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2. Similarly to the chemistry models using the law of mass action, you may want
to change the “reaction rate” by, e.g., introducing an exponent
α
such that
SICK RATE
=
CONTACT RATE * (CONTAGIOUS+SICK) *NON IM-
MUNE. Vary
α
in consecutive runs, for example, set
α =
0
.
9,
α =
1
.
1,
α =
1
.
3.
0 has
shown some historical veracity; i.e., the form has been sufficiently fit with
historical data.)
3. For the model of Section 2.1, connect the birth rate with the immune population
and try to reach a steady-state immune population.
4. Does the disease die out of the population?
5. Is it possible to wipe out the population with a variation in the parameters in
this form of the model?
6. Does your answer change if you allow for randomness of the contact rate?
7. Suppose that it takes anywhere between 1 and 5 days for someone to get so sick
that they choose to stay in bed. Should that variation be reflected in the model
as a random variable or be changed from model run to run?
8. Can you introduce an optimum (minimum number of sick) vaccination program
to stabilize the disease in this latter form of the model?
9. Can you model how the disease can frustrate the vaccination program through
mutation?
10. a) Can you break the population into age groups with different contact rates,
death rates, birth rates, initial populations, and disease-induced death rates for
each?
b) Show how some of these folks seem to be more resistant to the disease and
skip from contagious to immune directly.
c) Show how the result changes when immunity is slowly lost. (Note that in real-
ity, the immunes mingle with the nonimmunes and therefore dilute the original
effect of the contact rate coefficient. Can you fix this problem?)
11. Develop an epidemics model that captures the same population but distin-
guishes two regions. Immune and contagious people can travel but sick indi-
viduals cannot. People from the two regions have different contact rates and are
affected by the disease differently, i.e. the recovery rate differs between the two
subgroups of the population. What are the implications for an optimal vaccina-
tion program that does not restrict travel between the regions?
12. Let the virus in the model of Section 2.4 evolve to a less deadly form in humans
and show the impacts of this evolution.
13. Introduce a spatial component into the model of Section 2.5 by considering two
regions with different contact rates and different vaccination programs. Inves-
tigate the implications of travel restrictions imposed by one of the regions on
individuals originating in the other region.
14. Introduce longer delays in the CONTAGIOUS and SICK populations and elab-
orate on the effects of each on the trajectory of the IMMUNE population.
15. Show how the disease could be eliminated by reducing the CONTACT RATE
1 2 in the model of Section 2.5.
How do the model results differ? (Presumably, the basic form
α =
1
.
