We formalize the transfer of essential properties of the solution of a differential equation to the solution of a discrete scheme as qualitative stability with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of the denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schemes. Extensions of some models are considered, and numerical examples confirming the efficiency of the nonstandard approach are provided.

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Since its emergence late in 2019, the COVID-19 pandemic continues to exude major public health and socio-economic burden globally. South Africa is currently the epicenter for the pandemic in Africa. This study is based on the use of a compartmental model to analyze the transmission dynamics of the disease in South Africa. A notable feature of the model is the incorporation of the role of environmental contamination by COVID-infected individuals. The model, which is fitted and parametrized using cumulative mortality data from South Africa, is used to assess the impact of various control and mitigation strategies. Rigorous analysis of the model reveals that its associated continuum of disease-free equilibria is globally-asymptotically stable whenever the control reproduction number is less than unity. The epidemiological implication of this result is that the disease will eventually die out, particularly if control measures are implemented early and for a sustainable period of time. For instance, numerical simulations suggest that if the lockdown measures in South Africa were implemented a week later than the 26 March, 2020 date it was implemented, this will result in the extension of the predicted peak time of the pandemic, and causing about 10% more cumulative deaths. In addition to illustrating the effectiveness of self-isolation in reducing the number of cases, our study emphasizes the importance of surveillance testing and contact tracing of the contacts and confirmed cases in curtailing the pandemic in South Africa.

International audienceModelling the biomechanics of growing trees is a non-classical problem, as the usual framework of structural mechanics does not take into account the evolution of the domain geometry due to growth processes. Incremental approaches have been used in rod theory to bypass this problem and to model the addition of new material points on an existing deformed structure. However, these approaches are based on the explicit time numerical algorithm of an unknown continuous model, and thus, the accuracy of the numerical results obtained cannot be analysed. A new continuous space-time formulation has been recently proposed to model the biomechanical response of growing rods. The aim of this paper is to discretise the corresponding non-linear system of partial differential equations and the linearised system in order to compare the numerical results with analytical solutions of the linearised problem. The finite element method is implemented to compute the space boundary problem and different time integration schemes are considered to solve the associated initial value problem with a special attention to the forward Euler method which is the analogue of the previously used incremental approach. The numerical results point out that the accuracy of the time integration schemes strongly depends on the value of the parameters. The forward Euler method may present slow convergence property and errors with significant orders of magnitude. Nevertheless, attention must be paid to implicit methods since, for specific values of the parameters and large time steps, they may lead to spurious solutions that may come from numerical instabilities. Hence, the second order Heun's method is an interesting alternative even if it is more time consuming

We deal with the following question: Can the consumption of contaminated bush meat, the funeral practices and the environmental contamination explain the recurrence and persistence of Ebola virus disease outbreaks in Africa? We develop an SIR-type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses. We prove that the full model has one (endemic) equilibrium which is locally asymptotically stable whereas, it is globally asymptotically stable in the absence of the Ebola virus shedding in the environment. For the sub-model without the provision of Ebola viruses, the disease dies out or stabilizes globally at an endemic equilibrium. At the endemic level, the number of infectious is larger for the full model than for the sub-model without provision of Ebola viruses. We design a nonstandard finite difference scheme, which preserves the dynamics of the model. Numerical simulations are provided.