Mathematical Modeling in the Age of the Pandemic

William P. Fox

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Topic: Mathematics
• eBook ISBN:
  9781000395082
• Subtopic: Applied Mathematics
• Index: Mathematics

One cannot watch or read about the news these days without hearing about the models for COVID-19 or the testing that must occur to approve vaccines or treatments for the disease.

The purpose of Mathematical Modeling in the Age of a Pandemic is to shed some light on the meaning and interpretations of many of the types of models that are or might be used in the presentation of analysis. Understanding the concepts presented is essential in the entire modeling process of a pandemic.

From the virus itself and its infectious rates and deaths rates to explain the process for testing a vaccine or eventually a cure, the author builds, presents, and shows model testing.

This book is an attempt, based on available data, to add some validity to the models developed and used, showing how close to reality the models are to predicting "results" from previous pandemics such as the Spanish flu in 1918 and more recently the Hong Kong flu. Then the author applies those same models to Italy, New York City, and the United States as a whole.

Modeling is a process. It is essential to understand that there are many assumptions that go into the modeling of each type of model. The assumptions influence the interpretation of the results. Regardless of the modeling approach the results generally indicate approximately the same results. This book reveals how these interesting results are obtained.

Chapter 1

Modeling as a Process

1.1 Introduction

Why mathematical modeling? In the everyday news coverage of the present times, the word “model” and “models” are used as if everyone understands these concepts. The truth is most people do not. Not even everything mathematician gets the idea of a mathematical model.

Consider the importance of modeling for decision making in business (B), industry (I), and government (G), BIG. BIG decision making is essential to success at all levels. We do not encourage “shooting from the hip” or simply flipping a coin to make a decision. At times it might appears that either of these are happening every day. We recommend good analysis that enables the decision maker to examine and question results in order to find the best alternative to choose or decision to make. This book presents, explains, and illustrates a modeling process and provides examples of decision making analysis throughout.

Let’s describe a mathematical model as a mathematical description of a system using the language of mathematics. Why mathematical modeling? Mathematical modeling business analytics, operations research are all similar descriptions that represent the use of quantitative analysis to solve real problems. This process of developing such a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as, physics, biology, earth science, and meteorology), engineering disciplines (e.g., computer science, systems engineering, operations research, and industrial engineering), and in the social sciences (such as, business, economics, psychology, sociology, political science, and social networks).

The professional in these areas use mathematical models all the time. A mathematical model may be used to help explain a system and to study the effects of different components, and to make predictions about behavior (Giordano, Fox, & Horton, 2013).

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, optimization models, or game theoretic models. These and other types of models can overlap, of one output becomes the input for another similar or different model form. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments (Giordano et al., 2013). Any lack of agreement between theoretical mathematical models and experimental measurements lead to model refinements and better models. We do not plan to cover all the mathematical modeling processes here. We only provide an overview to the decision makers. Our goal is to offer competent, confident problem solvers for the 21st century. We suggest the books listed in the reference section in order to get familiar with many more modeling forms.

1.2 Background and the Modeling Process

1.2.1 Overview

Bender (1978) first introduced a process for modeling. He highlighted: formulate the model, outline the model, ask is it useful, and test the model. Others have expanded on this simple outlined process. Giordano, Fox, and Horton (2014) presented a six step process: identify the problem to be solved, make assumptions, solve the model, verify the model, implement the model, and maintain the model. Myer (1984) suggested some guidelines for modeling including formulation, mathematical manipulation, and evaluation. Meerschaert (1993) developed a five-step process: ask the question, select the modeling approach, formulate the model, solve the model, and answer the question. Albright (2010) subscribes mostly to concepts and processes described in previous editions of Giordano et al. (2014). Fox (2013) suggested an eight-step approach: understand the problem or question, make simplifying assumptions, define all variables, construct the model, solve and interpret the model, verify the model, consider the model’s strengths and weaknesses, and implement the model.

Most of these pioneers in modeling have suggested similar starts in understanding the problem or question to be answered and making key assumptions to help enable the model to be built. We add in this process the need for sensitivity analysis and model testing to help insure we have a model that is performing correctly to answer the appropriate questions.

For example, student teams in the Mathematical Contest in Modeling, were building models to determine the all-time best college sports coach. One team picked a coach that coached less than a year, went undefeated for the remaining part of the year, and won their bowl game. Thus, his season was a perfect season. Their algorithm picked this person as the all-time best coach. Sensitivity analysis and model testing could have shown the fallacy to their model.

Someplace bet...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• About the Author
• 1 Modeling as a Process
• 2 Discrete Dynamical System Models
• 3 Modeling Coupled Systems of Discrete Dynamical Systems
• 4 Modeling with Differential Equation
• 5 Systems of Differential Equations
• 6 Probabilistic Models
• 7 Hypothesis Tests
• 8 Two Samples Hypothesis Test (Means and Proportions)
• 9 Agent-Based Model with NetLogo
• 10 Concluding Remarks and Epilogue
• Index