One model for the way diseases die out when properly treated assumes that the ra

Question

One model for the way diseases die out when properly treated assumes that the rate dy/dt at which the number of infected people changes is proportional to the number y. The number of people cured is proportional to the number that have the disease. Suppose that m any given year, the number of cases of a disease is reduced by 30%. There are 10,000 cases today. How long will it take to reduce the number of cases to 1000? How long will it take to eradicate the disease, that is, reduce the number of cases to less than 1? It will take years to reduce the number of cases to 1000. (Round to the nearest hundredth.)

Explanation / Answer

Dy/Dt proportional to (-y)

y= ke-mt where t is in year.... putting boundary condition y becomes .7y in 1 year , we get m = .35667

a) to reduce it to 10% (10000 to 1000) time needed = 6.46 years

b) to reduce it to .01% (10000 to 1) time needed = 25.82 years

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