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Math 4600, Homework 8 1. Consider a process of molecules leaving a cell, independent of each other, with a constant probabilitic rate of 2 per second (same is in the previous homework) a) What is the expected waiting time between second molecule that leaves and the third one? b) What is he probability that you have to wait longer than one second for the first molecule to leave? 2. (computing) Generate times of 100 events of a Poisson process with λ = 1. (To help: the command t=-(1/lambda)*log(1-rand(m,n)) will generate m by n matrix of random numbers that are drawn from exponential distribution). a) Plot the event times as dots in a line at the appropriate times (horizontal axis should be time, dots, located on the same horizontal line, mark times of events). Look at different parts of the array. Does it look random? (to check that you did this correctly your last event should happen at t near 100.) b) Count the number of events up to time 10. Is it close to the number we expect from theory? Do the same thing up to time 50. Is it close this time? c) Now look at your generated array and pretend that you have forgotten λ that you used. Estimate it. 3. In the sickle cell anemia model, consider 2 different situations: a society with strong genetic counseling (R = 0.9, T = 0.1, S = 0.1) and a society in tropics where carrying a sickle cell allele makes people less likely to die from malaria (R = .2, T = 0.1, S = 0.4 and also see below). a) Explain why this choice of parameters fits the word description of each society b) What will happen to disease in each population in the long run (use cobwebbing)? c) How will the model change if individuals with bb genotype are sterile (cannot reproduce)? The precise mechanism by which sickle cell trait imparts resistance to malaria is unknown. Malaria parasites multiply within red blood cells. Red blood cells in trait-carrier individuals are mostly normal, but can become sickled in low-oxigen condition. It seems that infestation of a cell may create low-oxigen conditions and effectively destroy the cell. 1