Suppose o e R contains sizes of different populations carrying a particular infection in year 0; for example, when tracking malaria we might take Xo1 to be the number of humans with malaria and Xo2 to be the number of mosquitoes carrying the disease. By writing relationships like The average mosquito infects two humans, we can write a matrix M such that = Mo predicts populations in year 1, 72 = M2To predicts populations in year 2, and so on. = = (a) The spectral radius p(M) is given by maxi Nil, where the eigenvalues of M are 11,..., lk. Epidemiologists call this number the reproduction number Ro of M. Explain the difference between the cases Ro 1 in terms of the spread of disease. Which case is more dangerous? (b) Suppose we only care about proportions. For instance, we might use M E R50x50 to model transmission of diseases between residents in each of the 50 states of the USA, and we only care about the fraction of the total people with a disease who live in each state. If yo holds these proportions in year 0, give an iterative scheme to predict proportions in future years. Characterize behavior as time goes to infinity. Suppose o e R contains sizes of different populations carrying a particular infection in year 0; for example, when tracking malaria we might take Xo1 to be the number of humans with malaria and Xo2 to be the number of mosquitoes carrying the disease. By writing relationships like The average mosquito infects two humans, we can write a matrix M such that = Mo predicts populations in year 1, 72 = M2To predicts populations in year 2, and so on. = = (a) The spectral radius p(M) is given by maxi Nil, where the eigenvalues of M are 11,..., lk. Epidemiologists call this number the reproduction number Ro of M. Explain the difference between the cases Ro 1 in terms of the spread of disease. Which case is more dangerous? (b) Suppose we only care about proportions. For instance, we might use M E R50x50 to model transmission of diseases between residents in each of the 50 states of the USA, and we only care about the fraction of the total people with a disease who live in each state. If yo holds these proportions in year 0, give an iterative scheme to predict proportions in future years. Characterize behavior as time goes to infinity