Determine the outcome with the following parameters:

1. Initially 5 are sick, and 15 are sick the next week.

2. The flu lasts 1 week.

3. The flu lasts 4 weeks.

d. There are 4000 students in the dorm; 5 are initially infected, and 30 more are infected the next week.

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Consider a disease that is spreading throughout the United States, such as the new flu. The Centers for Disease Control and Prevention is interested in knowing and experimenting with a model for this new disease before it actually becomes a real epidemic. Let us consider the population divided into three categories: susceptible, infected, and removed. We make the following assumptions for our model: • No one enters or leaves the community, and there is no contact outside the community. • Each person is susceptible S (able to catch this new flu); infected I (currently has the flu and can spread the flu); or removed R (already had the flu and will not get it again, which includes death). • Initially, every person is either S or I. • Once someone gets the flu this year, they cannot get the flu again. • The average length of the disease is 5/3 weeks (1 and 2/3 weeks), over which time the person is deemed infected and can spread the disease. • Our time period for the model will be per week. The model we will consider is called the SIR model Let's assume the following definitions for our variables: S(n) = number in the population susceptible after period n I(n) = number infected after period n R(n) = number removed after period n Let's start our modeling process with R(n). Our assumption for the length of time someone has the flu is 5/3 weeks. Thus, 3/5 or 60% of the infected people will be removed each week: R(n + 1) = R(n) + 0.61(n) The value 0.6 is called the removal rate per week. It represents the proportion of the infected persons who are removed from infection each week. I (n) will have terms that both increase and decrease its amount over time. It is de- creased by the number of people removed each week: 0.6 * I(n). It is increased by the number of susceptible people who come into contact with infected people and catch the disease: aS(n)I(n). We define a as the rate at which the disease is spread, or the transmis- sion coefficient. We realize this is a probabilistic coefficient. We will assume, initially, that this rate is a constant value that can be found from the initial conditions. Let's illustrate as follows: Assume we have a population of 1000 students residing in the dorms. Our nurse found 5 students reporting to the infirmary initially: I(0) = 5 and S(0) = 995. After one week, the total number infected with the flu is 9. We compute a as follows: I (0) = 5, 1(1) = I(0) – 0.6 * I(0) + a1(0) * S(0) I(1) = 9 = 5 – 3+ a *5 * 995 7 = a(4975) a = 0.001407 Lets consider S(n). This number is decreased only by the number that becomes infected. We may use the same rate a as before to obtain the model: S(n + 1) = S(n) - aS(n)I(n) Our coupled model is R(n + 1) = R(n) + 0.61(n) I(n + 1) = I(n) - 0.61(n) + 0.0014071(n)S(n) S(n + 1) = S(n) – 0.001407S(n)I(n) I (0) = 5, S(0) = 995, R(0) = 0 (1.11) The SIR model Equation (1.11), can be solved iteratively and viewed graphically. Lets iterate the solution and obtain the graph to observe the behavior to obtain some insights. Consider a disease that is spreading throughout the United States, such as the new flu. The Centers for Disease Control and Prevention is interested in knowing and experimenting with a model for this new disease before it actually becomes a real epidemic. Let us consider the population divided into three categories: susceptible, infected, and removed. We make the following assumptions for our model: • No one enters or leaves the community, and there is no contact outside the community. • Each person is susceptible S (able to catch this new flu); infected I (currently has the flu and can spread the flu); or removed R (already had the flu and will not get it again, which includes death). • Initially, every person is either S or I. • Once someone gets the flu this year, they cannot get the flu again. • The average length of the disease is 5/3 weeks (1 and 2/3 weeks), over which time the person is deemed infected and can spread the disease. • Our time period for the model will be per week. The model we will consider is called the SIR model Let's assume the following definitions for our variables: S(n) = number in the population susceptible after period n I(n) = number infected after period n R(n) = number removed after period n Let's start our modeling process with R(n). Our assumption for the length of time someone has the flu is 5/3 weeks. Thus, 3/5 or 60% of the infected people will be removed each week: R(n + 1) = R(n) + 0.61(n) The value 0.6 is called the removal rate per week. It represents the proportion of the infected persons who are removed from infection each week. I (n) will have terms that both increase and decrease its amount over time. It is de- creased by the number of people removed each week: 0.6 * I(n). It is increased by the number of susceptible people who come into contact with infected people and catch the disease: aS(n)I(n). We define a as the rate at which the disease is spread, or the transmis- sion coefficient. We realize this is a probabilistic coefficient. We will assume, initially, that this rate is a constant value that can be found from the initial conditions. Let's illustrate as follows: Assume we have a population of 1000 students residing in the dorms. Our nurse found 5 students reporting to the infirmary initially: I(0) = 5 and S(0) = 995. After one week, the total number infected with the flu is 9. We compute a as follows: I (0) = 5, 1(1) = I(0) – 0.6 * I(0) + a1(0) * S(0) I(1) = 9 = 5 – 3+ a *5 * 995 7 = a(4975) a = 0.001407 Lets consider S(n). This number is decreased only by the number that becomes infected. We may use the same rate a as before to obtain the model: S(n + 1) = S(n) - aS(n)I(n) Our coupled model is R(n + 1) = R(n) + 0.61(n) I(n + 1) = I(n) - 0.61(n) + 0.0014071(n)S(n) S(n + 1) = S(n) – 0.001407S(n)I(n) I (0) = 5, S(0) = 995, R(0) = 0 (1.11) The SIR model Equation (1.11), can be solved iteratively and viewed graphically. Lets iterate the solution and obtain the graph to observe the behavior to obtain some insights.

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Answer A population is a number of all the organisms of the same group or species who live in a part... View the full answer