The Yellow Fever Model

Figure 1 shows a model based on a human population of 20,000. (Kalgraf mentions an epidemic in Veracruz, Mexico, a city of 20,000 poeple back in 1899. Yellow Fever spread through the Veracruz in 1899, and deaths reached 460 per month at the peak of the epidemic.) The model assumes that the humans live within contact of a mosquito population of around 500,000.

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 for 18 days and need four blood feeds during their lifetime. This means the average mosquito bites 0.2 persons per day, so there would be around 100,000 bites each day. If a mosquito avoids biting a contagious 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. The remaining 15 days is characterized by a 12 day incubation period and a 3 day infectious period.

The Humans: If an infectious mosquito bites a vulnerable human, the human contracts yellow fever. This person enters an incubation period for 4.5 days, followed by a contagious period of 4.5 days and a sick period of 2.5 days. After experiencing the sickness for 2.5 days, 90% of the humans recover and become immune to yellow fever. The model keeps track of the immune population and the cumulative number of deaths

Figure 1. Model of yellow fever with results reported for the peak day of the simulated epidemic.

Simulated Outbreak of Yellow Fever

Kalgraf's simulation begins with 100 people in the incubation stage. Figure 2 shows the simulation results over a 280 day period. The epidemic reaches its 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 as the reports from Veracruz.) The simulation shows the peak of the epidemic in the 140th day. Figure 1 focused on the peak day by providing a "snap shot" in what appears to be an equilibrium diagram. But you can see that system is not in equilibrium because there are over 8,000 people who remain vulnerable to the disease.

Figure 2. Simulated outbreak of yellow fever in a city of 20,000.

The mosquito stocks, on the other hand, are close to dynamic equilibrium. There are 81,259 mosquitoes in the "new" category. Each of these mosquitoes bites 0.2 humans per day, for a total of over 16,000 bites per day. But only 4.2% of these bites draw blood from a contagious human, so the become dangerous flow is 686 mosquitoes/day. Moving down the dangerous side of the mosquito stocks, we see 7,909 mosquitoes in the incubation stage and 1,945 in the infectious stage.The 1,945 infectious mosquitoes are responsible for the humans that contract yellow fever in the 140th day. Notice that 8,103 humans are in the vulnerable category at this point. The epidemic has reduced the total population to 19,014, so 42.6% of the humans are 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/day

This flow adds to the stock of people in the incubation stage. Figure 1 shows 781 people in the incubation stage, 803 in the contagious stage, and 450 sick. With a 2.5 day period of sickness, there would be 180 people reaching the final stage of the disease. Kalgraf assumes that 10% would die, so 18 people are simulated to die in the 140th day.