Spread of Disease in Africa Based on a Logistic Model By Christopher Morris

Scenario 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. 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 Part 1: Construct a model for the spread of this disease if no quarantine is imposed. 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 2: Calculate from this model the percent of the men who would have contracted the disease after 8 days.

Part 1 Assumptions as seen from Chapter 2.7 Epidemics: 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 (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.Assumption (J): The derivative of the function x(t) is a continuous function of t for t > 0.

Declaration of terms x(t) is the fraction of the population that is ill. 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. (1-x(t)) is the fraction of the population that is susceptible to the illness. t is time in units days. t is time in units days. k is a constant. k is a constant. dx/dt is the rate of change of the percent of ill people over time. dx/dt is the rate of change of the percent of ill people over time.

Initial setup Using Assumptions I and J we get the equation: 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 After only 8 days over 99.9% of the squadron is sick. X=

Limit of the equation Taking the limit of the equation shows that 100% of the population will become sick. Taking the limit of the equation shows that 100% of the population will become sick.

Conclusions In this model, since no quarantine was instated over 99.9% of the squadron became sick. 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. 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 T.P. Dreyer, Modelling with Ordinary Differential Equations, 1993 T.P. Dreyer, Modelling with Ordinary Differential Equations, 1993 Mathematica 5.2, Wolfram Research Mathematica 5.2, Wolfram Research

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