pipenv install --dev
pipenv shell
jupyter-lab

What is a model?

• a Model (M) for a system (S) and an experiment (E) is anything to which E can be applied in order to answer a question about S (Marvin Minski)
• in this course the focus is only on models that can be expressed as computer programs
• model's usefulness should be determined by model validation

What is a Simulation?

• Simulation is an experiment performed on a mathematical model

Types of models:

• (the actual system is not a model but could be used of course)
• mathematical model
  • there might exist an analytical solution, otherwise simulation must be used
• physical model

Types of mathematical models:

• Continuous time models (state changes continually over time)
  • state changes are usually represented by sets of differential equations
• Discrete time models (state changes continually but time axis is discretized)
  • states changes are usually represented by difference equations
• Discrete event models (like discrete time models, but the intervals between time steps does not have to be constant)

Types of simulations:

• Time-Driven Simulations
  • the time advances in fixed intervals of delta t
  • most often used for natural systems (continuous time models)
  • not ideal for discrete time models because delta t has to be small enough to capture every event in the discrete system
    • delta t could have to be infinitely small
• Event-Driven Simulations
  • the time advances whenever a new event occurs to account for the occurrence for that event
  • trade efficiency for precision

What are differential equations?

• Equations that describe changes rather than absolute values
• Often used when describing the change of values is easier than the formulation of a function that describes the absolute values
• Why do we need initial conditions?
  • Well, when we have a derivative and try to solve for a function that has this derivative, there can be multiple solutions
  • because of the constant term C
• usually a form of y' = f(x, y)
• g1(x) is a solution of y' = f(x,y)
  • <=> graph of g1(x) is an integral curve in the direction field associated with y' = f(x,y)
  • y'(x) = f(x, g1(x))

Types of differential equations:

• Orinary Differential Equations (ODE)
  • single independent variabele
• Partial Differential Equations (PDE)
  • multiple variables, e.g. the tempterature at each point in space

Types of solvers:

• solvers take a conceptual model (e.g. mathematical model) and map it to a virtual machine model to find solutions
• Numerical Solvers
• Direct Solvers
• Natural Solvers

• Epidemic
  • affecting many persons at the same time, and spreading from person to person in a locality where the disease is not permanently prevalent
• Pandemic
  • epidemic that has spread over a large area, that is, it’s “prevalent throughout an entire country, continent, or the whole world
• Endemic
  • disease that is prevalent in or restricted to a particular location, region, or population