31.2: Markov Models

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This is not the first time we have used Dynamical Linear Systems.

Do This

Review markov models in 10 Pre-Class Assignment: Eigenvectors and Eigenvalues. See how this is a special type of linear dynamical systems that work with state probabilities.

Example

The dynamics of infection and the spread of an epidemic can be modeled as a linear dynamical system.

We count the fraction of the population in the following four groups:

Susceptible: the individuals can be infected next day
Infected: the infected individuals
Recovered (and immune): recovered individuals from the disease and will not be infected again
Decreased: the individuals died from the disease

We denote the fractions of these four groups in \(x(t)\). For example \(x(t)=(0.8,0.1,0.05,0.05)\) means that at day \(t\), 80% of the population are susceptible, 10% are infected, 5% are recovered and immuned, and 5% died.

We choose a simple model here. After each day,

5% of the susceptible individuals will get infected
3% of infected inviduals will die
10% of infected inviduals will recover and immuned to the disease
4% of infected inviduals will recover but not immuned to the disease
83% of the infected inviduals will remain

Do This

Write the dynamics matrix for the above markov linear dynamical system. Come to class ready to discuss the matrix. (hint the columns of the matrix should add to 1).

# Put your matrix here

Do This

Review how we solved for the long term steady state of the markov system. See if you can find these probabilities for your dyamics matrix.

# Put your matrix here