# Deterministic to stochastic modeling

#### kiton

##### Member
Hello dear forum members,

I am seeking your feedback on the question that I have limited knowledge about (i.e., stochastic modeling).

Say, I am working with a variant of standard epidemiological compartment model -- SEIR (Susceptible --> Exposed --> Infected --> Recovered):

In my analysis I focus on the role of within- and between-county mobility in prediction of the new infections. Therefore, I add two components to this (deterministic) model: (1) within-county mobility parameter, denoted as rho, which pre-multiplies betas in E and I equations, and also (2) between-county mobility flows using origin-destination (OD) patterns. Building on prior research and available data science centered publications, I arrive at the following model for the newly exposed people:

where rho(jt) denotes the within-county mobility index. x(kt) denotes the proportion of infected population at county j at time t. m(jk) is the number of people traveling from county k to county j in one unit of time. To express the population flow, I use M by M OD-matrix where M is the number of locations in the simulated area.

My question is: What would be a plausible approach to "convert" this model to a stochastic one?

Your feedback and guidance would be greatly appreciated.

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#### Miner

##### TS Contributor
There are several approaches to stochastic modeling:
• Monte Carlo simulation - Start with any mathematical model. Generate random numbers from a specified distribution for each term in the model and calculate the response for each combination.
• Discrete Event simulation - Models a system with a discrete sequence of events in time with changes in state.
• Markov chains - probability based

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#### kiton

##### Member
Thank you for response @Miner. I will look into discrete event simulation.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Back in March-April i did a bunch of SIR compartment models (E) wasnt available at the time and conferred immunity unknown. I looked into monte carlo versions, where you typically provide distributions into the model and run multiple curves. Benefit obviously you can account for variability and get confidence/credible intervals. I will take a look if a made, notes. There was one particular method most people historically used, it was named after a person - i can almost see the name in my mind's eye!

#### kiton

##### Member
Thank you @hlsmith. Confidence intervals are exactly what I am after. I'd appreciate the update on the name.