Modeling the Spread of an Epidemic  

By developing a computer model of the spread of an infectious disease, the student develops an understanding of the role of the infection rate and the removal rate on the spread of the disease. The Threshold Theorem of Epidemiology claims that the extent of spread of an epidemic can be predicted if three values are known: initial number of susceptible people (S(0)), the infection rate (K), and the removal rate (by quarantine or cure) (Q). The extent of the spread of the epidemic is indicated by the percentage of susceptible who become infected. Diseases that are easily transmitted spread quickly unless measures are taken to quarantine or cure infected people quickly.

The three epidemic models below can each be used to experiment with various factors to see the impact on the populations. Vensim or STELLA software is needed to run the models. Right-click on the model link to download the model.

Simple Epidemic Model - Vensim
Simple Epidemic Model - STELLA
Assumes the infected people never leave the system

Infectious(t) = Infectious(t - dt) + (sick_per_day) * dt
   INIT Infectious = 1
Susceptible(t) = Susceptible(t - dt) + (- sick_per_day) * dt
   INIT Susceptible = 999
sick_per_day = Infectious*Susceptible*infection_rate
infection_rate = .0015

1. The simple epidemic model may be used to explore the impact of the infection rate variable on the healthy and infected populations. Change the infection rate and record the change in the output of the model. How does the shape of the s-shaped growth curve change? At what time does the model stabilize?

Infection Rate			Stabilization Time		Shape of Curve

2. Exponential growth usually occurs when the rate of change is proportional to the amount present. Steady-state occurs when the system reaches equilibrium. Why does this model yield an s-shaped curve?

3. This model could be used to represent the spread of a highly contagious disease in a population living in close quarters, such as a college dorm, a military barracks, or a tribal village. This model shows how quickly a new virus could spread in a population with no immunity. What are some examples that you could use in your class?

Second Epidemic Model - Vensim
Second Epidemic Model - STELLA
Assumes that infected people recover or die and therefore cannot continue infecting others

dead(t) = dead(t - dt) + (daily_deaths) * dt
   INIT dead = 0
Infectious(t) = Infectious(t - dt) + (sick_per_day - daily_recoveries - daily_deaths) * dt
   INIT Infectious = 1
recovered(t) = recovered(t - dt) + (daily_recoveries) * dt
   INIT recovered = 0
Susceptible(t) = Susceptible(t - dt) + (- sick_per_day) * dt
   INIT Susceptible = 999
daily_deaths = Infectious*(1-survival_rate)/disease_duration
daily_recoveries = Infectious*survival_rate/disease_duration
sick_per_day = Infectious*Susceptible*infection_rate
disease_duration = 5
infection_rate = .0015
survival_rate = .70

4. In a college dorm, the common cold or the flu would spread very quickly, but infected students would recover. This model could be used to experiment with disease duration and survival rate to see the effect on the population. By altering contact rate, duration or survival rate, try to achieve the following goals:

a) There are no susceptible people left

b) All people become immune

c) The entire population dies

Third Epidemic Model - Vensim
Third Epidemic Model - STELLA
Assumes that infected people become visibly sick and are removed from the general population so they can affect only a few caregivers

Infectious(t) = Infectious(t - dt) + (sick_per_day - quarantined_per_day) * dt
   INIT Infectious = 1
Quarantined(t) = Quarantined(t - dt) + (quarantined_per_day) * dt
   INIT Quarantined = 0
Susceptible(t) = Susceptible(t - dt) + (- sick_per_day) * dt
   INIT Susceptible = 999
quarantined_per_day = Infectious*quarantine_rate
sick_per_day = Infectious*Susceptible*infection_rate
infection_rate = .0015
quarantine_rate = .8

5. According to the Threshold Theorem of Epidemiology, the ratio of the quarantine rate to the infection rate determines whether an illness turns into an epidemic. Use the model above to find the ratio that will result in approximately 50% of the population getting the disease.

6. Design an epidemic model which incorporates any of the features below into the Third Epidemic Model.

a) Introduce births and natural deaths into the population

b) Introduce a vaccination program to prevent susceptible people from getting sick. How many must be vaccinated to reach steady state in the model?

7. Set up a STELLA model of the transmission of aids in the United States by identifying various segments of the population and their probability of engaging in behavior that puts them in danger of becoming infected.


  • Children
      Healthy, HIV Positive, Aids, Dead
  • Women
      Low Risk, Drug Abusing, Heterosexual, HIV Positive, Aids, Dead
  • Men
      Low Risk, Drug Abusing, Heterosexual, Bisexual, Gay, HIV Positive Heterosexuals, HIV Positive Bisexuals, HIV Positive Gays, Aids, Dead
Determine the flows to connect the stocks and do research to determine the appropriate values to use.

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