncertain SIR rumor spreading model is presented. Then an existence and uniqueness theorem concerning the solution is proved. Moreover, the stability of uncertain SIR rumor spreading differential equation is proved. In addition, the influence of different parameters on rumor spreading is analyzed through numerical simulation. This paper also presents a paradox of stochastic SIR rumor spreading model.

Abstract In this study, we develop a nonlinear ordinary differential equation to study the dynamics of syphilis transmission incorporating controls, namely prevention and treatment of the infected males and females. We obtain syphilis-free equilibrium (SFE) and syphilis-present equilibrium (SPE). We obtain the basic reproduction number, which can be used to control the transmission of the disease, and thus establish the conditions for local and global stability of the syphilis-free equilibrium. The stability results show that the model is locally asymptotically stable if the Routh–Hurwitz criteria are satisfied and globally asymptotically stable. The bifurcation analysis result reveals that the model exhibits backward bifurcation. We adopted Pontryagin’s maximum principle to determine the optimality system for the syphilis model, which was solved numerically to show that syphilis transmission can be optimally best control using a combination of condoms usage and treatment in the primary stage of infection in both infected male and female populations.

Abstract Epidemiological models have been playing a vital role in different areas of biological sciences for the analysis of various contagious diseases. Transmissibility of virulent diseases is being portrayed in the literature through different compartments such as susceptible, infected, recovered (SIR), susceptible, infected, recovered, susceptible (SIRS) or susceptible, exposed, infected, recovered (SEIR), etc. The novelty in this endeavor is the addition of compartments of latency and treatment with vaccination, so the system is designated as susceptible, vaccinated, exposed, latent, infected, treatment, and recovered (SVELITR). The contact of a susceptible individual to an infective individual firstly makes the individual exposed, latent, and then completely infection carrier. Innovatively, the assumption that exposed, latent, and infected individuals enter the treatment compartment at different rates after a time lag is also deliberated through the existence of time delay. The rate of change and constant solutions of each compartment are studied with incorporation of a special case of proportional fractional derivative (PFD). In addition, existence and uniqueness of the system are also comprehensively elaborated. Moreover, novel dynamic assessment of the system is carried out in context with the fractional order index. Succinctly, the manuscript accomplishes cyclic epidemiological behavior of the infectious disease due to the delay in treatment of the infected individuals.

Abstract In this paper we propose a stable finite difference method to solve the fractional reaction–diffusion systems in a two-dimensional domain. The space discretization is implemented by the weighted shifted Grünwald difference (WSGD) which results in a stiff system of nonlinear ordinary differential equations (ODEs). This system of ordinary differential equations is solved by an efficient compact implicit integration factor (cIIF) method. The stability of the second order cIIF scheme is proved in the discrete L 2 $L^{2}$ -norm. We also prove the second-order convergence of the proposed scheme. Numerical examples are given to demonstrate the accuracy, efficiency, and robustness of the method.

Abstract In this work, we develop and analyze a nonautonomous mathematical model for the spread of the new corona-virus disease (COVID-19) in Saudi Arabia. The model includes eight time-dependent compartments: the dynamics of low-risk S L $S_{L}$ and high-risk S M $S_{M}$ susceptible individuals; the compartment of exposed individuals E; the compartment of infected individuals (divided into two compartments, namely those of infected undiagnosed individuals I U $I_{U}$ and the one consisting of infected diagnosed individuals I D $I_{D}$ ); the compartment of recovered undiagnosed individuals R U $R_{U}$ , that of recovered diagnosed R D $R_{D}$ individuals, and the compartment of extinct Ex individuals. We investigate the persistence and the local stability including the reproduction number of the model, taking into account the control measures imposed by the authorities. We perform a parameter estimation over a short period of the total duration of the pandemic based on the COVID-19 epidemiological data, including the number of infected, recovered, and extinct individuals, in different time episodes of the COVID-19 spread.

Abstract In this paper, the dynamics of a general differential equation with neutral type are investigated. Under certain assumptions, the stability of positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Krawcewicz et al. Finally, by taking neutral Nicholson’s blowflies model and neutral Mackey–Glass model as two examples, some numerical simulations are carried out to illustrate the analytical results.

Abstract This work investigates the dissemination mechanism of pine wilt disease. The basic reproduction number is computed explicitly, and an ultimate invariable level of contagious hosts and vectors, without and with disease, is discussed by using this number. Highly effective techniques, Lyapunov functional and graph theoretic, are utilised to obtain the ultimate constant level of the whole population. The idea of complete disease eradication and reduction of endemic level is explored through the utilisation of two efficient methods. Using sensitivity analysis approach, necessary control measures are suggested to overcome the disease. Using the literature data, the robustness of control strategies is shown graphically.

Abstract Fractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful.

Abstract A swarming model is a model that describes the behavior of the social aggregation of a large group of animals or the community of humans. In this work, the swarming model that includes the short-range repulsion and long-range attraction with the presence of time delay is investigated. Moreover, the convergence to a consensus representing dispersion and cohesion properties is proved by using the Lyapunov functional approach. Finally, numerical results are provided to demonstrate the effect of time delay on the motion of the group of agents.

Abstract A class of inertial neural networks (INNs) with reaction-diffusion terms and distributed delays is studied. The existence and uniqueness of the equilibrium point for the considered system is obtained by topological degree theory, and a sufficient condition is given to guarantee global exponential stability of the equilibrium point. Finally, an example is given to show the effectiveness of the results in this paper.

Abstract In this research, we probe the influences of two common types of stochastic noises in the environment, namely, white noise and telephone noise, and put forward a stochastic differential equation population model with Allee effects. We analyze some asymptotic behaviors of the model, including extermination, persistence and invariant measure. Some vital functions of white noise and telephone noise on these asymptotic behaviors of the model are discovered and numerically expounded.

Abstract In this manuscript, we investigate a novel Susceptible–Exposed–Infected–Quarantined–Recovered (SEIQR) COVID-19 propagation model with two delays, and we also consider supply chain transmission and hierarchical quarantine rate in this model. Firstly, we analyze the existence of an equilibrium, including a virus-free equilibrium and a virus-existence equilibrium. Then local stability and the occurrence of Hopf bifurcation have been researched by thinking of time delay as the bifurcation parameter. Besides, we calculate direction and stability of the Hopf bifurcation. Finally, we carry out some numerical simulations to prove the validity of theoretical results.

Abstract In this paper, we introduce the functional equations f ( 2 x − y ) + f ( x + 2 y ) = 5 [ f ( x ) + f ( y ) ] , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + 4 f ( y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + f ( 2 y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 4 [ f ( x ) + f ( y ) ] + [ f ( − x ) + f ( − y ) ] . $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

Abstract Nowadays, online gambling has a great negative impact on the society. In order to study the effect of people’s psychological factors, anti-gambling policy, and social network topology on online gambling dynamics, a new SHGD (susceptible–hesitator–gambler–disclaimer) online gambling spreading model is proposed on scale-free networks. The spreading dynamics of online gambling is studied. The basic reproductive number R 0 $R_{0}$ is got and analyzed. The basic reproductive number R 0 $R_{0}$ is related to anti-gambling policy and the network topology. Then, gambling-free equilibrium E 0 $E_{0}$ and gambling-prevailing equilibrium E + $E_{ +} $ are obtained. The global stability of E 0 $E_{0}$ is analyzed. The global attractivity of E + $E_{ +} $ and the persistence of online gambling phenomenon are studied. Finally, the theoretical results are verified by some simulations.

Abstract This paper is motivated by the series of research papers that consider parasitoids’ external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study.

Abstract In this paper, we study and analyze the susceptible-infectious-removed (SIR) dynamics considering the effect of health system. We consider a general incidence rate function and the recovery rate as functions of the number of hospital beds. We prove the existence, uniqueness, and boundedness of the model. We investigate all possible steady-state solutions of the model and their stability. The analysis shows that the free steady state is locally stable when the basic reproduction number R 0 $R_{0}$ is less than unity and unstable when R 0 > 1 $R_{0} > 1$ . The analysis shows that the phenomenon of backward bifurcation occurs when R 0 < 1 $R_{0}

Abstract In this paper, we consider a diffusive predator–prey model with a time delay and prey toxicity. The effect of time delay on the stability of the positive equilibrium is studied by analyzing the eigenvalue spectrum. Delay-induced Hopf bifurcation is also investigated. By utilizing the normal form method and center manifold reduction for partial functional differential equations, the formulas for determining the property of Hopf bifurcation are given.

In this paper, we establish sufficient conditions for various stability aspects of a nonlinear Volterra integro-dynamic matrix Sylvester system on time scales. We convert the nonlinear Volterra integro-dynamic matrix Sylvester system on time scale to an equivalent nonlinear Volterra integro-dynamic system on time scale using vectorization operator. Sufficient conditions are obtained to this system for stability, asymptotic stability, exponential stability, and strong stability. The obtained results include various stability aspects of the matrix Sylvester systems in continuous and discrete models. [ABSTRACT FROM AUTHOR]

Abstract This research proposes and delves into a stochastic competitive phytoplankton model with allelopathy and regime-switching. Sufficient criteria are proffered to ensure that the model possesses a unique ergodic stationary distribution (UESD). Furthermore, it is testified that these criteria are sharp on certain conditions. Some critical functions of regime-switching on the existence of a UESD of the model are disclosed: regime-switching could lead to the appearance of the UESD. The theoretical findings are also applied to research the evolution of Heterocapsa triquetra and Chrysocromulina polylepis.

Abstract The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the L 2 $L_{2}$ error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

Abstract In this paper, we study the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji (BG) chaotic system. Since the current study on Hopf bifurcation for fractional-order delay systems is carried out on the basis of analyses for stability of equilibrium of its linearized approximation system, it is necessary to verify the reasonability of linearized approximation. Through Laplace transformation, we first illustrate the equivalence of stability of equilibrium for a fractional-order delay Bhalekar–Gejji chaotic system and its linearized approximation system under an appropriate prior assumption. This semianalytically verifies the reasonability of linearized approximation from the viewpoint of stability. Then we theoretically explore the relationship between the time delay and Hopf bifurcation of such a system. By introducing the delayed feedback controller into the proposed system, the influence of the feedback gain changes on Hopf bifurcation is also investigated. The obtained results indicate that the stability domain can be effectively controlled by the proposed delayed feedback controller. Moreover, numerical simulations are made to verify the validity of the theoretical results.

Abstract We study the dynamical properties of a discrete population model with diffusion. We survey the transcritical, pitchfork, and flip bifurcations of nonhyperbolic fixed points by using the center manifold theorem. For the degenerate fixed point with eigenvalues ±1 of the model, we obtain the normal form of the mapping by using the coordinate transformation. Then we give an approximating system of the normal form via an approximation by a flow. We give the local behavior near a degenerate equilibrium of the vector field by the blowup technique. By the conjugacy between the reflection of time-one mapping of a vector field and the model we obtain the stability and qualitative structures near the degenerate fixed point of the model. Finally, we carry out a numerical simulation to illustrate the analytical results of the model.

Abstract This work is devoted to a stochastic model on the spread and control of corona virus (COVID-19), in which the total population of a corona infected area is divided into susceptible, infected, and recovered classes. In reality, the number of individuals who get disease, the number of deaths due to corona virus, and the number of recovered are stochastic, because nobody can tell the exact value of these numbers in the future. The models containing these terms must be stochastic. Such numbers are estimated and counted by a random process called a Poisson process (or birth process). We construct an SIR-type model in which the above numbers are stochastic and counted by a Poisson process. To understand the spread and control of corona virus in a better way, we first study the stability of the corresponding deterministic model, investigate the unique nonnegative strong solution and an inequality managing of which leads to control of the virus. After this, we pass to the stochastic model and show the existence of a unique strong solution. Next, we use the supermartingale approach to investigate a bound managing of which also leads to decrease of the number of infected individuals. Finally, we use the data of the COVOD-19 in USA to calculate the intensity of Poisson processes and verify our results.

Abstract In this paper the problems of the existence and stability of periodic solutions of neutral-type inertial neural networks with time-varying delays are discussed by applying Mawhin’s continuation theorem and Lyapunov functional method. Finally, two numerical examples are given to illustrate our theoretical results.

Abstract In this paper, the dynamical behavior of a SEIR epidemic system that takes into account governmental action and individual reaction is investigated. The transmission rate takes into account the impact of governmental action modeled as a step function while the decreasing contacts among individuals responding to the severity of the pandemic is modeled as a decreasing exponential function. We show that the proposed model is capable of predicting Hopf bifurcation points for a wide range of physically realistic parameters for the COVID-19 disease. In this regard, the model predicts periodic behavior that emanates from one Hopf point. The model also predicts stable oscillations connecting two Hopf points. The effect of the different model parameters on the existence of such periodic behavior is numerically investigated. Useful diagrams are constructed that delineate the range of periodic behavior predicted by the model.

Abstract In this paper, we propose a new fractional-order financial model which is a generalized version of the financial model reported in the previous publications. By applying a suitable time-delayed feedback controller, we have control for the chaotic behavior of the fractional-order financial model. We investigate the stability and the existence of a Hopf bifurcation of the fractional-order financial model. A new sufficient condition that guarantees the stability and the existence of a Hopf bifurcation for a fractional-order delayed financial model is presented by regarding the delay as bifurcation parameter. The investigation shows that the delay and the fractional order have an important effect on the stability and Hopf bifurcation of involved model. Some simulations justifying the validity of the derived analytical results are given. The obtained results of this article are innovative and are of great significance in handling the financial issues.

Abstract The purpose of this article is to discuss the dynamics of the spread of Ebola virus disease (EVD), a kind of fever commonly known as Ebola hemorrhagic fever. It is rare but severe and is considered to be extremely dangerous. Ebola virus transmits to people through domestic and wild animals, called transmitting agents, and then spreads into the human population through close and direct contact among individuals. To study the dynamics and to illustrate the stability pattern of Ebola virus in human population, we have developed an SEIR type model consisting of coupled nonlinear differential equations. These equations provide a good tool to discuss the mode of impact of Ebola virus on the human population through domestic and wild animals. We first formulate the proposed model and obtain the value of threshold parameter R 0 $\mathcal{R}_{0}$ for the model. We then determine both the disease-free equilibrium (DFE) and endemic equilibrium (EE) and discuss the stability of the model. We show that both the equilibrium states are locally asymptotically stable. Employing Lyapunov functions theory, global stabilities at both the levels are carried out. We use the Runge–Kutta method of order 4 (RK4) and a non-standard finite difference (NSFD) scheme for the susceptible–exposed–infected–recovered (SEIR) model. In contrast to RK4, which fails for large time step size, it is found that the NSFD scheme preserves the dynamics of the proposed model for any step size used. Numerical results along with the comparison, using different values of step size h, are provided.

Abstract In this paper, a mathematical model for COVID-19 that involves contact tracing is studied. The contact tracing-induced reproduction number R q $\mathcal{R}_{q}$ and equilibrium for the model are determined and stabilities are examined. The global stabilities results are achieved by constructing Lyapunov functions. The contact tracing-induced reproduction number R q $\mathcal{R}_{q}$ is compared with the basic reproduction number R 0 $\mathcal{R}_{0}$ for the model in the absence of any intervention to assess the possible benefits of the contact tracing strategy.

Abstract This paper deals with a general SEIR model for the coronavirus disease 2019 (COVID-19) with the effect of time delay proposed. We get the stability theorems for the disease-free equilibrium and provide adequate situations of the COVID-19 transmission dynamics equilibrium of present and absent cases. A Hopf bifurcation parameter τ concerns the effects of time delay and we demonstrate that the locally asymptotic stability holds for the present equilibrium. The reproduction number is brief in less than or greater than one, and it effectively is controlling the COVID-19 infection outbreak and subsequently reveals insight into understanding the patterns of the flare-up. We have included eight parameters and the least square method allows us to estimate the initial values for the Indian COVID-19 pandemic from real-life data. It is one of India’s current pandemic models that have been studied for the time being. This Covid19 SEIR model can apply with or without delay to all country’s current pandemic region, after estimating parameter values from their data. The sensitivity of seven parameters has also been explored. The paper also examines the impact of immune response time delay and the importance of determining essential parameters such as the transmission rate using sensitivity indices analysis. The numerical experiment is calculated to illustrate the theoretical results.

Abstract Alcoholism is a social phenomenon that affects all social classes and is a chronic disorder that causes the person to drink uncontrollably, which can bring a series of social problems. With this motivation, a delayed drinking model including five subclasses is proposed in this paper. By employing the method of characteristic eigenvalue and taking the temporary immunity delay for alcoholics under treatment as a bifurcation parameter, a threshold value of the time delay for the local stability of drinking-present equilibrium and the existence of Hopf bifurcation are found. Then the length of delay has been estimated to preserve stability using the Nyquist criterion. Moreover, optimal strategies to lower down the number of drinkers are proposed. Numerical simulations are presented to examine the correctness of the obtained results and the effects of some parameters on dynamics of the drinking model.

Abstract This paper is concerned with traveling curved fronts of bistable reaction–diffusion equations with nonlinear convection in a two-dimensional space. By constructing super- and subsolutions, we establish the existence of traveling curved fronts. Furthermore, we show that the traveling curved front is globally asymptotically stable.

Abstract In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order O ( τ 2 − α + h 4 ) $O(\tau ^{2-\alpha }+h^{4})$ and O ( τ 3 − α + h 4 ) $O(\tau ^{3-\alpha }+h^{4})$ , respectively, with the temporal step size τ and the spatial step size h, and α ∈ ( 0 , 1 ) $\alpha \in (0,1)$ . For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

Abstract In this paper, a two-species competitive model with Michaelis–Menten type harvesting in the first species is studied. We have made a detailed mathematical analysis of the model to describe some important results that may be produced by the interaction of biological resources. The permanence, stability, and bifurcation (saddle-node bifurcation and transcritical bifurcation) of the model are investigated. The results show that with the change of parameters, two species could eventually coexist, become extinct or one species will be driven to extinction and the other species will coexist. Moreover, by constructing the Lyapunov function, sufficient conditions to ensure the global asymptotic stability of the positive equilibrium are given. Our study shows that compared with linear harvesting, nonlinear harvesting can exhibit more complex dynamic behavior. Numerical simulations are presented to illustrate the theoretical results.

Abstract No previous study has involved uncertain delay differential equations with jump. In this paper, we consider the uncertain delay differential equations with V-jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, we give the equivalent integral equation. Next, we establish an existence and uniqueness theorem of solution to the differential equations we proposed in the finite domain and the infinite domain, respectively. Once more, the concept of stability for uncertain delay differential equations with V-jump is proposed. In addition, the sufficient condition for stability theorem is derived. To judge existence, uniqueness, and stability briefly, we provide some examples in the end.

Abstract In this paper, we analyze the criteria for the stability of a method suited to the ordinary differential equations models. The relevant proof that the method satisfies the condition of stiff stability is also provided. The aim of this paper is therefore to construct an efficient two-point block method based on backward differentiation formula which is A-stable and converged. The new diagonally implicit scheme is formulated to approximate the solution of the pharmacokinetics models. By implementing the algorithm, the numerical solution to the models is compared with a few existing methods and established stiff solvers. It yields significant advantages when the diagonally implicit method with a lower triangular matrix and identical diagonal elements is considered. The formula is designed in such a way that it permits a maximum of one LU decomposition for each integration stage.

Abstract In this paper, we establish and examine a mathematical model that combines the effects of vitamins intervention on strengthening the immune system and its role in suppressing and delaying the growth and division of tumour cells. In order to accomplish this, we propose a tumour–immune–vitamins model (TIVM) governed by ordinary differential equations and comprised of two populations, namely tumour and immune cells. It is presumed that the source of vitamins in TIVM originates from organic foods and beverages, based on the food pyramid. The simulation of TIVM employs the fourth order Runge–Kutta method. It is found from the analysis and simulation results that one of the side effects of weakening the immune system is the possibility of transforming immune cells into immune cancer cells to prevent or delay the growth and division of tumour cells. Evidently, for regular intakes of vitamins, which is projected at 55% of vitamins per day, the immune system is strengthened, preventing the production of tumour cells.

Abstract In this paper, an explicit fourth-order compact (EFOC) difference scheme is proposed for solving the two-dimensional(2D) wave equation. The truncation error of the EFOC scheme is O ( τ 4 + τ 2 h 2 + h 4 ) $O({\tau ^{4}} + {\tau ^{2}}{h^{2}} + {h^{4}})$ , i.e., the scheme has an overall fourth-order accuracy in both time and space. Because the scheme is explicit, it does not need any iterative processes. Afterwards, the stability condition of the scheme is obtained by using the Fourier analysis method, which has a wider stability range than other explicit or alternation direction implicit (ADI) schemes. Finally, some numerical experiments are carried out to verify the accuracy and stability of the present scheme.

Abstract Virotherapy is a therapeutic treatment for cancer. It uses genetically engineered viruses to selectively infect, replicate in, and destroy cancer cells without damaging normal cells. In this paper, we present a modified model to include, within the dynamics of virotherapy, the interaction between uninfected tumor cells and immune response. The model is analyzed qualitatively to produce five equilibrium points. One of these equilibriums demonstrates the effect observed in virotherapy, where the immune system demolishes infected cells as well as viruses. Moreover, the existence and stability of the equilibrium points are established under certain criteria. Numerical simulations are performed to display the agreement with the analytical results. Finally, parameter analysis is carried out to illustrate which parameters in the model affect the outcome of virotherapy.

Abstract The fractional diffusion equations can be discretized by applying the implicit finite difference scheme and the unconditionally stable shifted Grünwald formula. Hence, the generating linear system has a real Toeplitz structure when the two diffusion coefficients are non-negative constants. Through a similarity transformation, the Toeplitz linear system can be converted to a generalized saddle point problem. We use the generalization of a parameterized inexact Uzawa (GPIU) method to solve such a kind of saddle point problem and give a new algorithm based on the GPIU method. Numerical results show the effectiveness and accuracy for the new algorithm.

Abstract In this work, we propose and study a new amensalism system with Allee effect on the first species. First, we investigate the existence and stability of all possible coexistence equilibrium points and boundary equilibrium points of this system. Then, applying the Sotomayor theorem, we prove that there exists a saddle-node bifurcation under some suitable parameter conditions. Finally, we provide a specific example with corresponding numerical simulations to further demonstrate our theoretical results.