Introduction
This part contains a python computer program that solves the differential equation for an exponential function. However, the exponential function has a simple mathematical form:
![\[ f(t) = A_o e^{\beta t} \]](https://i1.wp.com/davidalarrabee.com/wp-content/ql-cache/quicklatex.com-1a42132ba17f78eb8e7175e7669cce16_l3.png?resize=97%2C21&ssl=1 "Rendered by QuickLaTeX.com")
Here, 
is the value of the function at time zero and 
controls how fast the exponential grows. In order to compute the doubling time use the formula below.
![\[ T_{\frac{1}{2}} = \frac{ln(2)}{\beta} \]](https://i2.wp.com/davidalarrabee.com/wp-content/ql-cache/quicklatex.com-a24af2409d9b736f5ef3ff0f11b5996d_l3.png?resize=86%2C42&ssl=1 "Rendered by QuickLaTeX.com")
Unfortunately, in most cases there is no simple simple mathematical expression for the solution of our epidemiological models. Consequently, the solution of many models require some form of computer application.
Even though there is a simple solution for an exponential, simulating this simple case provides insight into how to proceed in more complicated cases. Also, solving problems where we know the answer also gives us some insight into the accuracy of the techniques.
Differential Equation for the exponential
Some models take the form of a differential equation. In this case, the differential equation for the exponential is just
![\[ \frac{df(t)}{dt} = \beta f(t) \]](https://i1.wp.com/davidalarrabee.com/wp-content/ql-cache/quicklatex.com-a47440ed12d1a3092c2f5f1218da89fe_l3.png?resize=103%2C38&ssl=1 "Rendered by QuickLaTeX.com")
The next three models we encounter will also have a description using differential equations. Formulas exist for the solutions of the first two models. For the third we will rely on a computer generated solution.
Probably the best way of solving this equation on a computer is to use a standard package. I will illustrate using the odeint package in the scipy python package.
Python Code
# ************** exponential differential equation Solver
import numpy as np
from scipy.integrate import odeint
def f(y, t, parm): # returns the derrivative of f(t) (y)
E = y[0] # get the current value of y
beta = parm[0] # get the parameter
derivative = [beta*E] # calculate the derrivative
print(E,beta,derivative) # if you want to see where odeint is looking
return derivative
beta = 0.10 # growth rate
A0 = 1.0 # Initial value
e=2.7182818284 # Euler's number (approx)
parm = [beta] #ODE solver wants a list of Parameters
y0 = [A0] #ODE also wants a list of initial values
# Make an array of the times at which we want a solution
Finaltime = 50.
Increment = 1.0
times = np.arange(0.0, Finaltime, Increment)
# Call the ODE solver
Solutions = odeint(f, y0, times, args=(parm,))
for i in range(len(times)): # printout and compare with actual solution
t = times[i]
F = Solutions[i,0]
val = A0*e**(beta*t)
error = (val-F)/val
print(t,F,val,error)
If you are new to Python I would recommend you install the anaconda package to start with. https://www.anaconda.com/products/individual There are many tutorials around the web and which you choose is largely dependent on how much programming experience you have.
When I first started using Python, I was skeptical. Having put Python to the test with some larger longer problems, I have been pleasantly surprised. Python will be used to illustrate the numerical evaluations of these models.
If you are not familiar with scipy, information on the scipy package can be found here. https://docs.scipy.org/doc/scipy/reference/index.html
Code output
Here is the results from the code output, without the print statement in the function definition. For now I have avoided putting in fancy formatting for the output, or dropping a .csv file, which we will do later. When you look at the output, you can see the ODE (ordinary differential equation solver) does quite well!
Next Step
Finally, here is a link to the next model, where everyone eventually gets infected. https://davidalarrabee.com/?page_id=717