Mathematical Biology 4010/6010 Fall 2015 Research Project: Smallpox Max points: 50 Due Tuesday, November 30 (40 pts if returned after Nov. 30) Name: . An outbreak of smallpox in Abakaliki in southeastern Nigeria in 1967 was reported by two researchers Bailey and Thomas. People living there belong to a religious group that is quite isolated and declines vaccination. Overall, there were 30 cases of infection in a population of 120 individuals. The time (in days) between newly reported cases is given in the following sequence: 13, 7, 2, 3, 0, 0, 1, 4, 5, 3, 2, 0, 2, 0, 5, 3, 1, 4, 0, 1, 1, 1, 2, 0, 1, 5, 0, 5, 5. Use the SIR model to develop a model which describes these data and analyze the epidemic outbreak. Use two Matlab files, integrate_and_plot_ODE.m and smallpox.m to simulate the model. Open integrate_and_plot_ODE.m and run it by hitting \"F5\". File \"smallpox.m\" contains the SIR model equations. Use trial and error to find approximate values of and , corresponding to this specific model. Hint: Match the graphs (t,S) by changing parameters and to fit the blue and green curves (Figure 3, graphs (t,S).) Find the epidemic threshold below which the (deadly) disease becomes extinct. Use graphs (t,S), (t,I) and (I,S) to find the maximum number of infected people alive at the same time. Attach the printed graphs. Why did the disease not kill the entire population? If allowed, what is the minimum number of people in the village that should have been vaccinated to stop the epidemic? Hint: Use the values and . function dU = net(t,U) global beta global nu %First have to name the function. dX and X are vectors. X = reshape(U,[1 2]); %dX = [dx,dy] dX(:,1) = -beta.*X(:,1).*X(:,2); dX(:,2) = (beta.*X(:,1).*X(:,2)) - (nu.*X(:,2)); %old formulas %dX(:,1) = c.*(X(:,2)+X(:,1)-X(:,1).^3/3 -I); %dX(:,2) = a-X(:,1)- b.*X(:,2); dU = reshape(dX, [2 1])