# You’ve Got to be Modeling Me: Analysis of Past Submissions

*Joining us today is Keshav Patel, who is a NSF Graduate Research Fellow at University of Utah. He was a part of the team that finished as runners-up in the 2015 MathWorks Math Modeling Challenge (M3C). Keshav will be following up on a previous blog post regarding Part 1 of the 2019 MathWorks Math Modeling Challenge (M3C). If you have not read part 1 yet, we encourage you to take a look here. Over to you Keshav..*

Curve Fitting Solutions

A Close Look at a “good” Regression Model

Assumptions

Model

Strengths and Weaknesses

Extra Practice

Examining Other Regression Models

Extra Practice

Other Mathematical Methods

Ordinary Differential Equations Model

Extra Practice

Closing Thoughts

## Curve Fitting Solutions

### A Close Look at a “good” Regression Model

#### Assumptions

- Assumption: The percent of the population that uses vaping products is an accurate measure of the spread of nicotine due to vaping products. Justification: It would be unreasonable to determine the exact amount of nicotine used over the past couple years and predict it for the coming years. Each vaping product has a different amount of nicotine in it and, as seen during our investigation, existing data does not record the amount of nicotine each user consumed against a time metric. However, the spread of nicotine can be measured by its popularity in the US market, as the more people use nicotine-based vaping products, the more nicotine is used.
- Assumption: There is no new pertinent information regarding the dangers of nicotine-based vaping products or laws that will affect its popularity. Justification: Many studies and reports have already been released advocating the negatives of using nicotine-based vaping products, but despite this, as our research showed, the popularity of nicotine-based vaping products has continued to increase. Additionally, although the introduction of comprehensive FDA legislation in the August of 2016 did cause a sharp decrease in the popularity of nicotine-based vaping products [citation], since most relevant legislation regarding the use of nicotine-based vaping products has already been passed, and since these products did regain popularity in the aftermath of the legislation with the surge of vaping use in 2018, it is reasonable to assume that future legislation will not have a significant impact on the popularity of nicotine-based vaping products.
- Assumption: The carrying capacity of the market size for nicotine-based vaping products can be estimated using the historical carrying capacity of the market size for cigarettes. Justification: When analyzing the trends in the popularity of cigarettes, the group noticed that the initial growth in popularity of cigarettes closely mirrored that of nicotine-based vaping products like e-cigarettes.

#### Model

#### Strengths and Weaknesses

### Extra Practice

- What similarities and differences do you note in the assumptions? If you had opposing assumptions to another team, consider a) how well you justified your assumption, b) how your model would have to change if you used a different assumption, and c) which assumption you would rather use.
- What similarities and differences do you note in the figures? How does the formatting look (i.e. is the text big enough, are the different curves clearly labeled, is there spacing between table entries, etc.)? If you looked only at a team’s figures and tables (and their captions), could you understand the team’s results?
- Consider each team’s model. Is it clear what they are trying to do? Are the variables clearly marked or labeled in some way? Is there an aspect of their model or their results that go against the team’s assumptions?

### Examining Other Regression Models

- Poor communication of their assumptions and variables
- Poor summary of the mathematical model
- Poor formatting or placement of important components

- Assumption: Nicotine/Tobacco product usage trends will have a linear pattern in the coming decade. Justification: Both Normal Average and Exponential lines-of-best-fit proved to be highly problematic in their ability to project nicotine product usage. Therefore, a linear trend must be assumed.

### Extra Practice

- Do you think the above assumption is a good assumption to make for this problem? If yes, rewrite the justification to improve the argument. If no, write down a different assumption and justification and consider how this team’s model might change as a result.

- Assumption: Teens had the same access to cigarettes as they do to vaping. Justification: This allows for equal comparisons of the two forms of nicotine transmission.
- Assumption: Once a health issue is discovered, the vaping growth rate will decrease similarly to the decrease of cigarette usage after 1964. Justification: This can be assumed because of the known detrimental effects of nicotine.

## Other Mathematical Methods

### Ordinary Differential Equations Model

Next, the team spends quite a bit of the submission (perhaps too much) explaining how they compute an important parameter in SIR models, called

R0, from the available data. This parameter is a measurement for how many people on average a single infectious person ends up infecting. Finally, they give the following plot as their main result:

As we mentioned earlier, a sensitivity analysis, particularly on this all-important

R0parameter, might be a good idea to include to show how much variability exists in your model. Also, it is a good idea to refer back to your assumptions and discuss how they match up with your results, and where you could alter your assumptions or conduct further testing in the future. Thinking about the logical arguments you are making as well as the problem outside of the context of the competition are things readers would love to see!

### Extra Practice

- What similarities and differences do you note in the assumptions? What assumptions are made in the SIR models that are not made in the regression models, and vice versa?
- What similarities and differences do you note in the figures? How does the formatting look (i.e. is the text big enough, are the different curves clearly labeled, is there spacing between table entries, etc.)? If you looked only at a team’s figures and tables (and their captions), could you understand the team’s results?
- Consider each team’s model. Is it clear what they are trying to do? Are the variables clearly marked or labeled in some way? Is there an aspect of their model or their results that go against the team’s assumptions?

## Closing Thoughts

**Category:**- Math Modeling,
- MATLAB

## Comments

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