Influenza Stella Model

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Name ______________________________________ Class __________ Date _______

UModeling Infectious Diseases Background:

Throughout history, infectious diseases have had a large impact on the human population. Although infectious diseases are present in human populations at all times to some degree, the effects of epidemics (def: a rapid spread of a disease) are the most noticeable and spectacular. For example, in 14th century Europe, one quarter of the total population died from the Black Death. In 1520, approximately half of the 3.5 million members of the Aztec population died of smallpox. (Cortez took advantage of the situation to overthrow their empire.) Between 1918 and 1921, the Soviet Union experienced about 25 million cases of typhus with a death rate of approximately 10 percent. Yet another, for the two decades starting in 1934, tropical Africa suffered a nearly 40% childhood mortality rate from malaria and malaria-related diseases, which still pose a major threat today.

Illustration of the Black Death from the HToggenburg H Bible (1411)

Currently, there is worldwide concern over the spread of many diseases, such as HIV, the virus which causes AIDS. In the United States, tens of thousands of deaths have been reported from this disease since the start of the epidemic in the late 1970’s. The impact has been even more severe in other parts of the world. In particular, central Africa has been severely affected by this disease with even more serious effects expected to be felt in the coming years. Epidemiology is the field of study which focuses on infectious diseases and their outbreaks. For centuries people have been attempting to understand and describe the phenomenon of infectious diseases. Daniel Bernoulli (1760) appears to have been the first person to apply mathematics to the study of infectious diseases. Ethical considerations as well as time constraints make it virtually impossible to do laboratory studies of the spread of an infectious disease throughout a population. However, mathematical models can be used to simulate the spread of a disease, and can therefore help to predict its impact.

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The SIR Model:

Imagine a community having a total population, N. This population can be divided into three classes of people who are:

- Susceptible (S) are those who don’t yet have it but can get it. - Infected (I) are those who have the infection and can spread it to others. - Removed (R) are those who had it and recovered or died.

N = S + I + R.

Given the nature of an infectious disease, the numbers in each group will change over time: Infected people come into contact with susceptible people, who then become infected as well. Infected people eventually recover and, depending on the disease, are now immune to it or temporarily so. These changes create what is called a dynamic (def: lively, changing, active) system. One mathematical model expresses that an epidemic can be predicted if three values are known:

1) the number of susceptible people at the start 2) the infection rate (disease specific) 3) the recovery rate (disease specific)

The severity of the outbreak is indicated by the percentage of susceptible people who get infected.

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The Control Scenario: • 500 total students at a school. • One student becomes sick with the flu. • 499 students are susceptible. • Once you catch the flu and recover from it, you are immune to it. • An infected person remains contagious and sick for an average of 5 days,

which means that on average 20% (0.20) recover every day. • An infection rate for the flu has been mathematically determined to be a 1/1000 chance.

1. a. For the formula, what values do we know at the start, given the standard scenario above?

N = S + I + R N= _____ S=_____ I= _____ R=_____

b. Based on the data above, sketch three lines that display your prediction of how each population - susceptible, infected, and recovered - would be affected as the days pass. Use the colors in the key to code each group.

c. Briefly explain your predicted trend for each population.

Susceptible trend: Infected trend: Recovered trend:

0

100

200

300

400

500

0 5 10 15 20 25 30 35 40 45 50

Num

ber o

f Peo

ple

Time in Days

Spread of the Flu

Susceptible People Infected People Recovered People

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d. Open the Stella program to view and use the model. Run the Control Simulation to see how the model predicts the populations will change over time.

e. Insert a copy of the model’s graph into the space below:

f. Explain how the model predicts how this infectious disease might spread through a population. Describe the trend for each population.

Manipulating the Variables

Use the modeling program to see how decreasing and increasing the variables below might affect how a disease spreads. You can manipulate the variables by adjusting the slider or dials. Minimum and maximum limits have been put in place so that realistic scenarios can be predicted.

Variables to Change: Population size (initial number of susceptible) Initial number of infected Infection rate Recovery rate Vaccination rate

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Complete the following for each variable you study. You will need to space the different sections out more when you add your content. An example will be done together before you and explore on your own. Manipulation Example:

a. Variable changed: _______________________________________________

b. Your prediction of the impact:

c. Inserted copy of model’s graph:

d. Graph analysis: Describe the model’s prediction about how this variable might impact disease transmission.

Manipulation 1:

a. Variable studied: _______________________________________________




Manipulation 2:





Manipulation 3:



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d. Graph analysis: Describe the model’s prediction about how this variable might impact disease

transmission.

Manipulation 4:





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Influenza Stella Model