Figure 3. Sensitivity analysis of the yellow fever model.
3. Sensitivity to the Initial Number of Incubating Humans
Yellow Fever: Introductory Exercises
Benenson (1990, p. 486) reports that "except
for a few cases in Trinidad in 1954, no outbreak of urban yellow fever has
been transmitted by Aedes aegypti in the Americas since 1942." Nevertheless,
Kalgraf's model is a useful instructional model for epidemiology. Also,
it may provide a launching point for models of more potentially epidemic
diseases such as hemorrhaging dengue (Dunham and Galvan). The following
exercises are designed for the student with an introductory interest in
epidemiology.
1. Build and verify
Build the model and verify that it performs as shown in Figures 1 and 2. Initialize the model with 19,900 vulnerable people and 100 incubating people. The mosquitoes may be initialized with 417,000 safe mosquitos and 83,000 new mosquitos. All other stocks may be set to zero.
2. Sensitivity to the Bites Per Day
Conduct the sensitivity analysis displayed in Figure 3.
The first simulation shows that the epidemic would never get started if
there were only 0.1 bites/day. The second simulation is the base case shown
previously. The remaining three simulations show more rapid increases to
higher peaks if the mosquitoes bite more frequently.
Figure 3. Sensitivity analysis of the yellow fever model.
3. Sensitivity to the Initial Number of Incubating Humans
Test the sensitivity of the Figure 2 results to a change in the initial value of the number of incubating people. Document your analysis with a comparative time graph like Figure 3 with the initial number of incubating people set at 10, 20, 50 and 100.
4. Reproductive Number
Review Hastings' (1997, p. 194) explanation of the reproductive
number of a basic epidemic model. This number combines several model parameters
to tell us the mean number of new infections caused by a single infective
individual. Derive an expression for a reproductive number for the yellow
fever model. Do the results in Figure 3 agree with the use of the number?
5. Causal Loop Diagram
Figure 4 shows two feedback loops in the epidemic model.
Complete this diagram by labeling each arrow as + or - and each loop as
(+) or (-). You will see a positive feedback loop which powers the initial
growth in the epidemic and a negative loop which applies the brakes to the
epidemic.
Figure 4. Two feedback loops at work in the yellow fever model.
6. The Logistic Function
Review the logistic function described in Chapter 6. Do you expect to see exponential growth in the number of people impacted by the epidemic during the early days? If so, estimate the value of "r", the intrinsic growth rate of the epidemic, based on model parameters (such as the bites/day and the number of days spent in each stage of the disease.) Does the negative feedback loop in Figure 4 apply the brakes to the epidemic in a linear manner, as required in the logistic function? What is the value of "K" to allow the logistic equation to give an accurate estimate of the number of people impacted by the epidemic?
7. Verify the Logistic Function
Review the method used to confirm the applicability of the logistic function in the flower model of Chapter 6. Apply the same method to confirm the relevance of the logistic function to the yellow fever. Define the total population impacted by subtracting the vulnerable people from the 20,000 people in the city at the start of the simulation. You should find a good match between the simulated value of total population impacted and the logistic equation with a K set to 17,721 and r set to 4.2%/day.