Final Review

# Final Review
### Mathew Kiang
### 3/6/2017

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## Agenda

1. Structure

1. `$R_0$`: The reproductive number

1. Case fatality rate

1. Vaccinations

1. General tips

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### What is the reproductive number?

`$R_0$` is "the expected number of secondary cases produced by a typical infected individual early in an epidemic".

That is, how many people are expected to become infected from a single infectious person in an otherwise susceptible population?

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### Why is it useful?

Provides a simple threshold:

- If `$R_0 > 1$`, disease is epidemic
- If `$R_0 = 1$`, disease is endemic
- If `$R_0 < 1$`, disease will die out

### What if the population is not entirely susceptible?

Effective reproductive number, `$R_e = R_0 x$` where `$x$` is the proportion of contacts that are susceptible.

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### How do we calculate it ($R_0$)?

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1. Directly fitting curve to epidemic trajectory
    - See Althaus Ebola paper where he used maximum likelihood to fit curves to the data
1. From equilibrium values
    - `$R_0 = N/S_{eq}$` (See Lecture 3, Slide 16)
1. From model parameters
    - In a simple SIR, `$$R_0 = \frac{infection}{contact} \times \frac{contacts}{time} \times \frac{time}{infection} = \frac{\beta}{v}=\frac{bk}{v}$$`
    - For more complex models, it is just the rate that people enter the infectious compartment over the rate people leave the infectious compartment (through death, recovery, or some other mechanism)
    - See Lecture 3, Slide 7
1. From age of first infection
    - For rectangular age structures, `$R_0 = L/A$`
    - For pyramidal age structures, `$R_0 = 1 + L/A$`
    - Where `$A$` is mean age of (first) infection) and `$L$` is average life span
    - See Lecture 3, Slide 19
1. (Also seroprevalence data, but don't use this for final)

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### When do we need to worry about mortality in estimates?

Discuss. (Recall Lecture 3, Slide 17 and 18 in comparing mumps and HIV.)

When in doubt, just calculate both and see what the percent difference is. For example,

`$$R_0 = \frac{bk}{v} \quad \textbf{vs} \quad R_{0}^* = \frac{bk}{v+d}$$`

Where `$v$` is recovery and `$d$` is death rate.

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### Should you use a fancy method?

No.
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### What is the case fatality rate?

Proportion of deaths from (some defined set of) cases over the course of the disease.

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### What are difficulties with estimating it?

Discuss.

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### Think about how CFR affects dynamics of your disease

For example, High CFR vs Low CFR in context of no recovery vs lifetime immunity.
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### Know how to include it in your equations

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### What is the critical vaccination threshold?

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Proportion of the population you need to vaccinate in order to eliminate a disease
`$$p_c = 1 - \frac{1}{R_0}$$`

### What is coverage and efficacy?

### Ways vaccines can be imperfect?

"Leaky", "all-or-nothing", or "waning". See Lecture 5, Slide 28.

### Know how to incorporate vaccination in your model

See Lecture 5, Slide 25.

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# Tips

Review all lectures (but especially Lecture 3 and 5)

Know the basic models (SI, SIR, SIER) and when they are appropriate

For more complicated models (e.g., SIWR, Ross-MacDonald), you should at least know roughly what data you would need to estimate parameters, points of intervention, etc. Review Lecture 6.

Use the simplest model that you can justify (i.e., don't make age-specific compartments if mass action assumption holds)

Remember: heterogeneity is bad. Spatial heterogeneity, network structures, age effects, etc. Compartmental models assume mass action.

Probably useful to form groups and just spend an hour going over the quiz and comparing answers. Given how open-ended the quiz was, it is useful to know how other people would address the same question.

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class: center, middle, inverse
# You're going to be fine