WorkSheetPrintOut.qmd

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title: " Pop. dynamics worksheet"
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# Introduction 

The aim of this demonstration is to show how we can use ideas from calculus to study dynamical systems.  

* It is *not* intended that you work through all the questions in the available time.
* You are encouraged to use your phone to explore the linked apps 



# Recap {#sec-background}

You might have previously encountered differentiation. Suppose that $y$ is some function of $x$. 

Consider the differential equation

$$
\frac{dy}{dx}=1.
$$

Upon integration

$$
y(x)=x+C
$$
where $C$ is an integration constant.


Now suppose that
$$
\frac{dy}{dx}=x.
$$

::: {.callout-note}
# Question

Can you integrate this ordinary differential equation and identify the solution $y=y(x)$?

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# Modelling population dynamics

## Formulating a model of population dynamics

Let's consider a model for the number of people in a room at a given time. Let $t$ represent time and $N(t)$ represent the number of people in the room at time $t$.

Suppose that there are initially no people in the room, but people enter at a constant rate, $k$.

We could formulate a model of population dynamics given by

$$
\frac{dN}{dt}=k, \quad N(0)=0.
$$ {#eq-constrhs}

::: {.callout-note}
# Question

* Can you integrate @eq-constrhs (Hint: it is mathematically equivalent to the ODE introduced in @sec-background)?

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* Can you use the solution of the model to determine the amount of time taken for the number of people in the room to reach some capacity, $N_C$.

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*  Can you use the app (see @fig-qrcode) to identify what the entry rate, $k$, needs to be such that the room reaches capacity of 40 people after 20 minutes?  

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![https://dundeemath.github.io/Admissions/posts/PopulationDynamicsIntro.html.](MathAdmissionsQRCode.png){#fig-qrcode width=10%}





## What if people enter the room at a constant rate but also leave the room at random?

Taking the previous model as a starting point, we now assume that people can also leave the room at a rate proportional to the number of people in the room

The model equation is now given by

$$
\frac{dN}{dt}=k - dN, \quad N(0)=0.
$$ {#eq-constrhsandrem}


::: {.callout-note}
# Question

It is possible to integrate @eq-constrhsandrem and show that the solution is
$$
N(t)=\frac{k}{d}(1-e^{-dt})
$$  {#eq-constrhsandremsol}

Can you do this? (hint: try using an *integrating factor*)?

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# Question

Can you use the model solution (@eq-constrhsandremsol) to determine the amount of time taken for the number of people in the room to reach capacity, $N_C$.
 Does a solution always exist?

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::: {.callout-note}
# Question

Can you use the app or the solution (@eq-constrhsandremsol) to identify the entry rate needs to be such that the room reaches capacity of 40 people after 20 minutes given $d=0.1$? 

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# The SIR model

The SIR model is used to study the spread of infectious disease.

In the SIR model a population is split into three groups:

- suspectible (S)
- infectious (I)
- recovered (R)

Unlike in the previous example, the population dynamics of each group depend on the levels of the other populations.



The governing equations are:
$$ 
\begin{aligned}
\frac{dS}{dt}&=-rIS, \\
\frac{dI}{dt}&=rIS-aI, \\
\frac{dR}{dt}&=aI. 
\end{aligned}
$$ {#eq-sir}

with initial conditions 

$$ 
\begin{aligned}
S(t=0)&=S_0, \\
I(t=0)&=I_0, \\
R(t=0)&=R_0.
\end{aligned}
$$


You can explore solution behaviour using this app in @fig-SIRMOdellink.




![https://dundeemath.github.io/Admissions/posts/TheSIRModel.html](MAthsadmissionsSIRQRCode.png){#fig-SIRMOdellink width=10%}



:::{.callout-note}

At Dundee, the mathematical tools needed are developed in modules:

* Maths 1A, 1B, 2A and 2B (Core maths modules)
* Computer algebra and dynamical systems
* Mathematical Biology I
* Mathematical Biology II

At Levels 2, 3 and 4 you will learn how to use computer programming to explore and communicate mathematical concepts.

You can find out more about these modules [here](https://www.dundee.ac.uk/undergraduate/mathematics-bsc/teaching-and-assessment).

:::