In IBAHCM/RPiR: Reproducible Programming in R

Models come in many types... {.flexbox .vcenter}

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The purpose of a model... {.flexbox .vcenter}

- A model captures the essence of an object or process - A model should be appropriate for the specific task or questions { width=60% }

The purpose of a model... {.flexbox .vcenter}

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"A model should be as simple as possible but no simpler..."

Albert Einstein

Why build mathematical models?

• To simplify
• To understand
• To predict

Uses of epidemiological models {.flexbox .vcenter}

- Understand disease dynamics - Estimate important parameters - Identify where we need more data - Explore control options - Make predictions { width=80% }

Implementing a model... {.flexbox .vcenter}

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Implementing a model... {.flexbox .vcenter}

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Simple population dynamics

Exponential growth or decline

• Rate of increase (or decrease) is proportional to current population size

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Exponential growth or decline

• Rate of increase (or decrease) is proportional to current population size

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Exponential growth or decline

• Rate of increase (or decrease) is proportional to current population size

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Exponential growth or decline

• Rate of increase (or decrease) is proportional to current population size

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Hamster population explosion {.flexbox .vcenter}

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Human population growth

• Define a variable to represent the number of individuals in the population at time $t$
• $N(t)$, where $N$ is a dependent variable and $t$ is an independent variable
• is the same as $N_t$
• or just $N$ (shorthand)

Initial conditions

• Define the value of the population at time $t=0$

$$\begin{eqnarray} N(t = 0) &=& N_0 \ &=& 7\ 000\ 000\ 000 \end{eqnarray}$$

Parameters

• Specify the rate of increase
• Human population increasing at approximately 1.5% each year
• Rate of increase $\lambda = 0.015$ per year

The difference equation model

• Incrementing the human population in 1 year time steps

$$N(t + 1) = \lambda \times N(t) + N(t)$$ - where $\lambda = 0.015$

Changing the time step

• Incrementing the human population in 1 month time steps

$$N(t + 1) = (\lambda / 12) \times N(t) + N(t)$$

• where $\lambda = 0.015$

Return to time steps of a year

• Incrementing the human population in 1 year time steps

$$N(t + 1) = \lambda \times N(t) + N(t)$$

• where $\lambda = 0.015$

How long until population doubles?

$$N(t + 1) = \lambda \times N(t) + N(t)$$

$$ \mathit{i.e.\quad} N(t + 1) = (\lambda + 1) \times N(t)$$

How long until population doubles?

$$\begin{eqnarray} N(t + 2) &=& (\lambda + 1) \times N(t + 1) \ &=& (\lambda + 1)\times(\lambda + 1) \times N(t) \ &=& (\lambda + 1)^2 \times N(t) \end{eqnarray}$$

How long until population doubles?

$$N(t + n) = (\lambda + 1)^n \times N(t)$$

So, the number of years ($n$) taken to double is the $n$ that satisfies

$$(\lambda + 1)^n = 2$$

Solving the equation

$$\begin{eqnarray} (\lambda + 1)^n &=& 2 \ \log((\lambda + 1)^n) &=& \log(2) \ n \times \log(\lambda + 1) &=& \log(2) \ n &=& \log(2) / \log(\lambda + 1) \ n &=& 46.6 \end{eqnarray}$$

Doubling time

• The doubling time ($n$) does not depend on the starting population size
• This is always true of an exponentially growing population

Fixed doubling time

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Model types

• Difference equation models
• Describe updating of population in chunks of time
• e.g. $N(t + 1) = \lambda \times N(t) + N(t)$

Model types

• Difference equation models
• Describe updating of population in chunks of time
• e.g. $N(t + 1) = \lambda \times N(t) + N(t)$
• Differential equation models
• Describe continuous updating of population
• e.g. $\frac{dN}{dt} = \lambda \times N$
• where $\frac{dN}{dt}$ is the rate of change of $N$

Model types

• Difference equation models
• Describe updating of population in chunks of time
• e.g. $N(t + 1) = \lambda \times N(t) + N(t)$
• Differential equation models
• Describe continuous updating of population
• e.g. $\frac{dN}{dt} = \lambda \times N$
• where $\frac{dN}{dt}$ is the rate of change of $N$
• Difference equations
• Good approximations to continuous models when time chunks are small
• No need to worry about differential equations today :)

Practicals

• Programming in R Practical:
• Learn some programming skills
• Build simple population dynamics models in R

IBAHCM/RPiR documentation built on Jan. 12, 2023, 7:41 p.m.