Math 3MB3 Assignment 2

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Math 3MB3 Assignment 2

Due: by 11:59pm on Monday, February 13, 2023 via Crowdmark

Instructions:  Please read the following background information and answer all ques- tions.  You may handwrite or type (e.g., Microsoft Word, LaTeX) your solutions, but if they are handwritten please make sure they are legible.  You are welcome to discuss this assignment with others in the class, but you must write up your own solutions; if your work too closely matches the work of another student, this will be treated as an act of academic dishonesty and appropriate measures will be taken. All work for this assignment is due by 11:59pm on Monday, February 13, 2023 through Crowdmark.  Late work is subject to the late work policy described in the official course outline.

Final Submission: Your final submission will be done through Crowdmark. You will need to upload a pdf file of your solutions for each question individually, either by scanning or taking a picture of your work. You will also need to upload pdf files of each of the plots you make for Problem 4 and screenshots of the R code used to generate those plots.

Background: Suppose we want to model the spread of an infectious disease through a population. One way we could do this would be to describe the number of people susceptible to the disease St  each week and the number of people infected with the disease It  each week (t ∈ {0, 1, 2, 3, . . . }). Susceptible individuals have a chance to become infected upon contact with an infected individual; we will assume that some fraction 0 < β < 1 of these contacts actually produce new infections. A fraction 0 < γ < 1 of infected individuals will also recover from the disease.  Often, it is also important to keep track of new individuals entering the population through births and individuals leaving the population due to death from natural causes (i.e., not dying from the disease).  We will assume that a fraction 0 < b < 1 of the total population N  (here, N is assumed to be constant) will produce new offspring.  To simplify our model, we will also assume that the same fraction 0 < b < 1 of susceptible and infected individuals die due to natural causes. We can summarize this description with the following discrete model:

St  = St − 1 + bN − βSt − 1 It − 1 − bSt − 1                     (1a)

It  = It − 1 + βSt − 1 It − 1 − γIt − 1 − bIt − 1 .                (1b)

Problems

Problem 1  [2 points]:  Do infected individuals who recover from the disease become im- mune (i.e., can they be infected again in the future)?  If you believe they don’t become immune, explain why. If you believe they do become immune, what term(s) would need to be added to the model to allow recovered individuals to be infected again in the future?

Note: All problems from here onward are referencing the model given above in equations (1a) and (1b) .   

Problem 2  [6 points]:  Find all equilibria of this model.  You should find that one equi-  librium involves no infected individuals (this is called the  “disease-free equilibrium”) and one involves some number of both susceptible and infected individuals (this is called the “endemic equilibrium”). Be sure to show your work.  

Problem  3  [7  points]:  Analytically determine conditions under which the disease-free equilibrium (the equilibrium with no infected individuals) is stable. You should state your  final answers in terms of inequalities on the quantity  . Be sure to show your work.  

Problem  4  [9  points]:   Suppose we know that in a population of N  =  50 people, a fraction b = 0.05 of susceptible and infected individuals die due to natural causes and a fraction γ = 0.2 of infected people recover from the disease.  Use R to simulate this model

and produce plots for various values of β  (Note:  beta and gamma are already used by R to reference some built-in functions, so you will need to name those parameters something different in your script; for example, using B for β and g for γ will work). Start with an initial state of 40 susceptible individuals and 10 infected individuals, and simulate the population over a period of 41 weeks (i.e., from t = 0 to t = 40). Explore the following set of values for

β:  {0.001, 0.01, 0.1} (Hint:  see Lab 4 for help with this).  Create three separate plots with descriptive axis labels and with titles that look like β = xx where xx is the value of β used to create that particular plot (you can just write out beta in the title, you don’t need to worry about getting the right symbol in there). With your final submission on Crowdmark, include each of these three plots and screenshots of the R code you used to produce each one. 

Problem 5  [4 points]:  Suppose someone from a public health agency has hired you to perform this modelling exercise for them. After you have completed all of the above analysis of your model, they come to you and ask you for a summary of your findings and predictions. Write a one-paragraph summary to convey your results to them. This should address what your analysis predicts will happen to the population in the long term and what the effects would be of lowering the fraction of contacts that cause infections, β (this is something they would have some control over through implementing various public health measures, so it is something they would be very interested in).


2023-02-10