5_Dynamic simulation models.Rmd

---
title: 'Dynamic simulation models'
author: "Vikram Dayal and M Rahul"
output: pdf_document
---

Nonlinear dynamics can be complex, as we can see in Figure \ref{fig:sim1} of a simulation with the equation
$X_{t + 1} = X_{t} + (rX_{t}(1 - (X_{t}/K)))$, $r = 2.6, K = 100, X_{0} = 40$. 

```{r, echo = F, message = F, warning=FALSE}
library(tidyverse)
num <- 40
X <- numeric(num)
K <- 100
r <- 2.6
X[1] <- 40
for (i in 2:num) {
  X[i] <- X[i - 1] + ( r*X[i - 1]* (
    1 - ( X[i - 1] /K)) )
}
Xdat <- data.frame(X, time = 1:num)
```

```{r, echo = F, fig.width = 4, fig.height = 3, fig.cap = "\\label{fig:sim1}simulated logistic difference equation"}
gg1 <- ggplot(Xdat, aes(x = time, 
                 y = X)) +
  geom_line()
gg1
```

A key issue in environmental economics concerns biological invasions. Biologically invasive species include *pests and pathogens*, plants and animals, which invade and take over ecological systems, harming the production of ecosystem services^[see Perrings et al. 2002 and Dayal 2007]

Epidemiologists, ecologists, economists, and ecological economists build on `workhorse' models of infectious disease epidemiology^[see Stevens 2010.].


We now see how we can simulate a difference equation in R. Consider a simple difference equation:

$X_{t + 1} = X_{t} + 2$. 

Let $X_{1}$ = 3. 

Then,

$X_{2} = X_{1} + 2 = 5$

$X_{3} = X_{2} + 2 = 7$


So we can numerically simulate a difference equation by iteration. This means we can use a for loop.


```{r}
X <- numeric(3)
X[1] <- 3
for(i in 2:3){
  X[i] <- X[i - 1] + 2
}
X

```

We consider a simple SIR model in R. Here we will focus on qualitative insight. An SIR model consists of: 

  **Susceptible** Individuals who are not yet infected, but may be tomorrow (S). 
  **Infected** Individuals who are infected (I).
  **Resistant** Individuals who are resistant (R). 
  

  
Note that total individuals, N is given by $N = S + I + R$. People move through the compartments: $S \rightarrow I \rightarrow R$. The dynamics are easier to think of if we consider two periods: $newI_{2} = \beta I_{1} S_{1}$, $R_{2} = \gamma I_{1}$. 

We now set up for our simulation in R. 
```{r}
num <- 100
N <- 1e6
S <- numeric(num)
S[1] <- N
I <- numeric(num)
I[1] <- 10
nI <- numeric(num)
Beta <- 4.5e-07
Measures <- 1
R <- numeric(num)
Gamma <- 1/5
```

We can use a 'for loop'

```{r}
for(i in 2:num) {

nI[i] <- Beta*Measures * S[i -1] * I[i -1]

S[i] <- S[i - 1] - nI[i]

R[i] <- Gamma * I[i - 1]

I[i] <- I[i - 1] + nI[i] - R[i]

}
```

We collect the values. 

```{r, message = F, warning = F}
Time <- 1:num
SIR <- data.frame(S, I, nI, R, Time)
library(tidyverse)
```

We plot the results.


```{r, fig.width = 3, fig.height=2, fig.cap = "Proportion of infected"}
ggplot(SIR, aes(x = Time, y = I/N)) +
  geom_line()
```

We create a function that allows us to vary $\beta$. 

```{r}
SIRfunc <- function(Measures = 1) {
  num <- 100
N <- 1e6
S <- numeric(num)
S[1] <- N
I <- numeric(num)
I[1] <- 10
nI <- numeric(num)
Beta <- 4.5e-07
#Measures <- 1
R <- numeric(num)
Gamma <- 1/5

for(i in 2:num) {

nI[i] <- Beta*Measures * S[i -1] * I[i -1]

S[i] <- S[i - 1] - nI[i]

R[i] <- Gamma * I[i - 1]

I[i] <- I[i - 1] + nI[i] - R[i]

}

Time <- 1:num
SIR <- data.frame(S, I, nI, R, Time)

library(tidyverse)

ggplot(SIR, aes(x = Time, 
                    y = I/N))  +
 geom_line()

}
```

We use the function. 

```{r, fig.width = 3, fig.height = 2, fig.cap = "Proportion of infected, with beta = 0.8"}
SIRfunc(0.8)
```

The basic model can be elaborated in different ways. 

  * Make it more detailed, add E, so SEIR. Two patches, mobility between patches. 
  * Provide a tool (Shiny) so non-specialists can play with different settings of parameters and see resulting scenarios. 
  * Use a calibration approach, take parameters from literature etc.
  * Explore mathematical properties: stability, see how dynamics change with parameters
  * Use for detailed predictions
  * Fit to data, likelihood or Bayesian approaches. 



# References

Dayal V. 2007. Social diversity and ecological complexity: How an invasive tree could affect diverse agents in the land of the tiger. Environment and Development Economics 12(04):553-571

Perrings C, Williamson M, Barbier E B, Delfino D, Dalmazzone S, Shogren J, Simmons P, Watkinson A. Biological invasion risks and the public good: an economic perspective. 2002. Conservation ecology, Vol. 6, No. 1, 1, 06.200

Stevens M H. 2010. A primer of ecology with R. Springer