An individual can be either susceptible (S) or infected (1), the probability of infection for a susceptible person is 0.05 per day, and the probability an infected person becoming susceptible is 0.12 per day.
An individual can be either susceptible (S) or infected (1), the probability of infection for a susceptible person is 0.05 per day, and the probability an infected person becoming susceptible is 0.12 per day.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:An individual can be either susceptible (S) or infected (I). The probability of infection for a susceptible person is 0.05 per day, and the probability of an infected person becoming susceptible is 0.12 per day.
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This text can be used as part of a lesson on epidemiology models, specifically focusing on the transition probabilities between states in a basic S-I (Susceptible-Infected) model.
Transcribed Image Text:**Understanding the Susceptible-Infected Model: Stationary Distribution**
In the study of epidemiology, the Susceptible-Infected (SI) model is a simplified way of understanding the spread of an infectious disease within a population. In this model, individuals can transition between two states:
1. Susceptible (S): Individuals who are not infected but can contract the disease.
2. Infected (I): Individuals who have contracted the disease and can spread it to others.
**Objective:**
To report the probability of the infected state in the stationary distribution for the two-state Susceptible-Infected model, rounding to two decimal places.
**Explanation:**
In a stationary distribution, the probabilities of being in each state (Susceptible or Infected) remain constant over time. Calculating these probabilities helps in understanding how the disease will likely settle within the population in the long run. For instance, figuring out the proportion of the population that will be infected at equilibrium is crucial for resource allocation and control strategies.
**Key Insights:**
- **Stationary Distribution:** The equilibrium state where the probabilities of being in each state (Susceptible or Infected) do not change over time.
- **Probability of Infected State:** The focus here is determining the long-term proportion of the population that remains infected.
**Steps to Calculate:**
1. **Model the Transition Dynamics:** Define the rates at which individuals move between Susceptible and Infected states.
2. **Solve for Stationary Distribution:** Use mathematical techniques to find the equilibrium probabilities.
This ultimately provides insight into the disease's persistence and helps guide public health interventions.
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