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2. Introduce a vaccination program into your chaotic epidemic model. How would
the program need to be structured to prevent “chaos” (in the modeling sense)?
3. In the model, can you find chaos for a fixed (constant) contagion rate simply by
varying SICK TIME COEFF? Interpret your results.
4. Change the exponent in the GETTING WELL equation to values
a. less than 2
b. larger than 2.
Interpret your findings.
BASIC EPIDEMIC MODEL WITH CHAOS
SICK(t)
=
SICK(t
−
dt)
+
(GETTING SICK
−
GETTING WELL) * dt
INIT SICK
=
10
INFLOWS:
GETTING SICK
=
CONTAGION RATE * SICK
OUTFLOWS:
GETTING WELL
SICK
∧
2/SICK TIME COEFF
AWARENESS LEVEL
=
=
.1*SICK
LAG SICK
DELAY(SICK,DT)
SICK TIME COEFF
=
=
150
CONTAGION RATE
GRAPH(AWARENESS LEVEL)
(0.00, 1.99), (1.00, 1.92), (2.00, 1.84), (3.00, 1.73), (4.00, 1.58), (5.00, 1.41),
(6.00, 1.19), (7.00, 0.94), (8.00, 0.7), (9.00, 0.32), (10.0, 0.00)
=
13.2 Chaos with Nicholson-Bailey Equations
2
13.2.1 Host-Parasitoid Interactions
In Chapter 3, we modeled the spread of a parasitic infection in an insect population
of two life stages. The focus of that model was the spread of the infection. Therefore,
we ignored the fate of the parasitoid. In this chapter however, we model explicitly
the interactions between the host and the parasitoid populations. Rather than setting
up our model in terms of population sizes, we specify host-parasitoid interactions
in terms of population densities.
In order to model the host-parasitoid interactions, we abstract away from the
fact that only specific life cycle stages exhibit those interactions. After you worked
through this chapter, you may want to refine the model to account for the fact that,
2
This chapter follows L. Ederstein-Keshet, Mathematical Models in Biology, Random House,
1988, 79-85, and D. Brown and P. Rothery, Models in Biology: Mathematics, Statistics and Com-
puting, Wiley, NY, 1993, pp. 399-406.
