Disease dynamics

class: center, middle, inverse, title-slide

.title[
# Disease dynamics
]
.author[
### Christopher Moore
]

---

# Diversity of disease models

There are many different models of disease dynamics, given the *enormous* diversity of pathogen, vector, and host life histories. (Notice that we are only considering the dynamics of the host and not the pathogen.)

As some examples:

I. Susceptibles become infected, and remain infected for their lives

II. Susceptibles become infected, fight off the disease, and again become susceptible

III. Susceptibles become infected, fight off the disease, and become resistant to subsequent infection

IV. Some diseases include a latent period before making the infected infectious

V. Some diseases can exist post-infected "carriers" that are no longer harmed, but can be indefinitely infectious

VI. Some diseases are maternally inherited

VII. Some diseases include vectors

---
# Diversity of disease models

As some examples:

I. Susceptibles become infected, and remain infected for their lives (**SI model**)

II. Susceptibles become infected, fight off the disease, and again become susceptible (**SIS model**)

III. Susceptibles become infected, fight off the disease, and become resistant to subsequent infection (**SIR model**)

IV. Some diseases include a latent period before making the infected infectious (**SEIR model**)

V. Some diseases can exist post-infected "carriers" that are no longer harmed, but can be indefinitely infectious (**Carrier state model**)

VI. Some diseases are maternally inherited (**MSI, MSIR, etc. model**)

VII. Some diseases include vectors

---
# Diversity of disease models

As some examples:

I. <font color="red"> Susceptibles become infected, and remain infected for their lives (**SI model**)</font>

II. Susceptibles become infected, fight off the disease, and again become susceptible (**SIS model**)

III. <font color="red"> Susceptibles become infected, fight off the disease, and become resistant to subsequent infection (**SIR model**)</font>

IV. <font color="red">Some diseases include a latent period before making the infected infectious (**SEIR model**)</font>

V. Some diseases can exist post-infected "carriers" that are no longer harmed, but can be indefinitely infectious (**Carrier state model**)

VI. Some diseases are maternally inherited (**MSI, MSIR, etc. model**)

VII. Some diseases include vectors

---
# Susceptible-infected (SI) model

The simplest disease model includes 2 state variables to represent a **s**usceptible portion of the population and an **i**nfected portion of the population.

First, let's assume that the host popualtion is *constant* over time. We can generally write down a system of equations to describe the dynamics:

`$$\begin{aligned}
\frac{\mathrm{d}S}{\mathrm{d}t} &= f_1(S, I) \\
\frac{\mathrm{d}I}{\mathrm{d}t} &= f_2(S, I)
\end{aligned}$$`

But also describe the simplest form with a transmission coefficient representing the probability that an infected contacts a susceptible and be transmitted:

`$$\begin{aligned}
\frac{\mathrm{d}S}{\mathrm{d}t} &= -\beta SI \\
\frac{\mathrm{d}I}{\mathrm{d}t} &= \beta SI
\end{aligned}$$`

---
# Susceptible-infected (SI) model

Let's exame the effect of transmission coefficient, `$\beta$` on disease dynaimcs. Graphically solve the model with different values of `$\beta$` and plot your results of infecteds as a times series with different values of `$\beta$` on the graph.

`$$\begin{aligned}
\frac{\mathrm{d}S}{\mathrm{d}t} &= -\beta SI \\
\frac{\mathrm{d}I}{\mathrm{d}t} &= \beta SI
\end{aligned}$$`

---
# Susceptible-infected (SI) model

```r
SI <- function(t, y, parameters) {
	S <- y[1]
	I <- y[2]
	beta <- parameters[1]
	dS <- -beta*S*I
	dI <- beta*S*I
	return(list(c(dS, dI)))
}
SI.init <- c(S = 0.99, I = 0.01)
SI.times <- seq(from = 0, to = 25, by = 0.1)
betas <- seq(from = 0, to = 1, by = 0.05)
cols <- colorRampPalette(c("red", "blue"))(length(betas))
for (i in 1:length(betas)){
  SI.parms <- c(beta = betas[i])
  SI.out <- ode(y = SI.init, times = SI.times, func = SI, parms = SI.parms)
  if (i == 1) {
      plot(x = SI.out[,1], y = SI.out[,3], ylim = c(0, 1), type = "l", col = cols[i])	
    } else {
    	lines(x = SI.out[,1], y = SI.out[,3], col = cols[i])
    }
}
```

---
# Susceptible-infected (SI) model
![](Wk11_Disease_files/figure-html/SITimeSeries-1.png)

---
# Susceptible-infected (SI) model

Notice that we are working in *proportion of a constant population*. This means `$S$` can be expressed in terms of `$I$` and vice versa becuase `$N = S + I$`, AND we are setting `$N = 1$`. This means that `$1 = S + I$`.

In terms of our equation, that means we can substitute for `$S$` or `$I$` and solve. This is done as eq. 1 in the book that shows that
`$$\frac{\mathrm{d}I}{\mathrm{d}t} = \beta SI$$`
can be written as
`$$\frac{\mathrm{d}I}{\mathrm{d}t} = \beta I(1 - I)$$`

That's logistic!

---
class: inverse, middle, center

![oprah](https://i.imgflip.com/288hiy.jpg)

---
# Susceptible-infected (SI) model
With vital rates, where `$\mu$` is the birth rate and `$\nu$` is the death rate.

`$$\begin{aligned}
\frac{\mathrm{d}S}{\mathrm{d}t} &= \mu N -\frac{\beta SI}{N} - \nu S \\
\frac{\mathrm{d}I}{\mathrm{d}t} &= \frac{\beta SI}{N}- \nu I
\end{aligned}$$`

Notice that the death rate is the same for susceptibles and infecteds. We could also include an infedted death rate. E.g.,

`$$\begin{aligned}
\frac{\mathrm{d}S}{\mathrm{d}t} &= \mu N -\frac{\beta SI}{N} - \nu S \\
\frac{\mathrm{d}I}{\mathrm{d}t} &= \frac{\beta SI}{N}- \gamma I
\end{aligned}$$`

---
# Susceptible-infected-recovered (SIR) model

We now add a state variable for a portion of the population who has **r**ecovered from being infected.

We also need to add a parameter for the rate of recovery; let's say `$v$`.

`$$\begin{aligned}
\frac{\mathrm{d}S}{\mathrm{d}t} &= -\beta SI \\
\frac{\mathrm{d}I}{\mathrm{d}t} &= \beta SI - vI \\
\frac{\mathrm{d}R}{\mathrm{d}t} &= vI
\end{aligned}$$`

---
# Susceptible-infected-recovered (SIR) model
Vandermeer and Goldberg describe the significance of the relationship between the two important parameters, recovery and transmission rates, `$v$` and `$\beta$`, respectively.

From the SI model, `$\frac{v}{\beta}$` was a critical ratio that determined whether the equilibrium for infecteds will be 0 or 1.

In the SIR model, the `$I$` derivative can be arranged algebraically where `$\frac{\beta}{v}$` acts as a threshold as to wether the infected portion will increase or decline. It's realted to `$R_0$`, a universal parameter known as the **the basic reproductive ratio** defined as *the average number of secondary infected cases  arising from an average primary case in an entirely susceptible population*. In other words, it's the rate of infection per recovery rate.

---
# Susceptible-infected-recovered (SIR) model

From the SI model, `$\frac{v}{\beta}$` was a critical ratio that determined whether the equilibrium for infecteds will be 0 or 1.

Create a time series for an SIR model. Pretend the suscetible population is Colby (roughly 2000 people on campus) and one person returns from winter break with a pathogen that is transmitted with a probability of 0.5. Create a time series to show how the infected Mules change over time for recovery rates both of 0.1 and 0.5. What and when are the peaks (maximum `$I$`) for each?

---

# Susceptible-infected-recovered (SIR) model
`$I$` is the red line and `$v = 0.1$`:
![](Wk11_Disease_files/figure-html/unnamed-chunk-2-1.png)

```r
max(SIR.out[,3]) # prop. I
## [1] 0.4782124
SIR.out[,1][which(SIR.out[,3] == max(SIR.out[,3]))] # Days
## [1] 22.9
```

---
# Susceptible-infected-recovered (SIR) model
`$I$` is the red line and `$v = 0.5$`:
![](Wk11_Disease_files/figure-html/unnamed-chunk-4-1.png)

```r
max(SIR.out[,3]) # prop. I
## [1] 5e-04
SIR.out[,1][which(SIR.out[,3] == max(SIR.out[,3]))] # Days
## [1] 0
```

---
# Susceptible-infected-recovered (SIR) model
### `$\frac{\beta}{v} > 1$`

![](Wk11_Disease_files/figure-html/unnamed-chunk-6-1.png)

---
# Susceptible-infected-recovered (SIR) model
### `$\frac{\beta}{v} < 1$`

![](Wk11_Disease_files/figure-html/unnamed-chunk-7-1.png)

---
# Susceptible-infected-recovered (SIR) model

---
# Nice model, Chris, but how'bout those data?