Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare-Bendixson theorem in combination with the Bendixson-Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number $\mathcal{R}_0 \leq 1$ and a disease-endemic equilibrium point is globally asymptotically stable whenever $\mathcal{R}_0 > 1$. Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorously mathematical analyses, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

A Reliable numerical scheme for a computer virus propagation model

In this paper, we apply the Mickens’methodology to construct a dynamically consistentnonstandard finite difference (NSFD) scheme for acomputer virus propagation model. It is proved thatthe constructed NSFD scheme correctly preservesessential mathematical features of the continuous-timemodel, which are positivity, boundedness and asymptotic stability. Consequently, we obtain an effectivenumerical scheme that can provide reliable approximations for the computer virus propagation model.Meanwhile, some typical standard finite differenceschemes fail to preserve the essential properties ofthe computer virus propagation model; hence, theycan generate numerical approximations which arenot only negative but also unstable. Finally, a setof numerical experiments is performed to supportthe theoretical results as well as to demonstrate theadvantage of the NSFD scheme over standard ones.As we expected, there is a good agreement betweenthe numerical results and theoretical assertions.

A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

Bifurcation and comparison of a discrete-time Hindmarsh-Rose model

In this paper, a discrete-time Hindmarsh-Rose model is obtained by a nonstandard finite difference (NSFD) scheme. Bifurcation behaviors between the model obtained by the forward Euler scheme and the model obtained by the NSFD scheme are compared. Through analytical and numerical comparisons, much more bifurcations and dynamical behaviors can be obtained and preserved by using the NSFD scheme, in which the integral step size can be chosen larger relatively due to its better stability and convergence than those in the forward Euler scheme. It means that the discretetime model obtained by the NSFD scheme is closer to the original continuous system than the discrete-time model obtained by the forward Euler scheme. These confirmed results can at least guarantee true available numerical results to investigate complex neuron dynamical systems.

A Reliable Numerical Analysis of Transmission Dynamics of Chicken Pox (Varicella Zoster Virus)

Solutions generated through numerical techniques are great in solving real-world problems. This manuscript deals with the numerical approximation of the epidemic system, describing the transmission dynamics of the Vercilla Zoster Virus (VZV) through the impact of vaccination. To discretize the continuous dynamical system, we proposed a novel numerical technique that preserves the true dynamics of the VZV epidemic model. The proposed technique is established in such a manner that it sustains all necessary physical traits depicted by the epidemic model under study. The designed technique is named a nonstandard finite difference (NSFD) scheme. Theoretical analysis of the designed NSFD technique is presented which describes its strength over the standard numerical procedures which are already being used for such purposes. The graphical solutions of all the numerical techniques are presented which verify the efficacy of the proposed NSFDS technique.

An Efficient Numerical Method for the Solution of the Polio Virus (Poliomyelitis) Epidemic Model with the Role of Vaccination

Mathematical modeling of a communicable disease is an effective way to describe the behavior and dynamics of the disease. It builds on our understanding of the transmission of a contagion in a population. In this paper, we explore the transmission dynamics of the polio virus (poliomyelitis) with vaccination using standard methods. We formulate an unconditionally stable Non-Standard Finite Difference (NSFD) scheme for a continuous system of the epidemic polio virus. The designed scheme to approximate the solution is bounded, consistent with the underlying model. The proposed numerical scheme preserves the positivity of the stated variables which is necessary for any population dynamical system. It is used to calculate the numerical solutions of the epidemic model for different step sizes “h”. Two other numerical schemes are enforced to find the solution of the proposed system. Finally, the comparison of the NSFD technique with these methods proves its validity and effectiveness.

Crowding effects on the dynamics of COVID-19 mathematical model

A disastrous coronavirus, which infects a normal person through droplets of infected person, has a route that is usually by mouth, eyes, nose or hands. These contact routes make it very dangerous as no one can get rid of it. The significant factor of increasing trend in COVID19 cases is the crowding factor, which we named “crowding effects”. Modeling of this effect is highly necessary as it will help to predict the possible impact on the overall population. The nonlinear incidence rate is the best approach to modeling this effect. At the first step, the model is formulated by using a nonlinear incidence rate with inclusion of the crowding effect, then its positivity and proposed boundedness will be addressed leading to model dynamics using the reproductive number. Then to get the graphical results a nonstandard finite difference (NSFD) scheme and fourth order Runge–Kutta (RK4) method are applied.

A reliable and competitive mathematical analysis of Ebola epidemic model

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