Ebola Worksheet

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# Ebola Worksheet
## From Wednesday lecture – But Slower
### Mathew Kiang
### 2/2/2018

---

# Goals for today

## Go over the Ebola handout.red[*]

.footnote[.red[*]Again, only providing you with enough code to finish it on your own.]

---
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# Question 1.red[*]
## Make an SEIR model that incorporates case fatality ratio f

.footnote[.red[*]Sort of -- Questions are unnumbered on the worksheet.]
---
# Start with code you already have

```r
SEIR <- function(t, x, parms){
    with(as.list(c(parms, x)), {
        N <- S + E + I + R 
        dS <- - (beta * k * S * I) / N
        dE <- + (beta * k * S * I) / N - (a * E)
        dI <- + (a * E) - (r * I)
        dR <- r * I
        der <- c(dS, dE, dI, dR)
        return(list(der))
    }) 
}
```
Here is your boilerplate `SEIR` code. Incoporate `$f$`, which is a case fatality ratio. Recall, this is the fraction of infectious who do not recover.
--

- Also, change `a` to `s` ($\sigma$) and `r` to `g` ($\gamma$) to be consistent with the Althaus

- We are not going to use `b * k`, so replace that with `$\beta$` as `B`
---
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# Work with a neighbor to make this model
## Remember:
### rename **a** to **s**
### rename **r** to **g**
### use **B** instead of **b * k**
---

```r
library(deSolve)

dt <- seq(0, 365, 1) 
inits <- c(S = 999999,E = 0, I = 1, R = 0) 
parms <- c(B = 0.45, g = 1/5.61, s = 1/5.3, f = 0.6)

SEIR_ex <- function(t, x, parms) {
    with(as.list(c(parms, x)), {
        
        N <- S + E + I + R
        dS <- - (B * S * I) / N 
        dE <- + (B * S * I) / N - (s * E) 
        dI <- (s * E) - (g * I) 
        dR <- (1 - f) * (g * I) 
        
        der <- c(dS, dE, dI, dR)
        
        return(list(der))
    }) 
}

data_out <- as.data.frame(ode(inits, dt, SEIR_ex, parms = parms))
```
Your code should now look something like this.

- This is almost **exactly** like our boilerplate code. 
--

- Use the `inits`, `dt`, and `parms` I specified

Should all be very familiar by now. Review previous slides if this is still unclear.
---
# Plot of all lines

```r
matplot(data_out[, 1], data_out[, 2:5], type = 'l', 
        ylab = 'People', xlab = 'Time (days)',lty = 1)
legend(x = "topright", legend = c('S', 'E', 'I', 'R'), col = 1:4, lty = 1)
```

![](index_files/figure-html/unnamed-chunk-3-1.png)
---
# Plot of infected

```r
matplot(data_out[, 1], data_out[, 4], type = 'l', 
        ylab = 'Infected', xlab = 'Time (days)')
```

![](index_files/figure-html/unnamed-chunk-4-1.png)
---
class: center, middle, inverse
# Question 2
## With a neighbor, add compartments **C** for total cases and **D** for total deaths

---
# Add new compartments

```r
SEIR_ex <- function(t, x, parms) {
    with(as.list(c(parms, x)), {
        
        N <- S + E + I + R
        dS <- - (B * S * I) / N 
        dE <- + (B * S * I) / N - (s * E) 
        dI <- (s * E) - (g * I) 
        dR <- (1 - f) * (g * I) 
        
        der <- c(dS, dE, dI, dR)
        
        return(list(der))
    }) 
}
```
Again, start with code you already have. Add:

- `dC` which is the cumulative cases

- `dD` which is the total number of deaths
---
# Add new compartments

```r
SEIR_alt1 <- function(t, x, parms) {
    with(as.list(c(parms, x)), {
        
        N <- S + E + I + R
        dS <- - (B * S * I) / N 
        dE <- + (B * S * I) / N - (s * E) 
        dI <- (s * E) - (g * I) 
        dR <- (1 - f) * (g * I) 
        
*       dC <- s * E
*       dD <- f * g * I
        
*       der <- c(dS, dE, dI, dR, dC, dD)
        
        return(list(der))
    }) 
}
```
Don't forget to return `dC` and `dD` and add them in `inits`.

---
# Full Solution

```r
inits_alt1 <- c(S = 999999,E = 0, I = 1, R = 0, C = 0, D = 0)

SEIR_alt1 <- function(t, x, parms) {
    with(as.list(c(parms, x)), {
        
        N <- S + E + I + R
        dS <- - (B * S * I) / N 
        dE <- + (B * S * I) / N - (s * E) 
        dI <- (s * E) - (g * I) 
        dR <- (1 - f) * (g * I) 
        
        dC <- s * E
        dD <- f * g * I
        
        der <- c(dS, dE, dI, dR, dC, dD)
        
        return(list(der))
    }) 
}

data_alt1 <- as.data.frame(ode(inits_alt1, dt, SEIR_alt1, parms = parms))
```

---
# Plot of all lines

```r
matplot(data_alt1[, 1], data_alt1[, 2:7], type = 'l', 
        ylab = 'People', xlab = 'Time (days)',lty = 1)
legend(x = "topright", legend = c('S', 'E', 'I', 'R', 'C', 'D'), 
       col = 1:6, lty = 1)
```

![](index_files/figure-html/unnamed-chunk-8-1.png)
---
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# Question 3
## Time-varying transmission probability
---
# Time-varying transmission
Althaus parameterizes transmission probability as: `$$\beta(t) = \beta e^{-k(t-\tau)}$$`

- Assume: 
    - `$k=0.0097$` 
    - `$\beta=0.45$`
    - `$\tau=0$` (immediate control measures)
--

With a neighbor, plot `$\beta$` as a function of time from `$t=0$` to `$t=120$`

- Hints:

--
    - Make a sequence
--
    - Vectorized formulas are your friend
---
# Solution

```r
## Set constants
beta0 <- 0.45
k <- 0.0097
tau <- 0

## Plug into formula
days <- 1:120
betas <- beta0 * exp(-k * (days - tau))

## Plot it
plot(x = days, y = betas, type = "l")
```

![](index_files/figure-html/unnamed-chunk-9-1.png)
---
# Solution

```r
## Set constants
*beta0 <- 0.45
*k <- 0.0097
*tau <- 0

## Plug into formula
days <- 1:120
betas <- beta0 * exp(-k * (days - tau))

## Plot it
plot(x = days, y = betas, type = "l")
```
Set some constants. Not necessary, but makes the formula clearer.
---
# Solution

```r
## Set constants
beta0 <- 0.45
k <- 0.0097
tau <- 0

## Plug into formula
*days <- 1:120
betas <- beta0 * exp(-k * (days - tau))

## Plot it
plot(x = days, y = betas, type = "l")
```
Set some constants. Not necessary, but makes the formula clearer.

Make a sequence of days (or `seq(0, 120, 1/24)` for calculate hourly `$\beta$`)
---
# Solution

```r
## Set constants
beta0 <- 0.45
k <- 0.0097
tau <- 0

## Plug into formula
days <- 1:120
*betas <- beta0 * exp(-k * (days - tau))

## Plot it
plot(x = days, y = betas, type = "l")
```
Set some constants. Not necessary, but makes the formula clearer.

Make a sequence of days (or `seq(0, 120, 1/24)` for calculate hourly `$\beta$`)

Make a new vector with the formula we want. Even though `k`, `tau`, and `beta0` are scalars, `R` will automatically vectorize (perform element-wise calculations) on `days` since it has length > 1. (Try `print(betas)` if this is unclear.)
---
# Solution

```r
## Set constants
beta0 <- 0.45
k <- 0.0097
tau <- 0

## Plug into formula
days <- 1:120
betas <- beta0 * exp(-k * (days - tau))

## Plot it
*plot(x = days, y = betas, type = "l")
```
Set some constants. Not necessary, but makes the formula clearer.

Make a sequence of days (or `seq(0, 120, 1/24)` for calculate hourly `$\beta$`)

Plot it.
---
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# Question 4
## Now calculate and plot the changing R0
---
# Solution

## Hint: This is (literally) one line of code to calculate and one line of code to plot.
--

```r
r0s <- betas / (1/5.61)
plot(x = days, y = r0s, type = "l")
```

![](index_files/figure-html/unnamed-chunk-14-1.png)
---
# On what day is R0 < 1?
--

- Try `help(which)`

- Combine that with indexing

```r
which(r0s <= 1)[1]
```

```
## [1] 96
```

---
# On what day is R0 < 1?

```r
plot(x = days, y = r0s, type = "l")
lines(x = days, y = rep(1, length(days)), col = 'red')
```

![](index_files/figure-html/unnamed-chunk-16-1.png)
Or we could do it visually.

(If this isn't clear, see `help(rep)` and consider why it is necessary.)
---
class: center, middle, inverse
# With a neighbor, add the time-varying beta to SEIR model
## Assume tau=0 for simplicity
---
# Solution

```r
SEIR_alt2 <- function(t, x, parms) {
    with(as.list(c(parms, x)), {
        
*       B  <- B_init * exp(-k * t)
        N  <- S + E + I + R
        dS <- -(B * S * I) / N 
        dE <- +(B * S * I) / N - (s * E) 
        dI <- (s * E) - (g * I) 
        dR <- (1 - f) * (g * I) 
        
        dC <- s * E
        dD <- f * g * I
        
        der <- c(dS, dE, dI, dR, dC, dD)
        
        return(list(der))
    }) 
}
```
Yes, that's it..red[*]

.footnote[.red[*] NOTE: This only works when tau=0. Need `ifelse()` if we incorporate tau.]
---
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# Examine one of the countries
## (Do this on your own or with a neighbor)
---
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# Where is the data?

###[Althaus's GitHub:](https://github.com/calthaus/Ebola) https://github.com/calthaus/Ebola.red[*]

.footnote[.red[*] NOTE: See the `Intro to R` tutorial if you don't know how to import `csv` files.]
---
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# That's it.
## Thanks