Dynamically consistent nonstandard finite difference schemes for epidemiological models

Abstract

This work is the numerical analysis and computational companion of the paper by Kamgang and Sallet (Math. Biosc. 213 (2008), pp. 1-12) where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens' rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.

Numerical methods in the context of compartmental models in epidemiology

“…[9] provides a large deviations principle which allows for such diminishing rates; however, their assumptions do often not apply to the more complicated models in epidemiology. 3 [10] provides an LDP on the space D([0, T ]; A) 4 for a large class of epidemiological models with compact domain A by generalyzing the results from [9]; the rate function I T,x of the LDP is given as follows: 5…”

“…In order to analyze the deviation of the stochastic model from its Law of Large Numbers limit, we first have to compute the solution of the deterministic models in a reliable way in section 2. In order to avoid instabilities of the solution (such as components becoming negative), we apply a non-standard finite difference scheme developed in [4]. 6 We want to remark here that we cannot consider the exit from the domain of attraction of the disease-free equilibriumx, i.e., O = {z ∈ A| lim t→∞ Z z (t) =x}.…”

“…In section 2.2 below, we consider the SIS model from Example 1.1 and illustrate that the solution of such a scheme can be oscillating around zero (and in particular become negative) if the parameters (particularly the step-size h) are not chosen with great care. For this reason, we apply a non-standard finite difference (NSFD) method for models in epidemiology which respects the qualitative behavior of the solution of the ODE (see [4]). This NSFD method follows two basic rules for NSFD methods (for a detailed description of the NSFD method see, e.g., [17,18]).…”

“…We address these questions in sections 2 -4 as follows. First we apply a non-standard finite difference scheme due to [4] in order to compute the solution of the deterministic model numerically. Second, in section 3, by solving a control problem via dynamic programming, we compute the leading coefficient in the large deviations analysis of the time of exit τ N by the solution of the SDE from the domain of attraction of a locally stable equilibrium of the ODE.…”

Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control

“…To develop such a scheme we will consider the nonstandard finite difference approach [39,3]. In recent works, it has been shown that this type of finite difference scheme is able to preserve the positivity, the equilibria, the local or global asymptotic stability (instability) of the different equilibria, as well as bifurcation property [2,4]. Nonstandard methods have been applied successfully on various problems in epidemiology [4,20,19,23], in ecology [22] and in mechanics [24].…”