---
title: "Linear Chain System (Cao et al., 2004)"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Linear Chain System (Cao et al., 2004)}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

<!-- github markdown built using 
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```{r, setseed, echo=FALSE}
set.seed(1)
knitr::opts_chunk$set(fig.width = 8, fig.height = 6)
```

The Linear Chain System consists of M chain reactions with M+1 species as follows:

```
  S_1 --c1--> S_2
  S_2 --c2--> S_3
       ...
  S_M --cM--> S_(M+1)
```

Load package
```{r}
library(GillespieSSA)
```

Define parameters
```{r}
parms <- c(c = 1)                # Rate parameter
M <- 50                          # Number of chain reactions
simName <- "Linear Chain System" # Simulation name
tf <- 5                          # Final time
```

Define initial state vector
```{r}
x0 <- c(1000, rep(0, M)) 
names(x0) <- paste0("x", seq_len(M+1))
```

Define state-change matrix
```{r}
nu <- matrix(rep(0, M * (M+1)), ncol = M)
nu[cbind(seq_len(M), seq_len(M))] <- -1
nu[cbind(seq_len(M)+1, seq_len(M))] <- 1
```

Define propensity functions
```{r}
a <- paste0("c*x", seq_len(M))
```

Run simulations with the Direct method
```{r direct}
set.seed(1)
out <- ssa(
  x0 = x0,
  a = a,
  nu = nu,
  parms = parms,
  tf = tf,
  method = ssa.d(),
  simName = simName,
  verbose = FALSE,
  consoleInterval = 1
) 
ssa.plot(out, show.title = TRUE, show.legend = FALSE)
```

Run simulations with the Explict tau-leap method
```{r etl}
set.seed(1)
out <- ssa(
  x0 = x0,
  a = a,
  nu = nu,
  parms = parms,
  tf = tf,
  method = ssa.etl(tau = .1),
  simName = simName,
  verbose = FALSE,
  consoleInterval = 1
) 
ssa.plot(out, show.title = TRUE, show.legend = FALSE)
```

Run simulations with the Binomial tau-leap method
```{r btl}
set.seed(1)
out <- ssa(
  x0 = x0,
  a = a,
  nu = nu,
  parms = parms,
  tf = tf,
  method = ssa.btl(f = 50),
  simName = simName,
  verbose = FALSE,
  consoleInterval = 1
) 
ssa.plot(out, show.title = TRUE, show.legend = FALSE)
```

Run simulations with the Optimized tau-leap method
```{r otl}
set.seed(1)
out <- ssa(
  x0 = x0,
  a = a,
  nu = nu,
  parms = parms,
  tf = tf,
  method = ssa.otl(),
  simName = simName,
  verbose = FALSE,
  consoleInterval = 1
) 
ssa.plot(out, show.title = TRUE, show.legend = FALSE)
```
