# Week 5

### Topics:

• Modeling continuous systems using Ordinary Differential Equations (ODE).
• Direction Fields
• Equilibria and their stability types, Linearization Theorem
• Logistic diferential equation
• Interacting populations

### Assignment:

1. Computing e: Read Tutorial 1 in PHASER for computing e using the initial-value problem x' = x, x(0) = 1. Recall that the exact solution of the initial-value problem x' = x, x(0) = x_0 is x(t) = x_0 e^t. Using the method in this tutorial, compute sqrt(e), 3.1*sqrt(e), 4.5e^2, and -1/e. Check your numbers against a calculator (Google calculator, for example). Report any differences. (Hints: stop your computations at the appropriate time; you can solve for negative time; you can change the initial condition.)

2. Harvest to extinction: Suppose that the size x(t) of a population is governed by the differential equation x' = 1.25 x - 0.16 x^2 - 0.45. This is the logistic model with constant harvesting. Using Phaser answer the following questions. Default Algorithm Dormand-Prince 5(4) and the Step Size 0.01 are fine to use.
If the initial population size is x(0) = 4.1, find the time at which the population doubles.
If the initial population size is x(0) = 0.4, find the time at which the population is exterminated.

3. Gompertz model of cancer growth: The differential equation
x' = a*(exp(-b*t))*x
is used to describe the growth of a tumor, where x(t) is a measure of its size (e.g. weight or number of cells), and a and b are parameters specific to a particular tumor. To get started, let us take a = 3 and b = 2, and x(0) = 5; the solution in the XivsTime view is shown in the image below.

Load the image into your PHASER by clicking on the picture. This picture is not to scale. Adjust the Window size to get a true aspect ratio and draw the Direction Field. Using the Direction Field and solution curves answer the following questions.
Explain what this model predicts for the tumor growth. Experiment with several different initial tumor sizes. How does the future size of tumor depend on its initial size?
Experiment with several values of the parameters a and b. Explain the roles of these parameters in the tumor growth.

Consult the famous article "Dynamics of Tumor Growth" by A.K. Laird in British Journal of Cancer (1964) 18, 490-502. doi:10.1038/bjc.1964.55
Verify that the formula for Gompertz curve x = x_0 exp((a/b)(1 - exp(-bt)) that she gives in this article satisfies Gompertz's differential equation above.

4. Epidemics: For certain infected diseases, a suceptible individual becomes infected and then either dies or recovers with immunity to the disease. Let S(t) denote the number of suceptible individuals and I(t) the number of infecteds in a population of fixed size. The change in the numbers of suceptibles and infecteds is often modeled (Kermack-McKendrick) with the pair of differential equations
S' = -r*S*I
I' = r*S*I - a*I
where the parameters r (infection rate) and a (death or recovery rate) take on nonnegative values.
The problem is to determine how the number of infected individuals change in time. If the number of infected individuals increase from the initial value I_0 at some some point in future time, then we say that an epidemic occurs.
• Enter these equations into Phaser. Set a = 0.6 and r = 0.003, and the initial number of infected individuals I_0 =20. Using the Xi-Values and Xi-vs-Time views, answer the following questions.
• If the initial number of Suceptibles S_0 is small enough, infecteds I(t) decreases from I_0 monotonically to zero --- no epidemics. If S_0 is suffciently large, the infected population first increases from I_0 to some maximum value before it dies out --- epidemics. What is the smallest number of suceptibles S_0 for which epidemics occur? This is called the threshold value for an epidemic to occur.
• How does the treshold value depend on the parameters a and r?