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Unformatted text preview: Math 302-050: Practice Final Assignment Directions: You are to work all problems on your own. You may use Maple to help with the analysis or to produce figures. Your solutions should include a copy of your Maple session if you do decide to use Maple. You are expected to budget two hours for the assignment. If you decide to spend more time, clearly separate what you accomplished during the first two hours from what you accomplished during the overtime, and indicate how much longer you spent. In 1927, Kermack and McKendrick proposed one of first mathematical models for the evolution of an epidemic. The model supposes that the population can be divided into three classes: x ( t ) = number of healthy people; y ( t ) = number of sick people; and z ( t ) = number of dead people. Note that by these definitions, we must have x ( t ) 0, y ( t ) 0, and z ( t ) 0. The model consists of three coupled differential equations: dx dt =- kxy dy dt = kxy- ry dz dt = ry We will assume the initial conditions: x (0) = x , y (0) = y , z (0) = 0 , with x + y = N, which says that we start with a total population of N , divided into x healthy people and y infected people. Furthermore, k and r are positive constants, whose interpretations are further discussed below. In their original paper, Kermack and McKendrick analyzed the model and showed that it gave a good fit to data from the 1906 outbreak of plague in Bombay. The equations are set up so that the total number of people, dead or alive, stays constant. All that can change is a persons status: healthy, infected, or dead. All changes in population due to immigration or emigration, births, or deaths from other causes are ignored, so the model is only appropriate for an isolated or quarantined population subject to a disease which develops rapidly. The first differential equation can be interpreted as follows. The disease is spread by encounters between healthy and infected people. The total number of encounters is proportional to the product of the numbers of these two subpopulations. At each encounter there is a constant probability of disease transmission. Therefore the overall rate of infection is proportional to xy , and the proportionality constant k depends on both the likelihood of encounters between the healthy and infected subpopulations and the rate of disease transmission. The second differential equation says that the number of infected indi- viduals grows at the rate at which people get infected, and once infected, sick people die at the constant rate r . Note that no one recovers from this disease. If you get sick, you stay sick and eventually die. This idealization is made to simplify the analysis. The possibility of recovery could be included without too much trouble....
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- Fall '08