Discretization of an eco-epidemiological model and its dynamic consistency

In this paper, we discretize a continuous-time eco-epidemiological model by non-standard finite difference (NSFD) scheme as well as standard Euler forward scheme. Dynamical properties of both the systems are explored and compared with their continuous-time model. We show that the solution of NSFD system remains positive for all positive initial values. Fixed points and their local stability properties are shown to be identical with that of the continuous model, indicating its dynamic consistency. Dynamics of the Euler model, however, depend on the step–size and therefore dynamically inconsistent. Solutions in this latter method may be negative and allows numerical instabilities, leading to chaos. Extensive numerical simulations have been performed to validate the theoretical results.     

EXISTENCE OF POSITIVE PERIODIC SOLUTION OF A DISCRETE TIME MUTUALISM SYSTEM WITH DELAYS

Discrete-time analogue of mutualism model is introduced. The discrete-time analogues is considered to be numerical discretization of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models.     

Topological dynamic consistency of non-standard finite difference schemes for dynamical systems

This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care.     

A Positivity-preserving Mickens-type Discretization of an Epidemic Model

A deterministic model for the transmission dynamics of two strains of an epidemic in the presence of a preventive vaccine is considered. Theoretical results on the existence and stability of the associated equilibria of the model are given. A robust, positivity-preserving, non-standard finite-difference scheme, having the same qualitative features as the continuous model, is constructed. The theoretical and numerical analyses of the model enable the determination of a threshold level of vaccination coverage needed for community-wide eradication of the epidemic.     

Dynamically consistent numerical methods for general productive–destructive systems

The dynamic consistency of a class of non-standard finite-difference schemes is analysed for general 2D and 3D productive–destructive systems (PDS). Based on those results a methodology for construction of positive and elementary stable non-standard numerical methods is developed. The numerical techniques are based on a non-local modelling of the right-hand side function and a non-standard treatment of the time derivative. This discretization approach leads to significant qualitative improvements in the behaviour of the numerical solutions. The explicit form of the proposed new schemes makes them a computationally effective tool in simulations of the dynamics of systems of biological, chemical and physical interactions that are naturally modelled by PDS. Applications to several specific biological systems are presented.     

Non-standard finite difference method applied to an initial boundary value problem describing hepatitis B virus infection

In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method.     

An NSFD scheme for a class of SIR epidemic models with vaccination and treatment

An non-standard finite difference scheme is employed to discuss a class of SIR epidemic model with vaccination and treatment. The dynamical properties of the discretized model are then analysed. The results demonstrate that the discretized epidemic model is dynamically consistent with the continuous model since it maintains essential properties of the corresponding continuous model, such as positivity property and boundness of solutions, equilibrium points and their local stability properties.     

Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks

We formulate discrete-time analogues of integrodifferential equations modelling bidirectional neural networks studied by Gopalsamy and He. The discrete-time analogues are considered to be numerical discretizations of the continuous-time networks and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the equilibria of the continuous-time networks. By constructing a Lyapunov-type sequence, we obtain easily verifiable sufficient conditions under which every solution of the discrete-time analogue converges exponentially to the unique equilibrium. The sufficient conditions are identical to those obtained by Gopalsamy and He for the uniqueness and global asymptotic stability of the equilibrium of the continuous-time network. By constructing discrete-time versions of Halanay-type inequalities, we obtain another set of easily verifiable sufficient conditions for the global exponential stability of the unique equilibrium of the discrete-time analogue. The latter sufficient conditions have not been obtained in the literature of continuous-time bidirectional neural networks. Several computer simulations are provided to illustrate the advantages of our discrete-time analogue in numerically simulating the continuous-time network with distributed delays over finite intervals.     

Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with a general contact rate. The model exhibits the traditional threshold behavior. We prove that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is globally asymptotically stable and when the basic reproduction ratio is great than unity, a unique endemic equilibrium exists and is globally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle’s invariant set theorem. Numerical simulations are provided to illustrate the theoretical results.     

Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems

General two-dimensional autonomous dynamical systems and their standard numerical discretizations are considered. Nonstandard stability-preserving finite-difference schemes based on the explicit and implicit Euler and the second-order Runge–Kutta methods are designed and analyzed. Their elementary stability is established theoretically and is also supported by a numerical example.     

Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

This paper considers a non-linear system of ordinary differential equations, which arises in the study of hantavirus epidemics. The system has the property that the total population obeys the logistic equation. For this system, we use linearization and the dynamical properties of the logistic equation to analyze the dynamics of the subpopulation system. In view of the usual numerical instabilities associated with standard finite difference methods used for simulating such systems, we construct non-standard finite difference (NSFD) schemes, which preserve the dynamic properties of the system, and may therefore be used for its simulation.     

Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease‐free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease‐free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh‐Volterra type, to be globally asymptotically stable for a special case.     

Multistrain edge‐based compartmental model on networks

Multistrain diseases, which are infected through individual contacts, pose severe public health threat nowadays. In this paper, we build competitive and mutative two‐strain edge‐based compartmental models using probability generation function (PGF) and pair approximation (PA). Both of them are ordinary differential equations. Their basic reproduction numbers and final size formulas are explicitly derived. We show that the formula gives a unique positive final epidemic size when the reproduction number is larger than unity. We further consider competitive and mutative multistrain diseases spreading models and compute their basic reproduction numbers. We perform numerical simulations that show some dynamical properties of the competitive and mutative two‐strain models.     

The effect of vaccines on backward bifurcation in a fractional order HIV model

In this paper, a homogeneous-mixing population fractional model for human immunodeficiency virus (HIV) transmission, which incorporates anti-HIV preventive vaccines, is proposed. The dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when there is no vaccine. However, it has been shown that when the efficacy or dosage of vaccines is low, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium point (DFE) coexists with a stable endemic equilibrium point (EE) when the associated reproduction number is less than unity. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. A new critical value at the turning point should be deduced as a new threshold of disease eradication. We have generalized the integer LaSalle invariant set theorem into fractional system and given some sufficient conditions for the disease-free equilibrium point being globally asymptotical stability. Mathematical results in this paper suggest that improving the efficiency and dosage of vaccines are all valid methods for the control of disease.     

Suppression of numerically induced chaos with nonstandard finite difference schemes

It has been previously shown that despite its simplicity, appropriate nonstandard schemes greatly improve or eliminate numerical instabilities. In this work we construct several standard and nonstandard finite-difference schemes to solve a system of three ordinary nonlinear differential equations that models photoconductivity in semiconductors and for which it has been shown that integration with a conventional fourth-order Runge-Kutta algorithm produces numerical-induced chaos. It was found that a simple nonstandard forward Euler scheme successfully eliminates these numerical instabilities. In order to help determine the best finite-difference scheme, it was found useful to test the local stability of the scheme by direct inspection of the eigenvalues dependent on the step size.     

Dynamics of a two-strain vaccination model for polio

A new deterministic model for the transmission dynamics of two strains of polio, the vaccine-derived polio virus (VDPV) and the wild polio virus (WPV), in a population is designed and rigorously analysed. It is shown that Oral Polio Vaccine (OPV) reversion (leading to increased incidences of WPV and VDPV strains), together with the combined effect of vaccinating a fraction of the unvaccinated susceptible and missed susceptible children, could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. Furthermore, the model undergoes competitive exclusion, where the strain with the higher reproduction number (greater than unity) drives the other (with reproduction number less than unity) to extinction. In the absence of OPV reversions (leading to the co-existence of both strains in the population), it is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the associated reproduction number is less than unity. Numerical simulations of the model suggest that the model undergoes the phenomenon of competitive exclusion, where the strain with the higher reproduction number (greater than unity) drives the other to extinction. Furthermore, co-existence of the two strains is feasible if their respective reproduction number are equal or approximately equal (and greater than unity).     

A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell-Whitehead-Segel equation

In this work, we propose a finite-difference scheme to approximate the solutions of a generalization of the classical, one-dimensional, Newell-Whitehead-Segel equation from fluid mechanics, which is an equation for which the existence of bounded solutions is a well-known fact. The numerical method preserves the skew-symmetry of the problem of interest, and it is a non-standard technique which consistently approximates the solutions of the equation under investigation, with a consistency of the first order in time and of the second order in space. We prove that, under relatively flexible conditions on the computational parameters of the method, our technique yields bounded numerical approximations for every set of bounded initial estimates. Some simulations are provided in order to verify the validity of our analytical results. In turn, the validity of the computational constraints under which the method guarantees the preservation of the boundedness of the approximations, is successfully tested by means of computational experiments in some particular instances.     

Nonstandard finite difference method revisited and application to the Ebola virus disease transmission dynamics

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.     

Stability and traveling waves of an epidemic model with relapse and spatial diffusion

An epidemic model with relapse and spatial diffusion is studied. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. By using the linearized method, the local stability of each of feasible steady states to this model is investigated. It is proven that if the basic reproduction number is less than unity, the disease-free steady state is locally asymptotically stable; and if the basic reproduction number is greater than unity, the endemic steady state is locally asymptotically stable. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem, the existence of a traveling wave solution which connects the two steady states is established. Furthermore, numerical simulations are carried out to complement the main results.     

Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor

In this work we investigate the dynamical behaviors of Van der Pol-Duffing circuit (ADVP) with parallel resistor. The model is described by a continuous-time three dimensional autonomous system. The stability conditions of the equilibria are analyzed. The existence of periodic solutions and their stabilities about the node equilibrium point of the system are studied by using Hopf's theorem and Hsü and Kazarinoff theorem. Lyapunov spectrum is calculated for the proposed system. Adaptive synchronization using backstepping design is applied successfully to the system. Chaotic behaviors and the efficiency of the synchronization method are verified by numerical simulations.