02 Intro Modeling

.title[Overview of Systems Modeling] .subtitle[BEE 4750/5750] .subtitle[Environmental Systems Analysis, Fall 2022] <hr> .author[Vivek Srikrishnan] .date[August 24, 2022]

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# Outline

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1. Questions
2. Review of Last Class
3. Classification of Systems Models
4. Example

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# Poll

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URL: [https://pollev.com/vsrikrish](https://pollev.com/vsrikrish)

Text: **VSRIKRISH** to 22333, then message]

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***Any questions?***

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# Last Class

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* Class Structure and Policies
  * What Is A System?

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# What Is A System?

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.left-column[A system consists of a set of components which are connected by pathways over which flows affect stocks.

**What are some examples?**]

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# What Is *Not* a System?

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We don't need a systems approach when components can be treated individually, e.g., *the whole is no more than the sum of its parts*.

**Can we think of some examples?**

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# Systems Analysis Needs

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To model a system, we need:

* A **definition** of the system ✅
  * A **model** of the system

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# What Is A Model?

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.centered[![Falling Water Miniature Model](https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Fallingwater_miniature_model_at_MRRV%2C_Carnegie_Science_Center.JPG/640px-Fallingwater_miniature_model_at_MRRV%2C_Carnegie_Science_Center.JPG)] .centered[.cite[Source: [Wikimedia](https://commons.wikimedia.org/wiki/File:Fallingwater_miniature_model_at_MRRV,_Carnegie_Science_Center.JPG>)]]] ]

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# Mathematical Models of Systems

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# Environmental Systems Example

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* Municipal sewage into lakes, rivers, etc.
 * Power plant emissions into air
 * Solid waste placed on landfill
 * CO2 into atmosphere

]

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# Types of Mathematical Models

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# Types of Mathematical Models

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Used primarily for describing or simulating dynamics.

* Intended for *simulations* and *exploratory* and/or *Monte Carlo analysis*.

]

Specify (prescribe) an action, decision, or policy.

* Intended for *optimization* or *decision analysis*.

]

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# An Example!

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Two factories are discharging a chemical, *chlororadiated ureadicarboxyl* (CRUD), into the Riley River.

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Environmental authorities have sampled water from the river and determined that concentrations exceed the legal standard (1 mg/L).

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We want to design a CRUD removal plan to get back in compliance.

**Where do we start? What do we need to know?**

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# What We Know About CRUD

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* CRUD decays in the river with first-order decay coefficient $k=0.45 \ \text{d}^{-1}$.

* Treating CRUD costs $ \$50 E^2 \text{ per } 1000 \  m^3,$ where $E$ is the treatment efficiency.
  * If $E_i$ is the treatment efficiency at factory $i$, the total treatment cost is

$$
C(E_1, E_2) = 50(100)E_1^2 + 50(60)E_2^2.
$$

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# How Much To Remove?

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Consider the problem at each point of release.

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(recall that $1 \ \text{mg/L} = 1 \ \text{g/m}^3 = 10^{-3} \ \text{kg/m}^3$)

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1. What is the inflow? What is the outflow for a given $E_i$?
2. How does this impact the concentration downstream?

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# Mass Balance at Release 1

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Factory 1 releases $100 \ \text{kg/d}$ without treatment.

**What is the outflow for a given $E_1$?**

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# Mass Balance at Release 1

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Total CRUD after factory 1 release: $\color{blue}\text{100} + \color{red} 1000(1-E_1) \color{black} \ \text{kg/d}$

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# Mass Balance at Release 2

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But this isn't the inflow for release 2...

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# Mass Decay

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Given a decay rate of $k = 0.45 \ \text{d}^{-1}$:

$$
\frac{dM}{dt} = -0.45 M \Rightarrow \frac{dM}{M} = -0.45 dt
$$

$$
\int_{M(0)}^{M(T)} \frac{dM}{M} = -0.45 \int_0^T dt
$$

$$
\ln\left(\frac{M(T)}{M(0)}\right) = -0.45 T
$$

$$
M(T) = M(0) \exp\left(-0.45 T\right)
$$

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# Mass Decay

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So now we know that after $t$ days,  $M(t) = M_0 \exp \left(-0.45 T \right).$

But we want to know $M(x)$, where $x$ is in distance, since we know the distance between factories 1 and 2 is $10 \ \text{km}$.

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# Mass Decay

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Since the velocity of the river is $25 \ \text{km/d}$:

$$
M(x) = M_0 \exp\left(-0.45 (x/25)\right), \quad x \leq 10 \ \text{km},
$$

and simplifying and plugging in $x = 10$ and $M_0$, the inflow of CRUD at the factory 2 release is:

$$
M(10) = (1100 - 1000E_1) \exp(-0.18) \ \text{kg/d}.
$$

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# Mass Balance at Release 2

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# Mass Balance at Release 2

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This means that after factory 2 releases CRUD, the concentration is:

$$
(1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2) \ \text{kg/d}.
$$

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# Mass Balance at Release 2

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We can use this as an initial condition for $M(x), x > 10$:

$$
M(x) = M_1 \exp\left(-0.45 (x/25)\right), \quad x > 10,
$$

where $M_1 = (1100 - 1000E_1) \exp(-0.18) + 1200(1 - E_2)$.

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# ...Now What?

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Our next steps depend on what we're trying to do: is our model descriptive or prescriptive?

* *Descriptive*: Use the dynamical equations to simulate concentrations for varying levels of $E_1, E_2$ and compare those to the costs and the legal limit.

* *Prescriptive*: Turn these equations and limits into constraints and find the values of $E_1, E_2$ which minimize costs while staying in compliance (we'll return to this later!).

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# Discussion of Oreskes et al (1994)

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> Models can corroborate a hypothesis by offering evidence to strengthen what may be already partly established through othermeans. Models can elucidate discrepancies in other models. Models can be also be used for sensitivity analysis –- for exploring "what if" questions –- thereby illuminating which aspects of the system are most in need of further study, and where more empirical data are most needed. **Thus, the primary value of models is heuristic: Models are representations, useful for guiding further study but not susceptible to proof.**
>
> .cite[– Oreskes et al, 1994]

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# "All Models Are Wrong, But Some Are Useful"

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> ...all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind....
>
> .cite[– Box & Draper, 1987]

A key consideration: how do we check/ensure that models we use are useful? How do we know when they're not?

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# A Leading Question

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Someone has developed a model of a complex process (say...number of cases of a particular infectious disease). They claim that their model has precisely predicted case counts for a few months, and use this to argue for the model's validity (and their skill as a modeler, presumably).

**Should we trust their claim? What might this imply about the model? Is it useful?**

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# Next Class

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* Descriptive modeling with systems simulation
  * What are *uncertainty* and *risk* and why are they important concepts?