2018

Instructions: Your report must address one, and only one, of the following problems.

"The Washington Ave Problem"

It can be argued that the street light system at intersections across Washington Avenue is suboptimal. That is, pedestrians often have to wait to cross at intersections with no vehicle activity, or cars wait at red lights needlessly. To that end, we are interested in measuring the effectiveness of a street light system at a given intersection. Note that, at Washington Avenue, there are more variables to consider given that its intersections are shared with bikers, university buses, and the light-rail train as well. In your report address, in a coherent narrative, as many of the following points as possible:

• Clearly define the variables and parameters you will incorporate into your model.

• Propose a way to measure the effectiveness of a street light system at a given intersection.

• Make a realistic, time-dependent model of activity in a typical day at an intersection of your choice.

• Propose a street light system that optimizes its effectiveness.

• Generalize your idea for measuring the effectiveness now, across multiple intersections.

"Video Game Engagement"

In the last decade, internet connectivity features, particularly online gaming, have seen huge acceptance in the video game industry, and in recent years it has allowed developers to profit from these online interactions in new ways, such as the selling of extra game features, collectibles, “pay-to-win” methods, etc.. However, in order that these profit strategies work, it is important that the player stays engaged in the online features. Now suppose you are helping to develop a video game whose online feature has one (human) player competing against one other (human) player in a match (for example, you are allowed to think of the video game as a chess game). We call the “match-making decision model” the way that the videogame decides which two human players to pit against one another in a match. For instance, many decision models strive to optimize fairness of each match, so they pit players of similar skills against each other. Your goal as a developer is to construct a match-making decision model which strives to maximize overall players’ engagement with the online feature. In your report address, in a coherent narrative, as many of the following points as possible:

• Clearly define the variables and parameters considered in your decision model.

• Propose a way to quantifiably define and measure player’s engagement with the online feature.

• Explain how your decision model maximizes player’s engagement.

• Note carefully that maximizing players’ engagement may not coincide with optimizing for fairness of each match. Investigate how your decision model differs, if at all, from one which optimizes fairness.

• Suppose now that your game has 2vs2 matches, where 2 (human) players are pitted against 2 (human) players in a match. How would you generalize your decision model to the setting of 2vs2 matches, where again the goal is to maximize overall players’ engagement?

"Nice LimeBird"

With the new trend of shared vehicles (e.g. bikes, and motorized scooters) around campus, you wish to understand the sustainability of such services in the long run. In other words, you aim to model the ratio of the population that uses one service over the other one (or both). In your report address, in a coherent narrative, as many of the following points as possible:

• Give an overview of how these services attract potential customers to use their service. Then, give a comprehensive summary of how their service works.

• Describe the parameters that affect an individual’s preference to choose one service over the other.

• Make a time-dependent model that incorporates your variables and parameters described in the previous point.

• Define a way to measure the sustainability of a service. Moreover, define criteria that identify a service as no longer profitable.

• Based on your model, predict which sets of parameters will benefit one service or the other. Moreover, identify parameters that will make a service go bankrupt in the long term.

"Do it for the Herd"

The University Health Services urges you to get your flu shot as to minimize the spread of flu. However, flu shots are limited by the amount of certified personnel and the availability of doses. Moreover, it is known that flu shots are not 100% effective and that some individuals may exhibit undesirable pain and side effects. Thus, some individuals prefer to not to get the flu shot. That being said, the task at hand is to model how flu shots affect the spread of the virus. Note that you may want to consider more parameters in your model. In your report address, in a coherent narrative, as many of the following points as possible:

• Give a comprehensive summary of the cycle of flu and how flu shots may affect such cycle.

• Identify relevant parameters to incorporate in your model (e.g, the ratio of healthy to infected individuals, initial conditions of the population, the effectiveness of flu shots, time to recover, acquired immunity to the disease, contagiousness etc)

• With the definitions above, state a quantitative measure for the flu spread.

• Build a mathematical model that incorporates your parameters above, and predicts the state of the population at future times.

• Make an argument as to which set of parameters minimize the flu spread.

• Argue about when is it the most critical time to increase the number of flu shots given.

2014: "The Hanta Virus"

The Hanta virus recently (1993) killed several people in the Four Corner regions of the United States. You are are a World Health Organization worker going to combat a new outbreak in a different region of the world and must determine the medical requirements for your unit. If just one copy of the virus enters a human body, it can start reproducing very rapidly. In fact, the virus can double its numbers in one hour! The human immune system can be quite effective, but this virus hides in normal cells. As a result, the immune response doesn't begin until the virus has one million copies floating around in the body.

One of the first actions of the immune response is to raise the body temperature, which lowers the virus replication rate to merely 150% per hour. The fever and then flu-like symptoms are usually the first indication of illness. Some people with the virus assume that they merely have the flu or a bad cold. This assumption leads to deadly consequences, since the immune response alone is not enough to combat the virus. At maximum reaction, the immune system can only kill 200,000 copies of the virus per hour.

To fully combat the illness, the infected person must receive an injection and hourly doses of a special antibiotic. The antibiotics do not affect the replication rate of the virus (the fever keeps it at 150%), but the immune system and the antibiotics together can kill 500,000,000 copies of the virus per hour. If these antibiotics are not started before the number of copies of the virus in the body reaches one billion, the virus cannot be stopped. When the virus reaches one trillion copies, the person will die.

Problem: Model the different phases of the illness. Namely, how long will it take for the immune response to begin? How much time does a person have to get to medical authorities?

Analyze your models. What assumptions have you made? What are the strengths and weaknesses of the models? How reliable are your results?

In addition to creating a general model, give a prediction for the following case: It's 2am and a worker comes to you because their body temperature just soared to 104 degrees. Thinking quickly they realize that they started feeling hot and achy at about 10pm the night before. Being the one of the first workers to become infected in the region, the medicine has to be flown in. While you're waiting, you can calculate the approximate time they were infected. When was that? The doctors work as quickly as possible, and but you don't get any medicine until 2pm the next day. Is it in time? How many copies of the virus are in the workers body?

Spring 2012: "Fish Farming"

In the recent years, fish farming has developed into an industry of about $US 60 billion. In particular, sturgeon is an important part of this industry due to the high market value of sturgeon filets and roe.

Find a harvesting policy in terms of population size and harvesting time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes this value over a realistic range of environmental conditions.

In order to do so, you will have to model the sturgeon’s natural interactions with its environment by expressing population levels of different groups in terms of the significant parameters. Do not forget to include any outside constraints imposed by food or space limitations that are supported by data that you found. Consider the value of the various quantities involved, the number of fish harvested, and the population size itself, in order to devise a numerical quantity that represents the overall value of the harvesting.

2010: "Museum Security"

Your security company has a contract with the Guild of Art Galleries to provide video surveillance at several locations. For aesthetic reasons, the museum curators want to have as few cameras as possible interfering with the art exhibitions. The camera resolution is good up to 25 feet when still, and up to about 8 feet when scanning horizontally. Besides, the angle covered by the camera lens is approximately 50 degrees.

Most museums try to maximize the area available to display paintings and other exhibits, so their floor-plans are quite convoluted, with several panes standing in the middle of a room at odd angles, and corridors connecting exhibitions. Although rooms can be long (up to 100 feet), opposite walls are never more than 30 feet apart so that visitors do not have to walk unnecessarily. All walls and panes are flat.

Your task is to design a procedure to place fixed and/or moving cameras in choice locations around the museum (curators prefer them in the corners) in order to provide the best possible coverage during the night. To validate, use your procedure on any mock museum floor-plan that includes the placement of panes.