Projected Surface Finite Elements for Elliptic Equations

In this article, we define a new finite element method for numerically approximating solutions of elliptic partial differential equations defined on “arbitrary” smooth surfaces S in RN+1. By “arbitrary” smooth surfaces, we mean surfaces that can be implicitly represented as level sets of smooth functions. The key idea is to first approximate the surface S by a polyhedral surface Sh, which is a union of planar triangles whose vertices lie on S; then to project Sh onto S. With this method, we can also approximate the eigenvalues and eigenfunctions of th Laplace-Beltrami operator on these “arbitrary” surfaces

Projected finite elements for systems of reaction-diffusion equations on closed evolving spheroidal surfaces

The focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is ``logically'' rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as the activator-activator and short-range inhibition; long-range activation

Structural identifiability analysis of epidemic models based on differential equations: A Primer

The successful application of epidemic models hinges on our ability to estimate model parameters from limited observations reliably. An often-overlooked step before estimating model parameters consists of ensuring that the model parameters are structurally identifiable from a given dataset. Structural identifiability analysis uncovers any existing parameter correlations that preclude their estimation from the observed variables. Here we review and illustrate methods for structural identifiability analysis based on a differential algebra approach using DAISY and Mathematica (Wolfram Research). We demonstrate this approach through examples of compartmental epidemic models previously employed to study transmission dynamics and control. We show that lack of structural identifiability may be remedied by incorporating additional observations from different model states or fixing some parameters based on existing parameter correlations, or by reducing the number of parameters or state variables involved in the system dynamics. We also underscore how structural identifiability analysis can help improve compartmental diagrams of differential-equation models by indicating the observed variables and the results of the structural identifiability analysis

Dynamics of a Vector-Borne model with direct transmission and age of infection

In this paper we the study of dynamics of time since infection structured vector born model with the direct transmission. We use standard incidence term to model the new infections. We analyze the corresponding system of partial differential equation and obtain an explicit formula for the basic reproduction number ℜ0. The diseases-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number is less than one, ℜ0 1. The disease will persist at the endemic equilibrium whenever the basic reproduction number is greater than one