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Question 2: Each Variable Changes over Time Next, let's make some assumptions about the rates of change of our dependent variables. No one is added to the susceptible group, since we ignore birth and migration. The only way an individual leaves the susceptible group is by becoming infected. The number of susceptibles depends on the number already susceptible, the number of individuals already infected and the contact between the two groups. Let us suppose that the coefficient beta, B, represents the contacts per day between the infected and the susceptible that are sufficient to spread the disease. If we assume homogeneous mixing of the populations, each infected individual generates B * s(t) new infected individuals per day. We also assume that a fixed fraction, mu, µ, of the infected group will recover during any given day. For example, if the average duration of infection is three days, then, on average, one-third of the currently infected population recovers each day. We will ignore the "recovered" (non- infectious) people, for now, who still feel miserable, and might even die later from other complications. As a result, we have the following differential equations: • The Susceptible Equation: ds = -ß * s(t) * i(t) dt • The Recovered Equation: dr = u * i(t) dt • The Infected Equation: di = B * s(1) * i(t) – µ * i(t) dt Answer the following questions in the space provided. 2a. di ds Explain why dt + af = 0 ? dt dt 2b. Describe both components of the infected equation. 2c. What do the minus signs represent?

Question 2: Each Variable Changes over Time Next, let's make some assumptions about the rates of change of our dependent variables. No one is added to the susceptible group, since we ignore birth and migration. The only way an individual leaves the susceptible group is by becoming infected. The number of susceptibles depends on the number already susceptible, the number of individuals already infected and the contact between the two groups. Let us suppose that the coefficient beta, B, represents the contacts per day between the infected and the susceptible that are sufficient to spread the disease. If we assume homogeneous mixing of the populations, each infected individual generates B * s(t) new infected individuals per day. We also assume that a fixed fraction, mu, µ, of the infected group will recover during any given day. For example, if the average duration of infection is three days, then, on average, one-third of the currently infected population recovers each day. We will ignore the "recovered" (non- infectious) people, for now, who still feel miserable, and might even die later from other complications. As a result, we have the following differential equations: • The Susceptible Equation: ds = -ß * s(t) * i(t) dt • The Recovered Equation: dr = u * i(t) dt • The Infected Equation: di = B * s(1) * i(t) – µ * i(t) dt Answer the following questions in the space provided. 2a. di ds Explain why dt + af = 0 ? dt dt 2b. Describe both components of the infected equation. 2c. What do the minus signs represent?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 2: Each Variable Changes over Time Next, let's make some assumptions about the rates of change of our dependent variables. No one is added to the susceptible group, since we ignore birth and migration. The only way an individual leaves the susceptible group is by becoming infected. The number of susceptibles depends on the number already susceptible, the number of individuals already infected and the contact between the two groups. Let us suppose that the coefficient beta, B, represents the contacts per day between the infected and the susceptible that are sufficient to spread the disease. If we assume homogeneous mixing of the populations, each infected individual generates B * s(t) new infected individuals per day. We also assume that a fixed fraction, mu, µ, of the infected group will recover during any given day. For example, if the average duration of infection is three days, then, on average, one-third of the currently infected population recovers each day. We will ignore the "recovered" (non- infectious) people, for now, who still feel miserable, and might even die later from other complications. As a result, we have the following differential equations: · The Susceptible Equation: ds = -ß* s(t) * i(t) dt dr • The Recovered Equation: = u * i(t) dt • The Infected Equation: di = B* s(1) * i(t) – u * i(t) dt Answer the following questions in the space provided. 2a. ds di dr + + dt dt dt Explain why = 0 ? 2b. Describe both components of the infected equation. 2c. What do the minus signs represent? Command Window Paul
Transcribed Image Text:Question 2: Each Variable Changes over Time Next, let's make some assumptions about the rates of change of our dependent variables. No one is added to the susceptible group, since we ignore birth and migration. The only way an individual leaves the susceptible group is by becoming infected. The number of susceptibles depends on the number already susceptible, the number of individuals already infected and the contact between the two groups. Let us suppose that the coefficient beta, B, represents the contacts per day between the infected and the susceptible that are sufficient to spread the disease. If we assume homogeneous mixing of the populations, each infected individual generates B * s(t) new infected individuals per day. We also assume that a fixed fraction, mu, µ, of the infected group will recover during any given day. For example, if the average duration of infection is three days, then, on average, one-third of the currently infected population recovers each day. We will ignore the "recovered" (non- infectious) people, for now, who still feel miserable, and might even die later from other complications. As a result, we have the following differential equations: · The Susceptible Equation: ds = -ß* s(t) * i(t) dt dr • The Recovered Equation: = u * i(t) dt • The Infected Equation: di = B* s(1) * i(t) – u * i(t) dt Answer the following questions in the space provided. 2a. ds di dr + + dt dt dt Explain why = 0 ? 2b. Describe both components of the infected equation. 2c. What do the minus signs represent? Command Window Paul
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