$40$ points$)$ Write a Matlab program which simulates the SZR model for initial condi$-$

tions

S$0=190,$ Z$0=10,$

and parameters

$\backslash$beta $=0\mathrm{.}01,\backslash$kappa $=0\mathrm{.}008,$

given in units of $($days$)1.$ The total simulation time should be $28$ days. Curves for

S$($t$),$ Z$($t$),$ and R$($t$)$ should all be shown on the same plot, and should be clearly labeled,

for example by using the legend command.

As part of a $.$zip file for this final project, please submit a file called szr$.$m for the main

program, and a file called szrf$.$m for the function which encodes the righthand side of

the SZR model, and which is called by the Matlab function ode$23$ in the main program.

b$)(20$ points$)$ It is shown in $[2]$ that if $\backslash$beta $>\backslash$kappa $,$ the zombies always win $($S $->0$ as t $->\backslash$infty $),$

for all initial conditions S$0$ and Z$0.$ On the other hand, if $\backslash$beta $<\backslash$kappa it is possible that either

the humans or the zombies could win, depending on the initial conditions.

Let$$s consider $\backslash$beta $=0\mathrm{.}008$ and $\backslash$kappa $=0\mathrm{.}01,$ and suppose that S$0+$ Z$0=200,$ where S$0$

could be as high as $199$ or as low as $1.$ Write a Matlab program which determines the

smallest integer value that S$0$ can take for which S$($t $=28)>1.$ As a hint, start with

S$0=1$ and Z$0=199,$ and run the simulation. Is S$($t $=28)>1?$ If not, try S$0=2$ and

Z$0=198.$ Is S$($t $=28)>1?$ If not, try S$0=3$ and Z$0=197,$ etc. Eventually you$$ll find

the smallest integer value of S$0$ for which S$($t $=28)>1$; your program should determine

and output this value.

As part of a $.$zip file for this final project, please submit your file szr$\_$ic$.$m which

solves this problem. This should call your function szrf from $($a$).$

c$)(40$ points$)$ While the results of the simulations of the SZR model might seem depressing,

particularly when $\backslash$beta $>\backslash$kappa $,$ there is another model called the Stochastic SZR $($Stoch$-$SZR$)$

model which, as we$$ll see, is more hopeful. Quoting $[2],$ this model suggests that $$even a

ferociously virulent zombie infestation might be killed early on by happy accident.$$ Note

that you can$$t use a built$-$in Matlab solver for this, so you$$ll have to write your own code

from scratch.

For the Stoch$-$SZR model, the variables S$,$ Z$,$ and R again denote the number of

individuals in the Susceptible, Zombie, and Removed groups, respectively, but they only

take non$-$negative integer values. There are two possible transitions:

Susceptible $+$ Zombie $\backslash$beta SZ

$->$ Zombie $+$ Zombie, $(1)$

Susceptible $+$ Zombie $\backslash$kappa SZ

$->$ Susceptible $+$ Removed. $(2)$

$2$