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Homework Set 1: Immune System

2022

Problem 1. Two Equation Model of the Immune System

Consider the paper discussed in class and provided in this assignment by Mayer et al. (A basic mathematical model of the immune response, Chaos, pp. 155 – 161, 1995).

Consider the plots in Figures 3a – c.  

A. Explain, physiologically, what these plots are representing.  In particular, explain what Figure 3c is showing

B. In this question, I am asking you to use ODE 45 in Matlab, the approaches that we have used this semester to solve a system of differential equations, and the equations in the Mayer paper that are used to represent the immune response of Effector (immune) cells to Target (e.g., virus or bacteria) in Equations (3) and (4) to recreate the plots in Figure 3a, b and c.  NOTE:  For Figure 3a and 3b, you will only be re-creating one of the curves shown (not two).

a. First, create an m-file (e.g., immune.m) from which to call ode45.  You may include parameter values for r, k, p, s, u, v, n, and d in this m-file since I am not asking you to change any of these parameters (the values to use are provided in Figure 3's caption)

b. In a separate m-file or the command window, consider a time range from 0 to 20 with an initial value of 1 for the Target cells and 0 for the Effector cells. NOTE:  All units are arbitrary.

c. Use ode45 and your immune.m file above (you will need to specify the options for odeset as in Homework 1 and the previous Matlab tutorial).

d. Create sub-plots for T vs time (subplot 311), E vs. time (subplot 312), and E vs. T (subplot 313). NOTE: You may wish to use the 'hold on' command or write your code in such as way that you can put the second set of plots (see below) on the same plots as these.

e. From your plots or the variables (preferable) determine the value referred to as E_m in the paper; this is the value of the Effector Cells when the Target cell population reaches 0 (i.e., when the infection is removed).

f. Now consider a time range from 20 to 40.  Use an initial value of 1 again for the Target cells and E_m for the Effector cells (note that this creates as infection at the same level as the initial infection from a – d above.

g. Again plot T vs time (subplot 311), E vs. time (subplot 312), and E vs. T (subplot 313) ON THE SAME PLOTS AS IN (d) above.  

h. Do your results match those in Figures 3a – c?  They should.

C. Discuss how accurate this model is at predicting response to infections.  What can you say about the second infection, given that the level of the infection (T = 1) was the same in both cases? Based on your knowledge of the immune system, is this reasonable?  Why or why not?  

NOTE:  In answering part (d), you should explain what would be happening at the cellular level.

Problem 2. Population model of immunity.

In class, we considered an SIR (Susceptibility, Infection, Removal) model of population response to pathogens (e.g, SARS-CoV-2).  Use this three-system set of equations to explore trends in susceptibility, infection, and removal.

A. Write an m-file and appropriate script commands for the SIR model of 3 differential equations.  Run the simulation three times for the parameters indicated below to give plots of S, I, and R as functions of time (use time of 0 to 1000; units can be arbitrary, but the values that I have chosen would correspond to days).  You should use 3 subplots to present your results (instructions on how to use the subplot function were in the previous tutorial).  Subplot 1 (top) should be for S (susceptibility), subplot 2 (middle) should be for I (infection) and subplot 3 (bottom) should be for R (removal). For these plots, you should also show all three runs of the simulation on the same plot (e.g., as blue, red and yellow lines for the different r values on each subplot) and you should include a legend.  You may do this by using a loop.  A simple, but not elegant, way to do this is to use the "hold on" command and then just re-run the simulation.  This will keep your previous results on the plot and overlay new results.  Be sure to label your x and y axes.

a. For each simulation, use a gamma value of 0.05

b. For the 3 different simulations, use the following r values:

i. r = 0.1

ii. r = 0.5

iii. r = 0.9

B. Consider the middle subplot (infection).  In terms of SARS-CoV-2 and COVID-19, we continue to hear about flattening the curve. In terms of your understanding of the immune system and the r parameter, what is needed in order to flatten the curve?

NOTE: For your answer, you might also consider the impact of the increasing r value on both the timing of the peak and its amplitude?  

C. What impact does r have on susceptibility and recovery?  Why?

D. Clearly, more complex models can be generated, but how might models such as these be used to shape public policy?  What are the positives and negatives of this?