In model (1.11) determine the outcome with the following parameters changed: a. Initially 5 are sick,...

 

 

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In model (1.11) determine the outcome with the following parameters changed: a. Initially 5 are sick, and 15 are sick the next week. b. The flu lasts 1 week. c. 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. 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: a S(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) S(0) = 995. After one week, the total number infected with the flu is 9. We compute a as = 5 and follows: I (0) = 5, I(1) = I(0) – 0.6 * I(0) + aI(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. In model (1.11) determine the outcome with the following parameters changed: a. Initially 5 are sick, and 15 are sick the next week. b. The flu lasts 1 week. c. 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. 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: a S(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) S(0) = 995. After one week, the total number infected with the flu is 9. We compute a as = 5 and follows: I (0) = 5, I(1) = I(0) – 0.6 * I(0) + aI(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.