Transcript CJM Math Modeling presentation

Spread of Disease in
Africa Based on a
Logistic Model
By
Christopher Morris
Scenario
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In a Massive military exercise with 6000 men in a
remote corner of the Kalahari in Africa, 6 men suddenly
report sick one morning. On examination the medical
staff finds that the men had contracted flu. This form of
flu is not fatal, but the patient is very weak and dizzy for
a few days. The disease spreads by personal contract
and once a person is infected, he stays infectious for
about 8 days, after which he is immune to the disease.
If the disease spreads at a fast rate, the whole exercise
may be jeopardized. On the other hand, extra tents
must be flown in for a quarantine area which might be
an unnecessary expense, since the exercise is finished in
a fortnight. The medical staff decides to send the 6 men
back to their barracks and to wait until the next morning
before a quarantine is imposed. The next morning 6
more men reported sick.
Instruction
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Part 1: Construct a model for the
spread of this disease if no
quarantine is imposed.
Part 2: Calculate from this model the
percent of the men who would have
contracted the disease after 8 days.
Part 1
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Assumptions as seen from Chapter
2.7 Epidemics:
• Assumption (I): The disease is spread
by contact between ill and healthy
members of a closed community and
there is no quarantine.
• Assumption (J): The derivative of the
function x(t) is a continuous function of
t for t > 0.
Declaration of terms
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x(t) is the fraction of the population
that is ill.
(1-x(t)) is the fraction of the
population that is susceptible to the
illness.
t is time in units days.
k is a constant.
dx/dt is the rate of change of the
percent of ill people over time.
Initial setup
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Using Assumptions I and J we get the
equation:
Derive to get a general solution
General solution
Applying initial conditions to solve
for constants
Model of the spread of disease in
this scenario
Part 2
X=0.999998
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After only 8 days over 99.9% of the
squadron is sick.
Limit of the equation
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Taking the limit of the equation shows
that 100% of the population will
become sick.
Conclusions
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In this model, since no quarantine
was instated over 99.9% of the
squadron became sick.
Additional modeling will need to be
conducted to see if a quarantine was
instated if the disease would not
have spread throughout the
encampment.
Sources
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T.P. Dreyer, Modelling with Ordinary
Differential Equations, 1993
Mathematica 5.2, Wolfram Research
Questions?