a$)$ Write a function in R to simulate epidemics using the discrete time SIR model in Section $2\mathrm{.}2\mathrm{.}2$ of the notes.

$$ The transition between compartments are described using the $$chain binomial$$ model:

I$*$ t$$ Bin$($St $,\backslash$pi $^($SI$))$

R$*$t $$ Bin$($It$,\backslash$pi $^($IR$))$

$$ The transmission probability is derived from a Poisson contact process and homogeneous mixing:

$\backslash$pi $($SI$)=1$exp$(\backslash$beta $($It$/$N$))$

$$ The removal probability assumes the length of time infectious is exponentially distributed: $\backslash$pi $($IR$)=1$exp$(\backslash$gamma $)$ The function should have inputs which include the population size $($N $),$ initial conditions $($S$0,$ I$0),$ model parameters $(\backslash$beta $,\backslash$gamma $),$ and the last time point at which to simulate new cases and removals $(\backslash$tau $).$

Output should include the numbers of individuals contained in the S$,$ I, R$,$ I$$ and R$$ states over time.

Include the function code in your writeup.

b$)$ Use your epidemic function to simulate an epidemic with initial conditions:

S$0=995$; I$0=5$; R$0=0($so$,$ population size, N $=1000)$

and model parameters:

$\backslash$beta $=0\mathrm{.}35$; $\backslash$gamma $=0\mathrm{.}1$

for $\backslash$tau $=60$ time points.

Produce plots of both the prevalence curve over time $($I$)$ and incidence curve over time $($I$).$

c$)$ We are now going to carry out a Monte Carlo simulation in order to estimate some character$-$ istic of the disease system described by this model. Specifically, we are going to look at the probability that a single infection results in a substantial epidemic, which we will define as more than $50$ out of $1000$ individuals being infected before the epidemic dies out.

We will use the same parameter values and population size, but change the initial conditions to:

S$0=999$; I$0=1$; R$0=0.$

You can either use your epidemic function, or adapt it if necessary, so that you can run multiple epidemics in order to estimate whether more than $50$ individuals are infected before the epidemic dies out.

Try to estimate this probability to $2$ decimal places,and provide evidence that your answer is correct to this degree of accuracy.