Problem 1 A theoretical SIRS model of infectious disease is presented. The model is represented by the following system of differential equations.
The values in the columns S, I and R represent the proportion of the population that are susceptible, infected and recovered, and N = S+I+R. The recovered class can become re-susceptible again, as defined by the alpha (α) term. Birth and death rate are equal, so the population size remains stable; this rate is defined by μ.
The simulation results begin at the outbreak of infection
a) Plot the proportion of susceptible, infected and recovered. Explain the graphs and dynamics of the outbreak
b) Define R0, calculate R0 for this infectious disease
c) State the relationship between the birth rate and life expectancy. Calculate the life expectancy in this population
d) Calculate the critical vaccination coverage for the range of vaccine efficiencies 0.1 to 1. Plot the critical vaccination coverage against vaccine efficiency. Interpret your graph
e) What does the beta value define in this system? Double the beta value to 0.6. Plot the proportions of the population in time. Describe the changes in the dynamics of the outbreak
f) Maintaining beta = 0.6, change the rate of re-susceptibility to 0.04. Plot the proportions of the population in time. Describe how the dynamics of the outbreak are altered
Problem 2 Provided in the Excel sheet named “Brains” are data of brain morphology and intelligence quotients sampled from five capital cities around the globe. Sex, total brain volume, head circumference and intelligent quotients (IQ) are given for 15 individuals. Brain measurements were inferred from MRI scans
a) Using city and sex as independent categorical variables, conduct an analysis of IQ
b) Measuring the brain requires expensive and time-consuming scanning procedures. Determine whether there exists a linear relationship between head circumference, which is quick and easy to measure and total brain volume. What conclusions do you draw?
c) Is there evidence that the head circumferences of males and females are drawn from the same population?
d) Comment on the sample used for the analysis. Given the opportunity to experiment further, would your sample be any different? How?
Problem 3 – Footrot in sheep Footrot in sheep is caused by the bacteria (Dichelobacter nodosus) that live on the feet of infected sheep, which may or may not be lame. The bacteria pass from one sheep to another via the surface the sheep are standing or walking on (e.g. the pasture or standing areas at gathering sites). D. nodosus can survive for a maximum of 7 –10 days on pasture and for up to 6 weeks in hoof horn clippings. A flock of 200 sheep had their feet swabbed and the presence of D. nodosus checked for by cell culture, the number of feet each sheep had positive for D. nodosus is shown below.
It is beneficial to know if the presence of D. nodosus among the feet of sheep follows a defined distribution. Compare the observed data to the appropriate expected data graphically, and formally test if the data follows that expected data. What do you conclude?