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We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms in the construction of nonstandard finite difference (NSFD) schemes. Firstly, we construct nonstandard one-stage and two-stage theta methods for a general dynamical system defined by a system of autonomous ordinary differential equations. We provide a sharp condition, which captures the dynamics of the continuous model. We discuss at length how this condition is pivotal in the construction of the complex denominator function. We show that the nonstandard theta methods are elementary stable in the sense that they have exactly the same fixed-points as the continuous model and they preserve their stability, irrespective of the value of the step size. For more complex dynamical systems that are dissipative, we identify a class of nonstandard theta methods that replicate this property. We apply the first part by considering a dynamical system that models the Ebola Virus Disease (EVD). The formulation of the model involves both the fast/direct and slow/indirect transmission routes. Using the specific structure of the EVD model, we show that, apart from the guidelines in the first part, the nonlocal approximation of nonlinear terms is guided by the productive-destructive structure of the model, whereas the choice of the denominator function is based on the conservation laws and the sub-equations that are associated with the model. We construct a NSFD scheme that is dynamically consistent with respect to the properties of the continuous model such as: positivity and boundedness of solutions; local and/or global asymptotic stability of disease-free and endemic equilibrium points; dependence of the severity of the infection on self-protection measures. Throughout the paper, we provide numerical simulations that support the theory. [ABSTRACT FROM AUTHOR]

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

Diseases, in particular, vector-borne diseases are very important issues in crop protection. However, despite their impact on food safety, very few mathematical models have been developed in order to improve control strategies. Motivated by existing literature, we begin by considering a temporal model of vector-borne diseases in (annual) crops. Using appropriate mathematical methods, we show the existence of threshold parameters and discuss several control strategies based on the model. Then, the model is modified to include a spatial component in order to take into account that vectors are moving. We study both theoretically and numerically the related system and other subsystems that are easier to handle using the theory of monotone dynamical systems. We show that traveling wave solutions may exist, with traveling wave speed dependent on the virus lifespan and the contact rate between the pest and the crop. Finally, we discuss the consequences in terms of control strategies. [ABSTRACT FROM AUTHOR]

A mathematical model presented in Berge T, Lubuma JM-S, Moremedi GM, Morris N Shava RK, A simple mathematical model for Ebola in Africa, J Biol Dyn 11(1): 42-74 (2016) for the transmission dynamics of Ebola virus is extended to incorporate vaccination and change of behavior for self-protection of susceptible individuals. In the new setting, it is shown that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than or equal to unity and unstable when . In the latter case, the model system admits at least one endemic equilibrium point, which is locally asymptotically stable. Using the parameters relevant to the transmission dynamics of the Ebola virus disease, we give sensitivity analysis of the model. We show that the number of infectious individuals is much smaller than that obtained in the absence of any intervention. In the case of the mass action formulation with vaccination and education, we establish that the number of infectious individuals decreases as the intervention efforts increase. In the new formulation, apart from supporting the theory, numerical simulations of a nonstandard finite difference scheme that we have constructed suggests that the results on the decrease of the number of infectious individuals is valid. [ABSTRACT FROM AUTHOR]

Two new explicit finite difference schemes for the solution of the one-dimensional Korteweg-de-Vries equation are proposed. This equation describes the character of a wave generated by an incompressible fluid. We analyse the spectral properties of our schemes against two existing schemes proposed by Zabusky and Kruskal (1965) and Wang et al. (2008). An optimisation technique based on minimisation of the dispersion error is implemented to compute the optimal value of the spatial step size at a given value of the temporal step size and this is validated by some numerical experiments. The performance of the four methods are compared in regard to dispersive and dissipative errors and their ability to conserve mass, momentum and energy by using two numerical experiments which involve solitons. [ABSTRACT FROM AUTHOR]

The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Planck's constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation term is approximated by nonstandard technique, the potential is approximated by its mean value on the cell and the complex dependent boundary conditions are handled by exact schemes. In the second scheme, the deficiency of classical schemes is corrected by the inner expansion in the boundary layer region. Numerical simulations supporting the performance of the schemes are presented. [ABSTRACT FROM PUBLISHER]

We design explicit nonstandard finite difference schemes for the nonlinear Allen–Cahn reaction diffusion equation in the limit of very small interaction length. In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimizing the restrictions normally associated with standard explicit finite difference schemes. The derivation involves splitting the equation into the space-independent and the time-independent different models. An exact nonstandard scheme is proposed for the space-independent model and energy conservative schemes are proposed for the time-independent model. We show the power of the derived scheme over the existing schemes through several numerical examples. [ABSTRACT FROM AUTHOR]

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation. [ABSTRACT FROM AUTHOR]

The tenth Ebola outbreak in the Democratic Republic of Congo (DRC) that occurred from 2018 to 2020 was exacerbated by long-lasting conflicts and war in the region. We propose a deterministic model to investigate the impact of such disruptive events on the transmission dynamics of the Ebola virus disease. It is an extension of the classical susceptible-infectious-recovered model, enriched by an additional class of contaminated environment to account for indirect transmission as well as two classes of hospitalized individuals and patients who escape from the healthcare facility due to violence and attacks perpetrated by armed groups, rebels, etc. The model is formulated using two patches, namely Patch 1 consisting of the three affected eastern provinces in DRC and Patch 2, a war- and conflict-free area consisting of the go-to neighboring provinces for escaped patients. We introduce two key parameters, the escaping rate from hospitals and the destruction of hospitals, in terms of which the effect of war and conflicts is measured. The model is fitted and parameterized using the cumulative mortality data from the region. The basic reproduction number [Formula: see text] is computed and found to have a complex expression due to the high nonlinearity of the model. By using, not a Lyapunov function, but a decomposition theorem in Castillo-Chavez et al.(in Castillo-Chavez et al. (eds) Mathematical approaches for emerging and reemerging infectious diseases: an introduction, vol 126. Springer Science & Business Media, Berlin, 2002), it is shown that the disease-free equilibrium is globally asymptotically stable when [Formula: see text] and unstable when [Formula: see text]. A nonstandard finite difference scheme which replicates the dynamics of the continuous model is designed. In particular, a discrete counterpart of the above-mentioned theorem on the global asymptotic stability of the disease-free equilibrium is investigated. Numerical experiments are presented to support the theoretical results. When [Formula: see text], the numerical simulations suggest that there exists for the full model a unique globally asymptotically stable interior endemic equilibrium point, while it is shown theoretically and computationally that the model possesses at least a one Patch 1 and a one Patch 2 boundary equilibria (i.e., Patch 2 and Patch 1 disease-free equilibrium) points, which are locally asymptotically stable. Some recommendations to tackle Ebola in a conflict zone are stated., (© 2022. The Author(s), under exclusive licence to Society for Mathematical Biology.)