Compartmental Model Analysis of Epidemiologic Processes
Your first task in infectious disease modeling deals with an article written by Herb Hethcote. This will give you the flavor of how mathematicians approach epidemiological modeling and how you can translate what they present into Stella™ models. Familiarity with the approaches taken by mathematicians should make it easier for you to establish collaborations with them.
Hethcote HW. "Three Basic Epidemiological Models" Biomathematics 1989;18:119-44. (Springer-Verlag)
Homework X5.1
Given the units and definitions of the parameters l, g, and m in model 4.1
Homework X5.2
Provide some rationale as to why the method of balancing births and deaths used in model 4.1 is acceptable. Suggest two alternative mechanisms that could have been used.
Homework X5.3
Are the definitions of the parameters l, g, andm in model 4.2 the same as those if 4.1? If not, how do they differ?
Homework X5.4
Explain why the average time spent in the infected state is 1/(g + m). To help in your explanation, consider the situation where you start with a fixed number of infectives and do not have any inflow of new infectives.
Homework X5.5
Construct Stella™ models of 4.1 (corresponding to Figure 1) and 4.2 in which your stocks now represent fractions of the total population. Use your models to empirically confirm theorem 4.1.
Homework X5.6
a
Use the "scatter" option in Stella's graph function to construct phase diagrams like those in Figure 2. The student version of Stella will not draw the arrows for you. You have to watch the graph develop and then add arrows in a drawing program. For example if you import the graph into word it is easy to add the arrows.
b
Explain why all of the phase plane is confined to a single diagonal line in the graph.
c
What information in a time plot of I and S is lost in a phase diagram?
d
Why or how is the phase diagram informative?
Homework X5.7
Construct models 5.1 (Fig. 6) and 5.2 in Stella™. It is commonly stated that epidemics end because of "exhaustion of susceptibles". Examine the tables and graphs of the various flows and stocks in the model to help you develop an explanation as to why an SIR epidemic comes to an end.
Homework X5.8
a
Use your Stella model to empirically confirm Theorem 5.2
b
If you find an instance in your Stella™ model where a Theorem does not hold, does that show that the theorem is wrong?
Homework X5.9
Construct the phase diagram in Fig. 7 using Stella™.
Homework X5.10
Construct the epidemic curve in Fig. 8 using Stella™.
Homework X5.11
Use Stella™ to confirm the relationships in equation 5.5. Use a dt value that does not make the Stella™ difference equation model perform as a differential equation model. What happens to the relationships in equation 5.5 in this case? Explain in words why this relationship holds.
Homework X5.12
Construct models 6.1 and 6.2 and use these to confirm theorem 6.1.
Homework X5.13
Construct phase plane plots like those in figures 15 and 16.