math4b-19 - Math 4B Lecture 19 June 5 2013 Doug Moore A...

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Math 4B Lecture 19 June 5, 2013 Doug Moore
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A pestilence isn’t a thing made to man’s measure; therefore we tell ourselves that pestilence is a mere bogy of the mind, a bad dream that will pass away. But it doesn’t always pass away and, from one bad dream to another, it is men who pass away. ( The plague , by Al- bert Camus, p. 36.) Today, we will consider a mathematical model for the spread of an infectious disease, as an illustration of the qualitative ap- proach to differential equations. The reader may wish to imagine an outbreak of bubonic plague, for example, as in the novel by Camus.
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Let x ( t ) = number of susceptible people at time t , y ( t ) = number of sick people at time t , z ( t ) = number of dead people at time t . In order to keep the model simple, we will assume that the num- ber of individuals in each of these categories is sufficiently large that the variables x , y , and z can be regarded as continuous rather than discrete. We will also assume that the epidemic lasts for a short time interval, so that we can neglect births and deaths due to other causes.
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Under these assumptions, it is reasonable to expect that the number of new cases per unit time will be separately propor- tional to x and y , and that the number of deaths will be propor- tional to y . We could then model the epidemic by the system of differential equations dx/dt = - Axy, dy/dt = Axy - By, dz/dt = By, where A and B are positive constants. Since the first two equa- tions do not involve z , we can solve them for x ( t ) and y ( t ) and then substitute into the third to find z ( t ).
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Thus our problem quickly reduces to solving a system of two equations in two unknowns: dx/dt = - Axy, dy/dt = Axy - By. The first step in analyzing this system consists of finding the constant solutions, i.e. the zeroes of the vector field F ( x, y ) = - Axy i + ( Axy - By ) j . It is quickly verified that the zeros of F are y = 0 , x = any constant .
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