Homeostasis refers to the biological robustness of some internal property even as external prop- erties change

1. Homeostasis refers to the biological robustness of some internal property even as external prop- erties change. One such example is our ability to maintain a steady body temperature in both cold and warm weather.
   Consider the homeostasis.mat file on Moodle, which contains t data describing external tem- peratures and corresponding y data describing the body temperature of lab mice (all in Celsius). Write a Matlab script that loads this data and find the cubic polynomial of best fit, i.e., the parameters a, b, c, and d for which the polynomial f (t) = a + bt + ct2 + dt3 fits the data. Do so by explicitly building the matrix and vector and solving for these parameters. (That is, DO NOT use the built-in polyfit function!) Your script should create a single plot that plots both the data and the best-fit function. Make sure to include all relevant labels and a legend!
2. Consider the following compartmental model for a zombie apocalypse, that tracks the fractions of individuals that are alive (A), zombies (Z), and dead (D):dA = −αAZγ dtdZ = αAZγ − βZ dtdD = βZ dtHere, α is the “zombification” rate that describes how efficiently zombies turn the living into the undead and β is the “death” rate that describes how quickly zombies waste away from lack of resources, and γ is a parameter that controls the “zombification” process. Consider the choices β = 1 and γ = 0.9.Write a Matlab script that uses ode45 to solve the differential equation for an initial condition representing a state where 99% of individuals are alive and 1% are zomibes. Find (i) a choice of α where at least half of the alive population survive in the long run and (ii) another choice of α where over 90% of the population eventually becomes a zombie and dies. Make a nice, neat plot for each case. In each figure plot the corresponding solutions for the alive, zombie, and dead populations, appropriately colored and with a nice legend.

Math 252: Mathematical Modeling Final (Take-home) Spring 2021 This test contains 2 problems. You are bound to the academic honesty statement below: “In accordance with Article II of the Trinity College Student Integrity Contract, I hereby confirm that the paper, exam, or other academic exercise I am submitting with this declaration represents my own work; that I have properly acknowledged and attributed any and all information and ideas that I have used from other sources; and that no collaboration unauthorized by the instructor of the course occurred in the course of its completion.” Instructions: The take-home portion of this test includes two programming tasks. You must submit a zip file labeled Yourlastname_Final to Moodle no later than noon on Wednesday, May 12. For example, if I were to complete this assignment, my zip file would be named Skardal_Final. This zip file should include the necessary m-files for the questions below. You may not work together. 1. Homeostasis refers to the biological robustness of some internal property even as external properties change. One such example is our ability to maintain a steady body temperature in both cold and warm weather. Consider the homeostasis.mat file on Moodle, which contains t data describing external temperatures and corresponding y data describing the body temperature of lab mice (all in Celsius). Write a Matlab script that loads this data and find the cubic polynomial of best fit, i.e., the parameters a, b, c, and d for which the polynomial f (t) = a + bt + ct2 + dt3 fits the data. Do so by explicitly building the matrix and vector and solving for these parameters. (That is, DO NOT use the built-in polyfit function!) Your script should create a single plot that plots both the data and the best-fit function. Make sure to include all relevant labels and a legend! 2. Consider the following compartmental model for a zombie apocalypse, that tracks the fractions of individuals that are alive (A), zombies (Z), and dead (D): dA = −αAZ γ dt dZ = αAZ γ − βZ dt dD = βZ dt Here, α is the “zombification” rate that describes how efficiently zombies turn the living into the undead and β is the “death” rate that describes how quickly zombies waste away from lack of resources, and γ is a parameter that controls the “zombification” process. Consider the choices β = 1 and γ = 0.9. Write a Matlab script that uses ode45 to solve the differential equation for an initial condition representing a state where 99% of individuals are alive and 1% are zomibes. Find (i) a choice of α where at least half of the alive population survive in the long run and (ii) another choice of α where over 90% of the population eventually becomes a zombie and dies. Make a nice, neat plot for each case. In each figure plot the corresponding solutions for the alive, zombie, and dead populations, appropriately colored and with a nice legend.