Model Validation
Read this material on mathematical modeling. Try to complete the exercise at the end of the chapters. Don't worry if you have not yet learned the math necessary to complete the exercises. Rather, notice how many physical systems can be represented by mathematical models. We can then implement these models using programming languages like Python and R. We will discuss those later. Think about the types of systems you interact with, personally or in business situations. How might you use a mathematical model to help you understand this system?
2. Bounded growth (Phase line, Stable equilibrium)
2.2 Practice problem (Flu!)
Problem
For this practice problem, you remember a story your favorite history teacher once told in class about a faraway island.
Flu on Ledom island
Ledom Island is inhabited by the native Htam. The island and its people was discovered in the 15th century by a group of 20 explorers. The explorers recorded that when they reached the island there were 980 Htam.
Unfortunately, the explorers all were sick with the flu when they arrived. They decided to stay for some time on the island and only to continue their travels after they had recovered.
The explorers infected the Htam people and all of them got the flu as well. In no time, the whole island was ill with the flu. Fortunately, no one died.
Problem statement
Now that you know a little about mathematical modelling, you want to impress your history teacher by building a mathematical model for the flu epidemic on Ledom Island. How could the number of sick people have grown from 20 to a full-blown epidemic?
Mathematical model
When the 20 sick explorers arrived, they infected the 980 Htam people on Ledom Island with the flu. You want to investigate how the resulting flu epidemic might have developed. So, now you construct a mathematical model. (Surprise!)
First define the variables:
t: the time in days after the arrival of the explorers in the 15th century.
F(t): the number of people infected with the flu.
X(t)=F(t)/1000: the fraction of the infected people relative to the entire population.
Just after the explorers arrived, 2 percent of the population on Ledom Island was sick with the flu, so X(0)=0.02.
In this model, we will assume that the flu epidemic spreads so quickly, that everyone has been infected before the first people get better again.
Calculation
The differential equation for the fraction of people X(t) on Ledom Island infected with the flu virus is
\(\dfrac{dX}{dt} = kX(t) (1-X(t))\),
where k is a positive parameter. The initial condition is X(0)=0.02.