---
title: "transition to multiple states"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{transition to multiple states}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r message=FALSE}
library(deSolve)
library(denim)
```

## 1. Transition to multiple states in denim

Modelers may encounter many situations where individuals can transition from one compartment to multiple others, such as the SIRD model. There are 2 main approaches to handle this:

-   Model transitions as competing risks

-   Model as multinomial transition (outgoing population is split by a fixed proportion to transition to new compartments).

`denim` allows the users to model both situations, while requiring minimal change to the code base to switch between 2 methods of modeling.

## 2. Multinomial in denim

### 2.1. Model definition

To specify the proportion that goes to each outgoing compartment, define the transition using the following syntax in model definition:

`proportion * from ->  to = [transition]`

Note that the `proportion` here is the proportion of `compartment` that will end up in `out_compartment` [at equilibrium]{.underline}.

***Example:*** An SIRD model where 90% of infected can recover while the remaining 10% will die.

![](./images/denim_website-mult-trans-multinomial.png)

```{r}
transitions <- denim_dsl({
  S -> I = beta * S * (I / N) 
  0.9 * I -> R = d_gamma(1/3, 2)
  0.1 * I -> D = d_exponential(0.1)
})
```

```{=html}
<details>
  <summary>
    Mathematical formulation for the model
  </summary>
```
$$
\begin{cases}
dS = -\beta S \frac{I}{N} \\
dIR_1 = 0.9 *\beta S \frac{I}{N} -\frac{1}{3}IR_1 \\
dIR_2 = \frac{1}{3}IR_1 - \frac{1}{3}IR_2 \\
dID = 0.1*\beta S \frac{I}{N} - 0.1*ID \\
dR = \frac{1}{3}IR_2 \\
dD = 0.1*ID 
\end{cases}
$$

Where:

-   $IR_1$ and $IR_2$ are the sub-compartments of $I$ that will transition to $R$. It is divided into 2 sub-compartments to model Erlang distributed time for $I \rightarrow R$ transition.

-   $ID$ is the sub-compartment of $I$ that will transition to $R$.

</details>

#### Proportion normalization

`denim` will automatically normalize the proportions if they don't sum up to 1.

***Example 2:*** The following model definition is equivalent to the one in ***Example 1***

```{r}
transitions <- denim_dsl({
  S -> I = beta * S * (I / N)
  36 * I -> R = d_gamma(1/3, 2)
  4 * I -> D = d_exponential(0.1)
})
```

### 2.2. Example model

To further demonstrate the implementation of multinomial in `denim`, we provide the equivalent model implemented in `deSolve` and compare the output of 2 implementations.

#### Model definition in denim

```{r}
# model in denim
transitions <- denim_dsl({
  S -> I = beta * S * (I / N) 
  0.9 * I -> R = d_gamma(1/3, 2)
  0.1 * I -> D = d_exponential(0.1)
})

denimInitialValues <- c(
  S = 999, 
  I = 1, 
  R = 0,
  D = 0
)
```

#### Equivalent model definition in deSolve

```{r}
# model in deSolve
transition_func <- function(t, state, param){
  with(as.list( c(state, param) ), {

      dS = - beta * S * (IR1 + IR2 + ID)/N
      # apply linear chain trick for I -> R transition
      # 0.9 * to specify prop of I that goes to I->R transition
      dIR1 = 0.9 * beta * S * (IR1 + IR2 + ID)/N - rate*IR1
      dIR2 = rate*IR1 - rate*IR2
      dR = rate*IR2 
      
      # handle I -> D transition
      # 0.1 * to specify prop of I that goes to I->D transition
      dID = 0.1 * beta * S * (IR1 + IR2 + ID)/N - exp_rate*ID
      dD = exp_rate*ID
      list(c(dS, dIR1, dIR2, dID, dR, dD))
  })
}

desolveInitialValues <- c(
  S = 999, 
  # note that internally, denim also allocate initial value based on specified proportion
  IR1 = 0.9,
  IR2 = 0,
  ID = 0.1, 
  R = 0,
  D = 0
)
```

#### Run simulation and compare

<details>

<summary>Model settings</summary>

```{r}
parameters <- c(
  beta = 0.2,
  N = 1000,
  rate = 1/3,
  exp_rate = 0.1
)

simulationDuration <- 200
timeStep <- 0.05
```

</details>

<details>

<summary>Run model</summary>

```{r}
# --- run denim model ---- 
mod <- sim(transitions = transitions, 
           initialValues = denimInitialValues, 
           parameters = parameters, 
           simulationDuration = simulationDuration, 
           timeStep = timeStep)

# run deSolve model
times <- seq(0, simulationDuration)
ode_mod <- ode(y = desolveInitialValues, times = times, parms = parameters, func = transition_func) 
ode_mod <- as.data.frame(ode_mod)
ode_mod$I<- rowSums(ode_mod[, c("IR1", "IR2", "ID")])
```

</details>

Output Comparison

```{r echo=FALSE}
# ---- Plot S compartment
plot(x = mod$Time, y = mod$S,xlab = "Time", ylab = "Count", main="S compartment",
     col = "#4876ff", type="l", lwd=3)
lines(ode_mod$time, ode_mod$S, lwd=3, lty=3)
legend(x = 150, y = 850,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))

# ---- Plot I compartment
plot(x = mod$Time, y = mod$I, xlab = "Time", ylab = "Count", main="I compartment",
      col = "#4876ff", type="l", lwd=2)
lines(ode_mod$time, ode_mod$I, lwd=3, lty=3)
legend(x = 150, y = 25,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))

# ---- Plot R compartment
plot(x = mod$Time, y = mod$R, xlab = "Time", ylab = "Count", main="R compartment",
      col = "#4876ff", type="l", lwd=2)
lines(ode_mod$time, ode_mod$R, lwd=3, lty=3)
legend(x = 150, y = 250,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))

# ---- Plot D compartment
plot(x = mod$Time, y = mod$D, xlab = "Time", ylab = "Count", main="D compartment",
     col = "#4876ff", type="l", lwd=2)
lines(ode_mod$time, ode_mod$D, lwd=3, lty=3)
legend(x = 150, y = 25,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))
```

## 3. Competing risks in denim

### 3.1. Model definition

When there are multiple transitions from one compartment and no proportions are specified, denim will automatically treat these transitions as competing risks.

***Example:*** the following model definition will treat `I->R` and `I->D` as competing risks.

![](./images/denim_website-mult-trans-competing-risks.png)

```{r}
transitions <- denim_dsl({
  S -> I = beta * S * (I / N)
  I -> R = d_gamma(rate = 1/3, shape = 2)
  I -> D = d_gamma(rate = 1/4, shape = 2)
})
```

### 3.2. Example model

To further demonstrate the implementation of competing risks in `denim`, we provide the equivalent model implemented in `deSolve` and compare the output of 2 implementations.

#### Model definition in denim

```{r}
transitions <- denim_dsl({
  S -> I = beta * S * (I / N)
  I -> R = d_gamma(rate = 1/3, shape = 2)
  I -> D = d_gamma(rate = 1/4, shape = 2)
})

denimInitialValues <- c(
  S = 950, 
  I = 50, 
  R = 0, 
  D = 0
)
```

#### Equivalent model definition in deSolve

```{r}
transition_func <- function(t, state, param){
  with(as.list( c(state, param) ), {
    dS = - beta * S * (I1 + I2 + IR + ID)/N

    dI1 = beta * S * (I1 + I2 + IR + ID)/N - (rate+d_rate)*I1

    dIR = rate*I1 - (d_rate + rate)*IR
    dID = d_rate*I1 - (d_rate + rate)*ID

    dI2 = d_rate*IR + rate*ID - (d_rate + rate)*I2

    dR = rate*IR + rate*I2
    dD = d_rate*ID + d_rate*I2

    list(c(dS, dI1, dIR, dID, dI2, dR, dD))
  })
}

desolveInitialValues <- c(
  S = 950, 
  I1 = 50,
  IR = 0,
  ID = 0,
  I2 = 0,
  R = 0,
  D = 0
)
```

#### Run simulation and compare

<details>

<summary>Model settings</summary>

```{r}
parameters <- c(
  beta = 0.2,
  N = 1000,
  rate = 1/3,
  d_rate = 1/4
)

simulationDuration <- 50
timeStep <- 0.05
```

</details>

<details>

<summary>Run model</summary>

```{r}
# run denim model
mod <- sim(transitions = transitions, 
           initialValues = denimInitialValues, 
           parameters = parameters, 
           simulationDuration = simulationDuration, 
           timeStep = timeStep)

# run deSolve model
times <- seq(0, simulationDuration)
ode_mod <- ode(y = desolveInitialValues, times = times, parms = parameters, func = transition_func) 
ode_mod <- as.data.frame(ode_mod)
ode_mod$I <- rowSums(ode_mod[,c("I1", "ID", "IR", "I2")])
```

</details>

```{r echo=FALSE}
# ---- Plot S compartment
plot(x = mod$Time, y = mod$S,xlab = "Time", ylab = "Count", main="S compartment",
     col = "#4876ff", type="l", lwd=3)
lines(ode_mod$time, ode_mod$S, lwd=3, lty=3)
legend(x = 30, y = 925,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))

# ---- Plot I compartment
plot(x = mod$Time, y = mod$I, xlab = "Time", ylab = "Count", main="I compartment",
      col = "#4876ff", type="l", lwd=2)
lines(ode_mod$time, ode_mod$I, lwd=3, lty=3)
legend(x = 30, y = 50,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))

# ---- Plot R compartment
plot(x = mod$Time, y = mod$R, xlab = "Time", ylab = "Count", main="R compartment",
      col = "#4876ff", type="l", lwd=2)
lines(ode_mod$time, ode_mod$R, lwd=3, lty=3)
legend(x = 30, y = 80,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))

# ---- Plot D compartment
plot(x = mod$Time, y = mod$D, xlab = "Time", ylab = "Count", main="D compartment",
     col = "#4876ff", type="l", lwd=2)
lines(ode_mod$time, ode_mod$D, lwd=3, lty=3)
legend(x = 30, y = 60,legend=c("denim", "deSolve"), col = c("#4876ff", "black"), lty=c(1,3))
```
