Mathematically Modeling PCR: An Asymptotic Approximation with Potential for Optimization

A mathematical model for PCR (Polymerase Chain Reaction) is developed using the law of mass action. Differential equations are written from the chemical equations, preserving the detail of the complementary DNA single strand being extended one bas e pair at a time. The equations for the annealing stage are solved analytically. The method of multiple scales is used to approximate solutions for the extension stage. A map is then developed from the solutions to simulate PCR. The advantage of this model is the ability to use the map to optimize the process. Our results suggest that dynamically optimizing the extension and annealing stages may significantly reduce the total time for a PCR run

Homogenization of Large-Scale Movement Models in Ecology with Application to the Spread of Chronic Wasting Disease in Mule Deer

A difficulty in using diffusion models to predict large-scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). In this work, we derive a homogenization procedure for ecological diffusion, which allows us to determine the impact of small-scale (10-100 m) habitat variability on large-scale (10-100 km) movement, and apply it to models for chronic wasting disease (CWD) in mule deer. CWD is an infectious prion disease that affects members of the Cervidae family. It is a slow-developing, fatal disease, which is rare in the free-ranging deer population of Utah. We first present a simple spatial disease model to illustrate our homogenization procedure and the use of ecological diffusion as a way to connect animal movement with disease spread. Then we develop a more disease-specific sex-structured model for the spread of CWD, incorporating both horizontal and environmental transmission pathways. We apply our homogenization technique to greatly reduce the computational load for a simulation of disease spread from the La Sal Mountains to the Abajo Mountains of Southeast Utah. We use the averaged coefficients from the homogenized model to explore asymptotic invasion speed and critical population size for portions of our study area. Lastly, we describe the estimation of motilities for the disease-specific model from GPS location data, using a continuous-time correlated random walk model

Priorities for synthesis research in ecology and environmental science

ACKNOWLEDGMENTS We thank the National Science Foundation grant #1940692 for financial support for this workshop, and the National Center for Ecological Analysis and Synthesis (NCEAS) and its staff for logistical support.Peer reviewedPublisher PD

Priorities for synthesis research in ecology and environmental science

ACKNOWLEDGMENTS We thank the National Science Foundation grant #1940692 for financial support for this workshop, and the National Center for Ecological Analysis and Synthesis (NCEAS) and its staff for logistical support.Peer reviewedPublisher PD