The Case of Yellow Fever

Extra Exercises for Chapter 8. Epidemic DynamicsThe Case of Yellow FeverYellow Fever has been described as a "major and terrifying scourge in the Americas, causingendemic disease in jungle and swamp areas from Canada to Chile and claiming tens of thousands oflives in periodic urban epidemics" (Garrett 1994, p. 66). Garrett reports that "a 1793 epidemic inPhiladelphia killed 15% of the city's population and sent one of three residents fleeing into thecountryside." Yellow fever is transmitted in urban areas by the bite of infective Aedes aegyptimosquitoes. It is "an acute infectious viral disease of short duration and varying severity" where the"case fatality rate may exceed 50% among no indigenous groups and in epidemics" (Benneson 1990,486).These exercises introduce you to a model to simulate the spread of Yellow Fever. The model wasadapted from a Dynamo model by Kjell Kalgraf's published in Goodman's (1983) Study Notes in SystemDynamics. Kalgraf describes the classical (urban) form of yellow fever in which a human contractsyellow fever from the bite of infectious mosquitos. The mosquitos, in turn, can only become infectiousby biting contagious humans. The model is more complex than the SIR model in chapter 8, so it is goodpractice to expand on what you have learned in the book.Figure 1 shows a model based on a human population of 20,000. Kalgraf mentions an epidemicin Veracruz, Mexico, a city of 20,000 people back in 1899. Yellow Fever spread through the Veracruz in1899, and deaths reached 460 per month at the peak of the epidemic. The model assumes that thehumans live within contact of a mosquito population of around 500,000. Mosquitoes are shown on theleft; humans on the right. Values are shown for the peak day of a simulated epidemic.The Mosquitoes: Adult mosquitoes are assumed to emerge from the pupae at the rate of 27,778 per day;the total population remains at around 500,000 for the duration of the epidemic. The mosquitoes live for18 days and need four blood feeds during their lifetime. This means the average mosquito bites 0.2persons per day, so there would be around 100,000 bites each day. If a mosquito avoids biting acontagious human during their first three days, they "become safe" for the remainder of their adult life.If they bite a contagious human during this initial period, they become dangerous to humans. Theremaining 15 days is characterized by a 12 day incubation period and a 3 day infectious period.The Humans: A human contracts yellower fever if bitten by an infectious mosquito. This person entersan incubation period for 4.5 days, followed by a contagious period of 4.5 days and a sick period of 2.5days. After experiencing the sickness for 2.5 days, 90% of the humans recover and become immune toyellow fever. The model keeps track of the immune population and the cumulative number of deathsAndrew Ford BWeb for Modeling the Environment 1

1. Build and Verify.Figure 1. Model of Yellow Fever with values shown at the peak of the epidemic.Build the model in Figure 1. Initialize the model with 19,900 vulnerable people and 100incubating people. The mosquitoes may be initialized with 417,000 safe mosquitos and 83,000 newmosquitos. All other stocks may be set to zero. Simulate the model for 280 days, and you should see theresults in Figure 2. The simulation begins with 100 people in the incubation stage. The epidemic reachesits peak by the 140th day with around 450 sick people. The peak death rate is around 18 people/day.(Kalgraf does not comment on the magnitude of the peak death rate, but it is approximately the same asthe reports from Veracruz.) The simulation shows the peak of the epidemic in the 140th day. Figure 1focuses on the peak day by providing a "snap shot" in what appears to be an equilibrium diagram. Butyou can see that system is not in equilibrium because there are over 8,000 people who remain vulnerableto the disease.The mosquito stocks, on the other hand, are close to dynamic equilibrium. There are 81,259mosquitoes in the "new" category. Each of these mosquitoes bites 0.2 humans per day, for a total of over16,000 bites per day. But only 4.2% of these bites draw blood from a contagious human, so they becomedangerous flow is 686 mosquitoes/day.Andrew Ford BWeb for Modeling the Environment 2

Moving down the dangerous side of the mosquito stocks, we see 7,909 mosquitoes in theincubation stage and 1,945 in the infectious stage. The 1,945 infectious mosquitoes are responsible forthe humans that contract yellow fever in the 140th day. Notice that 8,103 humans are in the vulnerablecategory at this point. The epidemic has reduced the total population to 19,014, so 42.6% of the humansare vulnerable. The flow of humans contracting yellow fever is calculated as follows:1,945 contagious mosquitos*0.2 bites/day*42.6% vulnerable = 166 persons/dayThis flow adds to the stock of people in the incubation stage. Figure 1 shows 781 people in theincubation stage, 803 in the contagious stage, and 450 sick. With a 2.5 day period of sickness, therewould be 180 people reaching the final stage of the disease. Kalgraf assumes that 10% would die, so 18people are simulated to die in the 140th day.Figure 2. Simulated outbreak of yellow fever in a city of 20,000.2. Sensitivity to the Bites Per Day: Conduct the sensitivity analysis displayed in Figure 3.Figure 3. Sensitivity analysis of the yellow fever model.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 morerapid increases to higher peaks if the mosquitoes bite more frequentlyAndrew Ford BWeb for Modeling the Environment 3

3. Sensitivity to the Initial Number of Incubating HumansTest the sensitivity of the Figure 2 results to a change in the initial value of the number of incubatingpeople. Document your analysis with a comparative time graph like Figure 3 with the initial number ofincubating people set at 10, 20, 50 and 100.4. Reproductive NumberReview 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 bya 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 DiagramFigure 4 shows two feedback loops in the epidemic model. Complete this diagram by labeling eacharrow as + or - and each loop as (+) or (-). You will see a positive feedback loop which powers theinitial growth in the epidemic and a negative loop which applies the brakes to the epidemic.new mosquitosemerging per daymosquitos in theincubation stagemosquitos in thecontagious stagefranction of humansthat are vulnerablevulnerable peoplebites per day thatinfect mosquitosbits per day thatinfect humansfraction of humans thatcan infect mosquitoshumans in theincubation stage6. The Logistic Functionhumans in thecontagious stageFigure 4. Key loops in the model of Yellow Fever. (complete this diagram)Review the logistic function described in Chapter 7. Do you expect to see exponential growth in thenumber of people impacted by the epidemic during the early days? If so, estimate the value of "r", theintrinsic growth rate of the epidemic, based on model parameters (such as the bites/day and the numberof days spent in each stage of the disease.) Does the negative feedback loop in Figure 4 apply the brakesto the epidemic in a linear manner, as required in the logistic function? What is the value of "K" to allowthe logistic equation to give an accurate estimate of the number of people impacted by the epidemic?Andrew Ford BWeb for Modeling the Environment 4

7. Verify the Logistic FunctionReview the method used to confirm the applicability of the logistic function in the flower model inexercise 7.7. Apply the same method to confirm the relevance of the logistic function to the yellowfever. Define the total population impacted by subtracting the vulnerable people from the 20,000 peoplein the city at the start of the simulation. You should find a good match between the simulated value oftotal population impacted and the logistic equation with a K set to 17,721 and r set to 4.2%/day.Policy SimulationsThe previous exercises require you to build and verify the Yellow Fever Model and to conductsome simple tests. These exercises should not take a lot of time, and you will often know when youhave the right answer. The next three exercises challenge you to expand model to simulate policies thatmight reduce the severity of the epidemic.8. Expand the Model to Simulate Mosquito Control ProgramsExpand the model to simulate the impact of a program to reduce the size of the mosquito population bythe application of larvacide. You may assume that the larvacide will reduce the number of emergingmosquitoes, but it will not directly affect the adult population. Include the start date and the percentreduction in emerging mosquitoes as program parameters that may be specified by the model user.9. Summary of Mosquito Control ProgramsUse the expanded model from the previous exercise to experiment with varying start dates and varyingmagnitudes. Summarize your results with a parameter space diagram, a two-dimensional diagram withthe start date on the horizontal axis and the magnitude of the mosquito reduction on the vertical axis.Label the space according to whether the simulated control programs would be successful orunsuccessful. For purposes of this exercise, define success as limiting the number of people impactedby the epidemic to 500 or less.10. Detection and Isolation ProgramsAssume that it is possible to detect the disease during the 4.5 day incubation period and that it is possibleto isolate the infected people before they reach the contagious stage. Add the new stocks and flowsneeded to move a fraction of these people into a separate category that is isolated from the mosquitopopulation. As in the previous exercise, allow the start date and the magnitude of isolation to bespecified by the model user. Experiment with the new model to learn the program parameters necessaryto limit the number of people impacted by the epidemic to 500 or less.Andrew Ford BWeb for Modeling the Environment 5