Question 2 : Consider a disease for which only 25% of individuals who have become exposed are progressing to become infectious. However, whether or not an individual becomes infectious can only be determined after a 5-day latent period. Individuals who are not progressing to become infectious will remain susceptible. The average time an individual stays infective is 2 days. Also, after an average of 10 days after having recovered, 10% of recovered individuals will revert to become susceptible again. 90% of the individuals, however, acquire lifelong immunity. (a) Develop the transition diagram that depicts the move of individuals between compartments. Hint: you can use PowerPoint or online tool such as https://app.diagrams.net/ to draw the diagram. The diagram should contain the labels of the stages and have arrows indicating from one stage to another stage. ******Susceptible, invectives, and recovered individuals in the SIR model are moving through "the compartments " S-I-R. The SIR model is referred to as compartmental model as you move individuals out of one compartment into another. *******
Question 2 :
Consider a disease for which only 25% of individuals who have become exposed are progressing to
become infectious. However, whether or not an individual becomes infectious can only be determined
after a 5-day latent period. Individuals who are not progressing to become infectious will remain
susceptible. The average time an individual stays infective is 2 days. Also, after an average of 10 days
after having recovered, 10% of recovered individuals will revert to become susceptible again. 90% of the
individuals, however, acquire lifelong immunity.
(a) Develop the transition diagram that depicts the move of individuals between compartments.
Hint: you can use PowerPoint or online tool such as https://app.diagrams.net/ to draw the diagram.
The diagram should contain the labels of the stages and have arrows indicating from one stage to
another stage. ******Susceptible, invectives, and recovered individuals in the SIR model are moving through "the compartments " S-I-R. The SIR model is referred to as compartmental model as you move individuals out of one compartment into another. *******
![**The Basic Reproduction Number \( R_0 \)**
To initiate an epidemic, an increase in the number of infectives is necessary:
\[
\Delta I = \alpha SI - \gamma I = \gamma \left(\frac{\alpha}{\gamma} S - 1\right) I
\]
**Graph Explanation:**
The accompanying graph illustrates the typical progression of an infectious disease outbreak over time. The y-axis represents the number of infectious individuals, while the x-axis marks the duration of the outbreak.
- **Peak of Outbreak:** Displayed by the highest point on the graph, signifying the maximum number of infectious individuals.
- **Measurement Area:** A small shaded region at the beginning of the outbreak, representing early data collection or initial cases.
- **Outbreak Duration:** The span from the start to the end of the outbreak, depicted by the graph's length on the time axis.
**Conditions for Epidemics:**
- \(\frac{\alpha}{\gamma} S > 1 \Rightarrow \Delta I > 0\): Indicates an increase in infectives, essential for an epidemic.
- \(\frac{\alpha}{\gamma} S = 1 \Rightarrow \Delta I = 0\): Suggests a stable number of infectives with no epidemic growth.
- \(\frac{\alpha}{\gamma} S < 1 \Rightarrow \Delta I < 0\): Reflects a decrease in infectives, leading to the decline of the epidemic.
This framework provides a mathematical basis for understanding how changes in the number of susceptible individuals and the transmission dynamics affect the potential onset of an epidemic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc353cc0f-0c4d-4d5b-a3e6-e746dbb90f94%2Fdcba9892-bdb4-4823-8b2c-63bafa1e1b8e%2Fy68z3tr_processed.png&w=3840&q=75)
![**Deriving \( R_0 \)**
The change in the number of infected individuals (\(\Delta I\)) over time is given by the equation:
\[
\Delta I = \gamma I_t \left( \frac{\alpha S_t}{\gamma} - 1 \right)
\]
Where:
- \(\Delta I\) is the change in the number of infected individuals.
- \(\gamma\) is the recovery rate.
- \(I_t\) is the number of infected individuals at time \(t\).
- \(\alpha\) is the transmission rate.
- \(S_t\) is the number of susceptible individuals at time \(t\).
The basic reproduction number \(R_0\) is defined as:
\[
R_0 = \frac{\alpha}{\gamma} S_0
\]
Where:
- \(R_0\) is the basic reproduction number.
- \(S_0\) is the initial number of susceptible individuals.
Interpretations of \(R_0\):
- If \(R_0 > 1\), there is a **possible epidemic**, meaning the infection will likely spread.
- If \(R_0 = 1\), the infection is considered **endemic**, remaining at steady levels.
- If \(R_0 < 1\), there will likely be **isolated cases**, and the infection might die out.
The diagram shows the relationship moving from understanding changes in infection levels to interpreting the significance of \(R_0\) for predicting the spread of an illness.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc353cc0f-0c4d-4d5b-a3e6-e746dbb90f94%2Fdcba9892-bdb4-4823-8b2c-63bafa1e1b8e%2Frco5kqt_processed.png&w=3840&q=75)
Susceptible Infectious Recovered model (SIR) helps in predicting the number of individuals which are susceptible or infected or been recovered from infection.
Infectious individuals: 25% of individuals are exposed to disease are said to be progressing to become infectious. Decision factor is the latent period of 5 days.
Susceptible individuals: Individuals who are not progressing to become infectious are susceptible to disease.
Average time an individual will be infective =2 days.
Recovered but yet susceptible: 10% of recovered individuals will revert to become susceptible again after 10 days.
Life long acquired immunity: Rest recovered 90% of the individuals will be recovered completely acquiring immunity.
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