(c) What conclusions can you draw about the behavior of the two dif- ferent epidemics?
(c) What conclusions can you draw about the behavior of the two dif- ferent epidemics?
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.4: Combining And Decomposing Functions
Problem 18E
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Have portions of part A and B complete, however I'm not sure on how to graph the phase potraits along with the inital conditions. Ultimately needing help with parts a through c thanks!
![### Understanding Epidemics: Susceptible-Infectious Model
In this section, we explore the dynamics of infectious diseases through a mathematical model. Specifically, we'll investigate a common epidemic model, which considers two vital populations: susceptible and infected individuals.
#### Key Assumptions:
1. **Immediate Infectivity:** Individuals become immediately infectious after contracting the disease.
2. **Average Death Rate:** On average, an infected individual has a lifespan of 10 days post-infection.
3. **Single Initial Infection:** The epidemic starts with just one infected person.
4. **Birth Rate of Susceptibles:** Susceptible individuals reproduce at a rate of 0.0003 per person per year, and their offspring are also susceptible.
#### Differential Equations Model
Let \( S(t) \) represent the number of susceptible individuals and \( I(t) \) represent the number of infected individuals over time \( t \). The following system of differential equations describes the interaction between these two populations:
\[
\frac{dS}{dt} = -\alpha SI + 0.0003S \quad \text{(2.2.7)}
\]
\[
\frac{dI}{dt} = \alpha SI - 0.1I \quad \text{(2.2.8)}
\]
Here, \( \alpha \) is a constant representing the relative infectivity of the disease.
#### Case Studies for Different Infectivity Rates
**Case (a): High Infectivity (\(\alpha = 0.05\))**
- **Task:** Draw the phase portrait for \(\alpha = 0.05\).
- **Instructions:** Identify and label all nullclines and equilibrium solutions. Assume initial conditions \( S(0) = 1000 \) and \( I(0) = 1 \).
- **Question:** What happens to the solutions as \( t \) approaches infinity?
**Case (b): Low Infectivity (\(\alpha = 0.000001\))**
- **Task:** Draw the phase portrait for \(\alpha = 0.000001\).
- **Instructions:** Identify and label all nullclines and equilibrium solutions. Assume initial conditions \( S(0) = 30000 \) and \( I(0) = 1 \).
- **Question:** What happens to the solutions as \( t \) approaches infinity?
#### Discussion
**Case (c): Comparative Analysis](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe39b72c6-6c1d-4e20-8eee-b58971cb1bcd%2Fa9ea945f-dbe3-4df1-a4c8-fe263656a9be%2Fphkb0sj_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Epidemics: Susceptible-Infectious Model
In this section, we explore the dynamics of infectious diseases through a mathematical model. Specifically, we'll investigate a common epidemic model, which considers two vital populations: susceptible and infected individuals.
#### Key Assumptions:
1. **Immediate Infectivity:** Individuals become immediately infectious after contracting the disease.
2. **Average Death Rate:** On average, an infected individual has a lifespan of 10 days post-infection.
3. **Single Initial Infection:** The epidemic starts with just one infected person.
4. **Birth Rate of Susceptibles:** Susceptible individuals reproduce at a rate of 0.0003 per person per year, and their offspring are also susceptible.
#### Differential Equations Model
Let \( S(t) \) represent the number of susceptible individuals and \( I(t) \) represent the number of infected individuals over time \( t \). The following system of differential equations describes the interaction between these two populations:
\[
\frac{dS}{dt} = -\alpha SI + 0.0003S \quad \text{(2.2.7)}
\]
\[
\frac{dI}{dt} = \alpha SI - 0.1I \quad \text{(2.2.8)}
\]
Here, \( \alpha \) is a constant representing the relative infectivity of the disease.
#### Case Studies for Different Infectivity Rates
**Case (a): High Infectivity (\(\alpha = 0.05\))**
- **Task:** Draw the phase portrait for \(\alpha = 0.05\).
- **Instructions:** Identify and label all nullclines and equilibrium solutions. Assume initial conditions \( S(0) = 1000 \) and \( I(0) = 1 \).
- **Question:** What happens to the solutions as \( t \) approaches infinity?
**Case (b): Low Infectivity (\(\alpha = 0.000001\))**
- **Task:** Draw the phase portrait for \(\alpha = 0.000001\).
- **Instructions:** Identify and label all nullclines and equilibrium solutions. Assume initial conditions \( S(0) = 30000 \) and \( I(0) = 1 \).
- **Question:** What happens to the solutions as \( t \) approaches infinity?
#### Discussion
**Case (c): Comparative Analysis
Transcribed Image Text:### Understanding Epidemics in Isolated Populations
Consider an epidemic that moves through an isolated population. We will make the following assumptions about the epidemic:
- **Infection Rate:** Individuals are infected at a rate proportional to the product of the number of infected and susceptible individuals. We assume that the constant of proportionality is \( \alpha \).
- **Incubation Period:** The length of the incubation period is negligible, and infectious in-
This excerpt provides a foundational understanding of how an epidemic might spread in an isolated population by considering specific assumptions related to the infection rate and incubation period. These assumptions are critical for constructing mathematical models to predict the behavior of such epidemics over time.
**Note:** The text seems to be incomplete at the end, and it abruptly terminates after mentioning the incubation period.
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