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# Activity: 28 Days Later – Zombies Zombies

## Purpose

To explore differential equations as a means to model infection outbreak.

## Achievement objectives

• M 8-4 Use curve fitting, log modelling, and linear programming techniques
• M 8-10 Identify discontinuities and limits of functions
• M 8-11 Choose and apply a variety of differentiation, integration, and anti-differentiation techniques to functions and relations, using both analytical and numerical methods
• M 8-12 Form differential equations and interpret the solutions

## Indicators

Solves problems using curve fitting and log modelling.

• Links features of graphs with the limiting behaviour of functions.
• Uses limiting features of functions to sketch graphs.
• Finds limits algebraically, graphically, and numerically by considering behaviour as:
• x approaches a specific value from above and below
• x tends towards +∞ or -∞
• Uses a variety of integration techniques for functions including:
• Applications include:

• rates of change
• Uses knowledge of anti-differentiation to form and solve differential equations of the type:
• with the rate of change directly or inversely proportional to the variable of interest
• first and second order with variables easily separated for functions including those listed in M8-11.
• Applies boundary or initial conditions to solutions of differential equations.
• Distinguishes between families of solutions and exact solutions using given boundary or initial conditions and interprets these solutions.
• Links families of solutions to the gradient field of the differential equation.

## Specific learning outcomes

Students will be able to:

• use numerical methods and spread sheets to model rates of change
• set up and solve differential equations.

## Diagnostic snapshot(s)

Students will need prior understanding of differential equations and separation of variables. This could be a good summation exercise following teaching on these topics.

Students will need prior knowledge of M7-2 graphs.

## Planned learning experiences

Show the trailer for the film ’28 Days Later’. Please be advised that this film is R16, check suitability before showing. The trailer is rated M.

You may also wish to show the related links which simulate zombie infections and university simulations.

### Numerical modelling an outbreak of zombies

Discuss with the students what will happen if:

1. On day 1 there is 1 zombie and 999 humans in the world.
2. The proportion of humans that will become zombies on any one day is 0.2.
3. The proportion of zombies that will recover and become human is 0.1.
4. No one will die, so that there are always 1000 zombies and humans in the world altogether.

### Modelling numerically

This can be modelled in a pencil and paper exercise. Keep a table of the number of zombies and humans on each of days 1,2,3,4….etc.

Number of zombies in day n = zn.

Number of humans on day n = hn

Then zn+1 = 0.9zn + 0.2hn and zn + hn = 1000

Students can then move to modelling this on a spread sheet. One possible spread sheet is provided.

Graph the population of zombies against the number of days.

Fit an appropriate model (exponential to all but the first day or two.)

### Modelling by simulation

The spread of the zombies could be modelled by a simulation, linking with the level 7 simulation work.

The simulation requires a large number of people in the initial population (at least over 100) and thus requires a collaborative exercise from the class.

Initial population could 10 zombies and 90 humans. Use random numbers to simulate the conversion from zombie to human and vice versa, and record the total number of zombies for each day in a table. Graph the number of zombies against the day.

Fit an appropriate model (an exponential function to all but the first day or two).

Repeat the simulation with different conversion probabilities.

### Extension

Change the initial populations and conversion rates in the spread sheet and investigate how this affects the graph and model.

Investigate how the limit of the number of zombies is informed by the two conversion rates.

### Modelling the outbreak of zombies with a differential equation.

Given the initial conditions above:

• let z = number of zombies in the population
• t = number of days after the initial infection

The rate of change of zombies in the population:

1. decreases in proportion to the number of zombies in the population: -0.1z

and

2. increases in proportion to the number of humans in the population: +0.2(1000-z)

therefore dz/dt = -0.1z + 0.2(1000-z)

Use this information to solve the differential equation and compare the solution with the graphs and functions found when using the numerical modelling process.

### Possible adaptations to the activity

You may wish to make this activity appeal to a different audience by varying the type of infection e.g. to avian flu, measles or another topical infection.

Health, Biology, Media studies.

## Extension/enrichment ideas

What if:

• you could change the initial number of zombies?
• you could change the initial population
• you could change one or both proportion rates
• how do these changes the model?
• how do these changes affect the limit?you could apply these principles to social networking and ‘viral’ clips?

## Planned assessment

This teaching and learning activity could lead towards assessment in the following achievement standards:

## Spotlight on

### Pedagogy

• Creating a supportive learning environment
• Listening to all student responses.
• Valuing student contributions.
• Respecting and valuing the mathematics and cultures that students bring to the classroom.

### Key competencies

Thinking

• Students select appropriate methods and strategies when solving problems.
• Students make deductions, they justify and verify, interpret and synthesis and they create models.
• Students hypothesise, investigate, analyse and evaluate.
• Students design investigations, explore and use patterns and relationships in data and they predict and envision outcomes.
• Students deal with uncertainty and variation, they seek patterns and generalisations.

Using language, symbols and texts

• Students use symbols and diagrams to solve problems.

Relating to others

• Students listen to others, they accept and value different viewpoints.
• Students work in groups, they debate solutions, negotiate meaning and communicate thinking.

Managing self

• Students develop skills of independent learning.