Some diseases (such as typhoid fever) are spread largely by carriers, individuals who can transmit the disease but who exhibit n
o overt symptoms. Let x and y denote the proportions of susceptibles and carriers, respectively, in the population. Suppose that carriers are identified and removed from the population at a rate β, so dy/dt = −βy. Suppose also that the disease spreads at a rate proportional to the product of x and y; thus dx/dt = −αxy. a) Determine y at tany time t by solving eq(i) subject to the initial condition y(0)=y0
b) Use the result of part (a) to find x at any time t by solving Eq.(ii) subject to the initial condition x(0)=x0
c) Find the proportion of the population that escapes the epidemic by finding the limiting value of x as t approaches infinity
b) Use the result of part (a) to find x at any time t by solving Eq.(ii) subject to the initial condition x(0)=x0
c) Find the proportion of the population that escapes the epidemic by finding the limiting value of x as t approaches infinity
1 answer:
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Answer:
Step-by-step explanation:
To solve this, all we need to do is draw a triangle.
From the we can deduce that
From that, we can draw our triangle as we know that tan(x) is the opposite side over the adjacent side. Attached is that triangle. Through the Pythagorean Theorem, we can find that the hypotenuse of this right triangle is
Now all we need to do is take the sin of this triangle, which is the opposite side over the hypotenuse
This gives us the value of which is our answer.
