Math project (mathematic modelling)

Brief Project Descriptions

1

Badger Culling

Culling is the process of removing breeding animals from the population based on certain
criteria. In the case of badgers, the culling is typically done in an effort to reduce the number
of cows infected by bovine tuberculosis (Mycobacterium bovis). Badgers are a vector for this
disease and farmers in England want to reduce the costs due to bovine TB since infected
cattle are typically destroyed. There is strong opposition to badger culling since there are
questions on whether culling is effective and what damage is done to the badger population.
In this project, you will build a population model for the badgers and determine under what
conditions bovine TB would be endemic to the population as well as predicting the effect of
culling on the badger population.

2

Native vs. Invasive Species

The introduction of an invasive species into a new environment can have catastrophic con-
sequences for the native species. Examples of non-native species that out compete native
wildlife include asian carp, kudzu, and zebra mussels. In this project, you will develop a pop-
ulation model between two competitive species and determine under what conditions would
one population dominate the other and when both populations would coexist. Additionally,
you will examine how predation of one species will effect the population dynamics.

3

Harvesting

When harvesting from a population, it is important to understand the effect of removing
members has on the stability of the population. This is important for both farm grown and
wild populations since over-harvesting can be a big concern for many animals species. In
this project, you will develop a population model subject to logistic growth with harvesting.
How does different harvesting strategies effect the total population? Can harvesting collapse
the population? What is the maximum number of members that can be removed from the
population without collapsing the entire population?

4 Age-Based Predation

Populations exhibit age structures, where the population is segregated into different age
groups. For example, immature animals are unable to reproduce but age into mature animals
that are capable of reproducing. Additionally, predators prey more often upon the immature
members of the population so predation can exhibit age based structures. In this project, you
will develop an age-structured population model of immature and mature members subject
to predation. How does the population evolve in the absence of predation? How does age
based predation affect the stable population sizes?

1

5 Zombie Model with Vital Dynamics

In this project, you will develop a model of the interaction between a human population
that exhibits logistic growth with a zombie outbreak. Under what conditions will a zombie
outbreak occur and how does births/deaths affect the dynamics of the epidemic?

6 SIR Vaccination Model with Vital Dynamics

Vaccination of young children can be an effective measure against the outbreak of numerous
diseases. Diseases like smallpox and polio have been effectively eradicated in the United
States by the use of vaccines. In this problem, you will develop an SIR model with Vital
Dynamics that models the vaccination of susceptible population. Using this model, you will
try to determine the effect vaccination has on the disease and make suggestions on effective
vaccination strategies.

7 Vertical Transmission in a SIR Model with Vital Dynamics

Vertical transmission is the infection of offspring at birth by the mother. Examples of this
type of infection might be HIV infections (vertical transmission is not the primary mode
of infection for HIV and AIDS) where a fraction of births by an HIV-positive mother are
infected. How does vertical transmission affect the dynamics of the infection?

8 Chytridiomycosis Infections in Amphibians

Chytridiomycosis is a fungal disease in amphibians and is linked to dramatic declines and
extinctions of various species. In this project, you will develop a basic model of a SIR-type
infection in a logistic population. How does the infection affect the stable population size
and under what conditions would the infection cause the population to go to zero?

9 Latent Infection SIS Model

For some diseases, there is a time period between the time the patient becomes infected
and when the can infect others. This latent period can have effects on the dynamics of the
infection. In this project, you will develop a SIS-type model that incorporates a latent period
and compare it to a SIS model without a latent period. A SIS-type model is similar to a
SIR model except that there is no acquired immunity after recovering from the infection.

10 SIS Quarantine Model

A possible control mechanism during an epidemic is quarantining infected individuals. In
this project, you will develop a SIS-type model with an additional quarantined group. How
does a quarantine program affect the epidemic? Could it prevent the infection from becoming
an epidemic and/or an endemic infection?

2

11 Mathematical Model of Vector Infection (Malaria)

Malaria is a mosquito-borne infection of humans caused by a type of microorganism called a
protist. In this project, you will develop and analyze a mathematical model that incorporates
the following aspects of a malarial outbreak: external source of infection and partial recovery.

12 Coupled SIS Models

The epidemiological models that we have seen so far have involved a homogeneous, well-
mixed population. We can also develop a model that incorporates the interaction between
two distinct, homogeneous, well-mixed populations. An example of where this type of model
might be a isolated population (say on an island) that is visited by a different population.
In this project, you will develop a coupled SIS model for each population. If a disease is
endemic in one population but not in the other, under what conditions would the coupling
cause the non-endemic population to become endemic?

13 Model of a Stocked Pay Fishing Lake

A stocked pay fishing lake is a commercial enterprise where fishermen pay to fish at a lake
that is stocked with attractive species of fish (catfish, carp, trout, bass, etc.). In this project,
you will develop a basic model for the dynamics of both the fish population in the lake and
the number of fishermen. Does the model predict a stable steady fish population? How does
stocking fish change the dynamics?

14 Warring Species

Many social species exhibit warfare where members of one group battles with one or more
opposing groups. This behavior is see in many different species of insects such as ants as
well chimpanzees and, of course, humans. In this project, you will develop a basic model
that describes the population dynamics of two warring populations when the war lasts longer
than a single generation. Which side wins? Do both populations kill each other off? Can
the war last perpetually?

3

MATH 377 Projects

In your final project you will pick one topic and do a deep dive into that topic.

COVID-19. This is an extremely fluid and challenging global situation we are in. I will try to support you

as best I can and be as accommodating as I can, so please reach out if you need something or if, say, the

dates are unreasonable.

Things you have to do:

Stage 1 – Due Mar 27th – Choose A Topic.

Stage 2 – Due April 3rd – Provide a preliminary update

Stage 3 – Due April 10th – Submit Report

Stage 1 – Choosing a topic. There are two options.

1) Pick from the list. I’m providing a list of topics you can choose from with relatively well defined

problems and questions to answer. The list and descriptions are on CourseSpaces.

2) Pick your own topic. You can write your report on any topic you wish. I’d love to hear about

something interesting I don’t know about, particularly if you are a dual major something from

your discipline would be very interesting. You want to choose a topic where the mathematics is

not completely trivial, nor so challenging you can’t say anything meaningful. I can see two

possible subcategories

a. You think of something novel and want to try it out yourself. I’d be very excited to see

these.

b. You want to take an established paper – or a couple papers – in a field and do more of

an “explanatory” project where your goal is to explain all the aspects of the model that

others have performed. You will be carefully citing their contributions and delineating

what is your contributions and what is the paper’s contribution. You will share a link to

the papers so I can read it in comparison. You are graded on the value you bring

explaining this complicated topic at the level of the Report Standards.

The expectation is that there is a random distribution of topics selected, so choose something

DIFFERENT than what other people you know are doing. The “Pick your own topic” should all be

different, but the “pick from the list” I am going to UPDATE this list to remove options after a few people

have chosen a specific topic. So it is in your best interest to choose on the sooner side and submit the

form with your topic choice.

When you have chosen a topic, use the CourseSpaces form to let me know. Did you recheck the Project

List to make sure I haven’t eliminated it due to too many people selecting it?

The due date for this is Mar 27th. You are not absolutely locked to this topic, if you investigate it further

and it turns out to suck, that’s fine just email me and we can change it. I will accept late submissions, up

to April 17th, if you email me by April 10th explaining the reason why you can’t reasonably complete by

April 10th.

Working together? You may work together with ONE other friend. If you choose to work together, my

standards will be a bit higher. Not twice as high, a bit higher. You’re welcome to “solicit” a classmate

through the online forum by saying what project you are working on. If so, your “preliminary update”

should be CCed with both emails and your single report will have both names.

Stage 2 – Preliminary Update. The due date for this is April 3rd. Over this week, I want you to try and

think about the contour of your project. I want to touch base with each students. You can either:

a) Send me an email. In the email you should type a couple paragraphs telling me the big idea of

the project that you have learned thus far AND any questions you have.

b) Have an online meeting with me. We can use blackboard collaborate to have a short discussion

about what you know thus far AND any questions you have. Send me an email with a few times

that work for you and I’ll reply with the link.

Stage 3 – The report. The report must include the following components.

Introduction: The report begins with an introduction section which explains the background of the

problem. Normally, you use this section to motivate why the subject of study is important and to

provide background knowledge, including prior work by yourself or others. This should be very readable

and nontechnical.

Model Derivation: In this section, you will describe the derivation of the model you will use. You should

discuss how you translate the problem described in the introduction into a mathematical model, stating

assumptions you have made with justifications and possible limitations the model might have. You

should also state the model equations explicitly, using LaTeX, Microsoft Equation, etc., along with

defining all the parameters and variables in the

model.

Results: In this section, you will conduct analysis and discuss the results of your analysis of the model

given in the previous section. You should state the mathematical methods used to obtain your results.

Your results should be stated in context of the original problem. For example, if your model shows the

population goes to zero, you should state that the analysis shows that the population collapses for the

specific case given. Graphs or other visuals will often be useful to aid the exposition of the analysis.

Conclusion: In this section, you will summarize your results from the two previous sections in context of

the original problem. You should state the general conclusions along with assumptions made and any

limitations of the current model or analysis. If relevant, your conclusion could also include a short

discussion of future approaches for a more sophisticated model that addresses limitations in the current

model.

Report Standards:

 Audience: Write the report as if you are writing to a classmate who knows nothing about your

specific model but knows the basics of what we learned in class. You don’t have to explain what

Linearization is, for instance, but some new formula relevant to your model you definitely do

need to explain where it comes from. Your classmate should be able to follow along through the

entire report. You are welcome to share your report with a classmate – if they are working on a

different problem – to see if it satisfies this test.

 The report should be typed. If you know Latex, use that, if you don’t use Word or similar word

processor where there is an equation editor functionality. In word, for instance, Alt + is the

command to bring up equations. If there are graphics or numerical results, try to use an

appropriate math software like MATLAB to draw or compute these. I’m happy to help if need

be.

 Mathematical Sophistication. The mathematics involved should not be so completely trivial

there is nothing of meaningful value added, nor so hard that all you can reasonably say is “other

people got this result” without being able to explain how. Feel free to bring this point up with

me in an email/meeting if you are unclear.

Grading:

 5% for completing the Stage 1 and Stage 2 Check in. I will grade this on the basis of a binary “Did

the student meaningfully participate in the check in”.

 20% for the Introduction

 20% for the Model Derivation

 25% for the Results

 20% for the Conclusion

 10% for “flow and polish”. The four sections should work together. Everything should be neatly

presented.

Note: Choosing a topic based on 1, 2a, or 2b will likely all look quite different. That’s ok. My goal is to

fairly evaluate regardless of these choices so I don’t think either of these three is automatically “easier”

or “harder”.

Academic Integrity: We are all doing the best we can given COVID-19, and part of all of our role is not

exploiting the challenges this poses. Specifically, the standards of academic integrity remain in place and

I am hoping that this just something we don’t have to think about. Nevertheless, let me be clear on two

rules:

1) If you use a published work you must reference it. If you copy something without referencing it

you will get a zero and be referred to the dean for academic dishonesty. I plan to google

anything that sets off my spidey senses as possibly being copied.

2) Similarly, the expectation is that students – beyond the pairs that are allowed to work together

– are working independently.

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