Information:

a Y Peak infection Peak S value at the peak Theoretical Speak level infection infection level (from Eq. 3) (% total time (days) (Speak) (% total (Hint 1: use the fact that population) population) Speak occurs when Ro= 1.) (Hint 2: divide by total population size to get Speak as a % total population) 0.00015* 0.04* 0.0003 0.04 0.0005* 0.04* 0.001 0.04 0.0015* 0.04*W: You will be using versions of these three models implemented as R programs. Simply copy and paste the different programs into notepad or the script editor in m e dit the param ete rs as necessary. Note that when these programs are ru n, they will automatically prod uoe graphs of the percent of the population that is suscepti ble [black line on graph), infected (red line on gra phi or, in part III. resistant [blue line on graph]. Part I. SI Model (0.1!) l \fPart I. Susceptible-Infected (SI) Model: In this Part you will be studying the effect of different parameters on the time it takes the infection to spread through the population within the Susceptible-Infected (SI) model. The time point you will be recording is the time it takes the infection to reach 50% of the population (a very high rate of infection) which we'll designate as Yearso. For this study you will manipulate two different parameters: the transmission coefficient (a], and the initial Susceptible Population (So). Procedure Using the appropriate R program for the SI model given above, adjust the parameters in the program and record the year in which 50% of the population becomes infected (Yearso). Note that you can read this off of the graph created by the R program (this is the point where the two lines cross). Study 1 Study 2 So=500; 10=1 lo=1; a=0.0002 Yearso So Yearso 0.000025 49 25 12500 0.00005 49 50 24000 0.0001 48 100 48000 0.0002 45 200 96000 Analysis Questions 1. How does the speed of the infection spread change as the transmission coefficient is altered? Look at the values in your table above to determine if this is a linear or inverse linear relationship. Le. Does doubling a double the time until 50% of the population is infected or does it halve the time until 50% of the population is infected? Study 1 reveals that the transmission coefficient increases as the number of susceptible populations decrease over time, from 0.0025 percent to 0.02 percent at 50% of the population. As a result, the transmission coefficient has an inversely linear relationship with the number of susceptible populations. According to study 2, the higher the transmission coefficient, 0.02 percent, the greater the number of susceptible populations over time until it reaches 50% of the population. As a result, the transmission coefficient is proportional to the number of vulnerable populations. When the transmission coefficient is doubled, the time it takes for 50% of the population to get infected is also doubled. 2. How does the speed with which the infection spreads change with the size of the initial Susceptible Population? Look at the values in your table above to determine if this is a linear or inverse linear relationship. Le. Does doubling So double the time until 50% of the population is infected, or halve the time until 50% of the population is infected? During times of war, famine, or other crises, people often move from rural areas to cities. How would this affect the speed at which an epidemic will affect a population? In many respects, movement is linked to the transmission of disease. The introduction of a new microorganism into a new geographic location is one way that travels aid the spread of infectiousdiseases. The most severe disease is caused by a novel pathogen that enters a population that has never encountered it before. Initiating outbreaks of acute infections, changing the prevalence of infectious diseases at a given region, and modifying the face of chronic disease resulting from prior infection, population mobility plays a vital role in the transmission of disease. Historical note: The ancient Greek historian Thucydides describes a great plague that struck Athens during the Peloponnesian War starting in 430 BC. Not only does he describe the disease itself in great detail (and mentions that he himself was sick], but he also makes the following comment about the impact of the epidemic: "An aggravation of the existing calamity was the influx from the country into the city." [Book 2.52 Thueudidea, The Landmark Thucydides, R.B. Scapelex, ed. The Free Press, New York, 1996.]Part II. Susceptible-Infected-Susceptible (SIS] Model: In this part you will be studying the effect of different parameters on the endemic level of infection in the population within the Susceptible-Infected-Susceptible model. The end point you will be calculating is the fraction of the total population that is constantly carrying the infection after the infection has reached steady state (i.e. isn't changing anymore). This fraction will be denoted %Is which is just the asymptotic number of infected individuals Ix divided by the total population size, and you will investigate how this response varies with the transmission coefficient and removal rate. Procedure 1. Using the appropriate R program given above for the SIS Model, adjust the values of the transmission coefficient (a) and the removal rate (y) in the program to get the %Ix for each combination of parameters in the Tables below. Record in the tables below the values %In (that is, the asymptotic value of %I, read from your graph). Then use equation 2 (given in the Theoretical Background) to calculate the theoretical values and fill those results into the tables too in the column labelled "% I s ( theory)". Study 1 Study 2 So=500; 10=1; v=0.05 So=500; 10=1; a=0.0016 CL Moko (theory) Y Volo (theory) 0.0002 251 3.984 x 10^- 13.5 0.7407 3 0.02 0.0004 126 7.9365 x 26 0.03846 104-3 0.04 0.0008 63.5 0.01575 51 0.08 0.01961 0.0016 32.25 0.031 101 9.901 x 10^-3 0.16 Analysis Questions 1) In your models, how does varying the transmission coefficient (a) affect the fraction of the population that ends up infected after the epidemic has reached equilibrium? After the epidemic has reached equilibrium, changing the transmission coefficient () increases the proportion of the population that contracts the disease. A bigger proportion of the population is infected at equilibrium as the transmission coefficient rises because more people become infected and the likelihood of infection rises. 2) In your models, how does varying the removal rate (y) affect the fraction of the population that ends up infected after the epidemic has reached equilibrium? When the elimination rate is changed, the proportion of the population that contracts the disease after the epidemic has reached equilibrium decreases. At equilibrium, a lesser proportion of thepopulation is infected because as the clearance rate rises, more people wand fewer people are infected