Numerical Modeling of COVID-19 Neurological Effects

ODE/PDE Analysis in R

William Schiesser

• Publisher: CRC Press
• Year: 2021
• ISBN:
  9781000509984
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Covid-19 is primarily a respiratory disease which results in impaired oxygenation of blood. The O 2 -deficient blood then moves through the body, and for the study in this book, the focus is on the blood flowing to the brain. The dynamics of blood flow along the brain capillaries and tissue is modeled as systems of ordinary and partial differential equations (ODE/PDEs).

The ODE/PDE methodology is presented through a series of examples,

1. A basic one PDE model for O 2 concentration in the brain capillary blood.

2. A two PDE model for O 2 concentration in the brain capillary blood and in the brain tissue, with O2 transport across the blood brain barrier (BBB).

3. The two model extended to three PDEs to include the brain functional neuron cell density.

Cognitive impairment could result from reduced neuron cell density in time and space (in the brain) that follows from lowered O 2 concentration (hypoxia).

The computer-based implementation of the example models is presented through routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Rather, the presentation is through detailed examples that the reader/researcher/analyst can execute on modest computers. The PDE analysis is based on the method of lines (MOL), an established general algorithm for PDEs, implemented with finite differences.

The routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the blood/brain hypoxia models, such as changes in the ODE/PDE parameters (constants) and form of the model equations.

1 Source of Neurological Effects One PDE Model

DOI: 10.1201/9781003243052-1

1.1 Introduction

COVID-19 neurological effects are generally described in the following introductory statement [2]:

While primarily a respiratory disease, COVID-19 can also lead to neurological problems. The first of these symptoms might be the loss of smell and taste, while some people also may later battle headaches, debilitating fatigue, and trouble thinking clearly, sometimes referred to as “brain fog”. All of these symptoms have researchers wondering how exactly the coronavirus that causes COVID-19, SARS-CoV-2, affects the human brain.

This book focuses on a mathematical model describing a reduction in oxygen (O₂) to the brain resulting from impaired respiratory function of the lungs. The reduced brain O₂ (hypoxia) decreases the population density of normal (healthy) neuron cells that then leads to brain cognitive impairment.

Patients who experience diminished O₂ in the capillary blood to the brain resulting from respiratory impairment are sometimes termed “Covid long haulers”, long haul Covids”, or “long Covids”.

Additional background concerning COVID-19 neurological effects is given in [1, 3, 4].

1.1.1 One Pde Model Formulation

A mass balance on the O₂ concentration in the capillary blood flowing in the brain follows (the balance is derived in chapter appendix A1):

$$\frac{\partial u_1}{\partial t}=-v_z\frac{\partial u_1}{\partial z}-(2/r_l)k_{m1}(u_1-u_{2n})$$(1.1-1)

Equation (1.1-1) states that the time rate of change of the blood O₂ concentration $\left(\frac{\partial u_1}{\partial t}\right)$ equals the sum of the rate of convection of O₂ along the capillary $(-v_z\frac{\partial u_1}{\partial z})$ and the rate of transfer of O₂ across the BBB (−(2/r_l)k_m1(u₁ − u_2n)).

Table 2.1 PDE model variables, parameters

u1(z,t)	oxygen concentration in capillary blood
z	distance along the blood capillar...