Exploring Epidemiology: Modeling
the Spread of a Disease
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Ebola breaks out in Zaire. The Hot Zone describes a
close call with Ebola in Reston, Virginia. The movie, Outbreak, depicts a
fictional hemorrhagic fever outbreak in the U.S. AIDS is spreading rapidly in many
parts of the world. Given the threats from emerging diseases and concerns over the
resurgence of older diseases, exacerbated by the prospects for global climatic change
and the evolution of drug-resistant pathogens, the threat of global epidemics seems
to have replaced nuclear war as this generationís Armageddon.
When a new disease enters a population, we can envision several
different kinds of outcomes: (1) the disease could quickly die out, (2) the disease
could remain in the population at more or less stable levels, perhaps "settling
down" after a major outbreak (i.e., become endemic), (3) the disease
could cycle in incidence, causing periodic epidemics (the cycles could increase or
decrease in amplitude, or remain about the same), (4) epidemics could come and go
at more or less random intervals, perhaps exhibiting "chaotic" behavior,
or (5) the disease could cause the population to go extinct.
Which of these outcomes can we expect from diseases like AIDS
or Ebola? Examining simulation models may give us some insights. We might also learn
something about the factors that influence the severity of epidemics, or the levels
of endemism. Finally, models may help us understand the impact of public health policies
and practices.
How would we go about building a model of the spread of a disease?
One important step is to identify the variables that we want to track through time.
For example, a demographer tracks the size of populations through time, perhaps divided
into population categories, such as gender groups, racial/ethnic groups, or age groups.
To track changes in population size, we must look at the processes that could bring
about change. In this simple case, there are only three things that can happen (1)
individuals can come into the population via birth or immigration, (2) they can leave
the population through death or emigration, or (3) they can move from one category
to another (e.g., by getting older, or by having a sex change operation, or by changing
their economic status). Thatís it! So if we could model the processes by which the
number of individuals in each category change, then we could make predictions about
the future state of the population. Of course, the challenge is to figure out what
influences birth rates, death rates, etc., and that may be very hard, especially
when all of the influencing factors may interact in complex ways.
In the case of epidemiological models, the population variables
can be divided into individuals that are susceptible to infection, those that are
infected currently, and those that have recovered (assuming that these individuals
are now immune to infection). This is the simplest view, and is the basis for the
so-called SIR models. Of course, in more complex models, each of these categories
could be divided by gender, age, socioeconomic status, etc. Other complications could
include reservoir populations, intermediate host populations, or vector populations.
But to start with, lets keep it simple. Letís concentrate now on a simple disease,
like smallpox, that is transmitted only from individual to individual, and which
seems to be an "affirmative action" disease, showing no gender, age or
social preferences. Once we understand aspects of this simple model, we can consider
how various complications might affect our results.
The computer program that we will be using (Epidemiology)
allows you to control factors such as the probability of infection, the death rate
due to a disease, the proportion of a population vaccinated, and the average length
of infection, so that you can study the way these factors interact to influence the
spread of a disease.
The next few pages introduce you to the computer program. They
are followed by some questions to help guide your explorations.
The flow chart describes the model of disease transmission
in graphical form. Individuals in the population belong to one of three categories.
They are either susceptible to infection, infected, or recovered (and immune to further
infection) . The arrows in the flow chart show how individuals enter and leave these
categories. For the disease graphed here, all individuals are born susceptible. They
stay in that category unless they become infected or they die (from causes other
than the disease).
The population vs. time display tracks the number of individuals
in a population through time in each of the three categories: susceptible (blue line),
infected (purple line), and recovered (green line).
Describe in words what happened to this population from time
0 to time 100.
To learn more about how to set up and run simulations and about
viewing your results, click on the appropriate link in the Table of Contents on the
left side of the page.
The slider bars in the "Epidemiology
Settings" display area can be used to change the factors that influence the
change the rates at which individuals enter or leave each of the population categories
(or you can enter values by typing in the boxes on the top right of each slider.
Make sure that you understand what each of these sliders represents. The simple
model that we will be exploring allows you to set values for the following factors:
Birth rates -- You enter the number of individuals born,
per individual in the population, per time unit. For example, if the time unit is
years, and we are dealing with a human population that has a birth rate of 20/1000
per year, you would enter 0.02.
Death rates -- You enter the number of individuals dying,
per individual in the population, per time unit. For example, if the time unit is
years, and we are dealing with a human population that has a death rate of 10/1000
per year, you would enter 0.01. This is a "background" death rate due to
factors other than the disease.
Disease death rate -- You enter the proportion of the
infected population that dies of the disease each time interval. For a disease such
as the common cold, this would be 0.0. For a disease like smallpox it might be something
like 0.25.
Number of contacts per time interval or Contact Rate--
How many contacts (of the sort that might result in transmission) does an individual
make with other individuals during the course of a time interval, on average? This
will depend on how the disease is transmitted and the nature of living conditions.
For example, the value would be much lower for an STD than for the common cold. It
would also be lower in a rural area than in an urban area.
You enter the average number of contacts per thousand individuals
that each individual in the population has during one time interval. In our model
the number of contacts depends on the population density. Thus a contact rate of
50 (per 1000) yields 500 contacts for each individual when the population size is
10,000, and 5000 contacts for each individual if the population is 100,000. Start
off with a number like 50.
Probability of transmission -- Given a contact between
a susceptible individual and an infectious individual, what is the probability that
the susceptible individual will become infected? Values entered must be below 1.
Something like 0.01 might be reasonable to start with.
Recovery rate -- This represents the proportion of infected
individuals who recover each time interval. Infected individuals who do not die or
recover remain in the infected category. Try a value of 0.333 to start out with.
Immunity Loss Rate -- This represents the proposrtion
of recovered individuals who lose their immunity each time interval. When individuals
lose their immunity, the move from the recovered category to the susceptible category.
Try starting with a value of 0.05.
Now that you have entered these settings, go ahead and run
the simulation. Let it run for about 50 time intervals. Examine and interpret the
resulting population graphs. Did anything unexpected happen?
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Posing a Question / Making a
Prediction
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Working with your partner, choose one of the factors from the
previous page, and make a prediction about how you expect the outcome of the simulation
to change if you change that factor.
Which factor did you pick? _____________________________________
What is your prediction? (If I increase/decrease the value
of ____, I expect...)
Now make the change and run the simulation. Describe the outcome
of the simulation and how it compares with your predictions.
What hypotheses can you come up with that might explain the
differences between the expected and the actual outcomes?
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Investigating Virulence and Ease
of Transmission
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We will concentrate our modeling efforts in this exercise on
examining how the virulence of the disease agent (how deadly it is) and its ease
of transmission affect its spread. We should be able to use our results to help assess
the potential threats of newly emerging diseases like Lassa Fever and Ebola.
The approach that we will take is to systematically explore
"parameter space", seeing what happens for different combinations of parameter
values. Since we will start by just examining two parameters, we will need to choose
values for the other parameters, and keep those constant. We'll use the values that
we had when we initially started the program.
The first thing you should do is to make some predictions about
what you expect to happen. Suppose that we start out with a host population in which
most individuals are susceptible to a particular disease, but a few are infected.
If the probability of transmission was very high and the level of virulence was very
low, what would you expect to happen to this population through time? Possible outcomes
might include "the host population goes extinct", "disease levels
fluctuate", "the disease and the host population come to an equilibrium",
"the disease dies out", etc. Fill in the appropriate spot in the table
below. Also, think about various human diseases that you know about and see if you
can identify one that has a high transmission and low virulence, and enter it in
the right spot in the table. Fill out the entire table in a similar way.
Transmission
Probability |
Low
Virulence
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Medium
Virulence
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High
Virulence
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Low
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Predicted Outcome:
Disease:
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Predicted Outcome:
Disease:
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Predicted Outcome:
Disease:
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Medium
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Predicted Outcome:
Disease:
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Predicted Outcome:
Disease:
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Predicted Outcome:
Disease:
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High
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Predicted Outcome:
Disease:
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Predicted Outcome:
Disease:
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Predicted Outcome:
Disease:
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Give the rationale for your predictions below.
Your Rationale:
Once we've all thought about and written down our expectations,
we can start our experiment. Basically, we need to fill in a chart similar to the
graph above. Rather than all of us doing all of the simulations, we can divide up
the workload and collate all our results.
Class discussion: What data should we be gathering for
each simulation run? Before you start doing your simulations, we will need to discuss
this as a class.
Class Experiment: Each group in the class will be assigned
certain simulations to perform. Please collect all the information that was agreed
on in the class discussion (as well as anything else that you think might be important,
and fill in the appropriate cells in the table below:
Death rate->
Trans-mission |
Very Low
0.0 |
Low
0.1 |
Medium
0.2 |
High
0.3 |
Very High
0.5 |
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Very Low
0.001
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Low
0.005
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Medium
0.01
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High
0.1
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Very High
0.5
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As a control, data for each cell in the table will be collected
by more than one group. You should compare your results with others to make sure
that you are all in agreement. If not, why?
Once we have collated all our results, each of you should compare
your results with your predictions and attempt to explain what we have observed.
Fill in the table on the next page. Are there major surprises?
Death rate->
Trans-mission |
Very Low
0.0 |
Low
0.1 |
Medium
0.2 |
High
0.3 |
Very High
0.5 |
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Very Low
0.001
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Low
0.005
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Medium
0.01
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High
0.1
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Very High
0.5
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Testing Your Understanding
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Do you really understand what is going on with these simulations?
If your immediate answer is no, don't just blow this assignment off. This stuff is
hard and requires persistence. Try working out the problems below; come talk
to us if you have questions. If you think you DO understand what is going on, the
questions below will be a good way to make sure. Some are quite challenging!
1. How do the class results compare with your predictions?
Are there any major surprises or discrepancies? Explain the differences.
2. Why does the disease die out when transmission rates are
very low, even when the disease doesn't ever kill its host?
3. For a fixed transmission rate (say 0.01), why does the equilibrium
proportion of susceptibles (S) increase as the disease death rate increases?
4. For a fixed death rate, the proportions S, I and R do not
change as the probability of transmission increases, but the equilibrium population
size decreases. Furthermore, it seems that when the transmission rate is doubled,
the equilibrium population size decreases by half, etc. Why?
5. Why doesn't the host population go extinct, even when the
probability of transmission is very high and the disease death rate is high?
6. For a given probability of transmission, the equilibrium
population size first decreases with disease death rate, then increases. How can
you account for the increase?
7. How would these results have differed if 25% of the population
was immunized at birth? How about if the disease had an asymptomatic infectious stage
(like HIV)? Explain your rationale. You might try your predictions out using the
computer simulation.