Combinations of nonstandard finite difference schemes and composition methods with complex time steps for population models

We propose an efficient numerical method for two population models, based on the nonstandard finite difference （NSFD） schemes and composition methods with complex time steps. The NSFD scheme is able to give positive numerical solutions that satisfy the conservation law, which is a key property for biological population models. The accuracy is improved by using the composition methods with complex time steps. Numerical tests on the plankton nutrient model and whooping cough model are presented to show the efficiency and advantage of the proposed numerical method.

On a new fractional-order Logistic model with feedback control

In this paper,we formulate and analyze a new fractional-order Logistic model with feedback control,which is different from a recognized mathematical model proposed in our very recent work.Asymptotic stability of the proposed model and its numerical solutions are studied rigorously.By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function,we show that a unique positive equilibrium point of the new model is asymptotically stable.As an important consequence of this,we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability.Furthermore,we construct unconditionally positive nonstandard finite difference(NSFD)schemes for the proposed model using the Mickens’methodology.It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model.Finally,we report some numerical examples to support and illustrate the theoretical results.The results indicate that there is a good agreement between the theoretical results and numerical ones.

Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change.The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment.Firstly,we provide a theoretical study of the nonlinear differential equations model obtained.More precisely,we derive the effective reproduction number and,under suitable conditions,prove the stability of equilibria.Afterwards,we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one.We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models.In fact,when a backward-bifurcation occurs,the disease-free equilibrium may be globally stable.Numerically,we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak,and perform the sensitivity analysis.To control Ebola epidemics,our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy.Secondly,we propose and analyze a fractional-order Ebola epidemic model,which is an extension of the first model studied.We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme,and show its advantages.