91Ó°ÊÓ

Mit dem folgenden Modell hat man die Ausbreitung ansteckender Krankheiten untersucht. Man nehme an, \(\mathrm{da} \beta\) es \(\mathrm{N}\) Personen gibt, von denen einige an Grippe erkrankt sind. Folgende Annahmen sollen zutreffen: ( a) wenn eine kranke und eine gesunde Person in Kontakt kommen, dann ist \(\alpha\) die Wahrscheinlichkeit dafür, daß letztere angesteckt wird; ( b) alle Kontakte finden zwischen zwei Personen statt ( \(\mathrm{c}\) ) alle möglichen Kontakte sind gleichwahrscheinlich ( \(\mathrm{d}\) ) in jeder (so gewählten) Zeiteinheit kommt es zu einem solchen Kontakt. Man definiere eine Markow-Kette für die Ausbreitung der Epidemie und schreibe die Übergangsmatrix auf. (Sind Sie niedergeschmettert durch alle diese zu sehr vereinfachenden Annahmen? Angewandte Mathematik beruht auf der klugen Auswahl und Auswertung solcher vereinfachter Modelle.)

Step by step solution

01

- Define the States

Define the possible states of the Markov chain. Let the state be the number of infected individuals in the population, denoted by \(X_t\) where \(t\) is the time step. Each state can be represented by \(X_t = i\), where \(i\) indicates the number of infected people (\(i = 0, 1, 2, \text{...}, N\)).

02

- Identify Possible Transitions

Determine the possible transitions between the states. From state \(X_t = i\), the number of infected individuals can increase, decrease, or remain the same in the next time step \(X_{t+1}\). This means the transitions are: \(X_t = i \rightarrow X_{t+1} = i + 1\): An uninfected person becomes infected. \(X_t = i \rightarrow X_{t+1} = i - 1\): An infected person recovers (though recovery is not included in the problem, it may be assumed for completeness). \(X_t = i \rightarrow X_{t+1} = i\): No change in the number of infected individuals.

03

- Calculate Transition Probabilities

Given the probability \(\alpha\) that a healthy person becomes infected upon contact with an infected person, compute the transition probabilities.:- **\(P(i \rightarrow i+1)\)**: The probability that the number of infected individuals increases by 1. This can be computed as: $$P(i \rightarrow i+1) = \alpha \, \frac{i}{N} \, (N-i)$$ where \(\, \frac{i}{N}\) is the probability for an infected person and \((N-i)\) denotes the number of susceptible individuals.- **\(P(i \rightarrow i-1)\)**: Assuming recovery has the probability \(\gamma\), the transition probability to a decrease in the number of infected individuals is: $$P(i \rightarrow i-1) = \gamma \, \frac{i}{N}$$- **\(P(i \rightarrow i)\)**: The probability that the number of infected individuals remains the same is the complement of the above probabilities:$$P(i \rightarrow i) = 1 - (P(i \rightarrow i+1) + P(i \rightarrow i-1))$$

04

- Construct Transition Matrix

Based on the calculated transition probabilities, construct the transition matrix \(P\). This matrix is \((N+1)\times(N+1)\) with each element \(P[i][j]\) representing the probability of transitioning from state \(i\) to state \(j\). For a small population, say \(N=2\), the matrix would be: $$P = \begin{bmatrix} 1 - \alpha & \alpha & 0 \ \gamma & 1 - (\alpha + \gamma) & \alpha(N-1) \ 0 & \gamma & 1 - \gamma(N-1) \end{bmatrix}$$

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Epidemic Modeling

Epidemic modeling is a method used to understand and predict the spread of infectious diseases within a population. When we create these models, we often make simplifying assumptions to manage complexity. For instance, consider a population of \(N\) individuals, among whom some are infected with a virus, like the flu. We might assume:

• Infected and healthy persons have equal chances of contact (equal contact likelihood)
• Each contact between an infected and a healthy person carries a probability \(\alpha\) of transmitting the virus.

These assumptions help us to set up a framework where we can apply mathematical and statistical tools to predict how the disease spreads over time. Using these assumptions, we can define a Markov chain—a sequence of random events where the future state depends only on the current state, not on the sequence of events that preceded it. In epidemic modeling, the `state` usually represents the number of infected people at a specific time.

Transition Matrix

In the context of epidemic modeling, a transition matrix is a key tool used to describe the probabilities of moving from one state to another in a Markov chain.

For instance, suppose \(X_t=i\) represents the number of infected individuals at time step \(t\). The transition matrix \(P\) then encompasses the probability of transitions:

• \(P(i \rightarrow i+1)\): Infection probability, \(\alpha \, \frac{i}{N} \, (N-i)\)
• \(P(i \rightarrow i-1)\): Recovery probability, \(\gamma \, \frac{i}{N}\) (if a recovery mechanism exists)
• \(P(i \rightarrow i)\): Probability that the number of infected individuals remains the same, computed as \(1 - (P(i \rightarrow i+1) + P(i \rightarrow i-1))\)

Each entry in the transition matrix represents the probability of moving from one state to another. For example, in a small population of \(N=2\), the transition matrix might look like:
\[P = \begin{bmatrix} 1 - \alpha & \alpha & 0 \ \gamma & 1 - (\alpha + \gamma) & \alpha(N-1) \ 0 & \gamma & 1 - \gamma(N-1) \end{bmatrix}\] This matrix allows us to analyze how probabilities evolve over time as individuals move between being infected and healthy, providing a powerful tool for predicting disease spread.

Infection Probability

The infection probability, denoted as \(\alpha\), is a crucial parameter in epidemic modeling. It represents the likelihood that a healthy person will become infected upon contact with an infected person. The calculation for this takes into account various factors:

• Factor of \(\alpha\): Probability of disease transmission per contact
• Number of infected individuals \(i\)
• Total population size \(N\)

The formula \(\alpha \frac{i}{N} (N-i)\) describes the probability that any contact between an infected and a healthy individual results in the transmission of the disease. This model assumes all contacts are equally probable and occurs once per time step. Keep in mind this is a simplification, real-world interactions can be more complex. This probability is foundational in our transition matrix and helps us create detailed predictions about how quickly a disease can spread through a population. Understanding this concept is key to implementing effective preventative measures in public health.