SOLVE THIS IN MATLAB!!!!

Consider the following simplified scenario of the Covid$-19$ pandemic. We divide the entire population into

three classes:

Susceptible $($S$)$

Infected $($I$)$

Recovered$/$Immune $($R$)$

The susceptible class has either not been exposed to the virus or has not been given the vaccine. The

infected class will recover eventually but can infect others in the mean time. The recovered class has either

had either been exposed to the virus and has recovered or has been successfully vaccinated. Lets suppose

in this scenario that we do not consider the otherwise grim reality of this pandemic. We observe that it is

difficult to change human behavior$/$interaction but suppose we can produce vaccinations. So$,$ we study the

effect of vaccination behavior of the outbreak.

Suppose today is day zero and we capture the proportion of the population in each of the classes $S,$I,$R$ in

a column vector

$x_0=\left[\begin{array}{c}0\mathrm{.}9\\ 0\mathrm{.}09\\ 0\mathrm{.}01\end{array}\right]$

It is observed that each day $1/1,000$ of the infected individuals recover. Each day, $1$ out of every $200$ susceptible

individuals becomes infected. Because of mutations in the virus, $1/10,000$ of the recovered individuals become

susceptible again. Through vaccination, we are able to move some fraction of the individuals in $S$ directly to

$R$; call this fraction $p$ where $0p1.$ Find a matrix $M$ such that

$x_1=M{x}_0$

describes the change in the population make up from day $0$ to day $1.$ This is the same matrix that describes

the transition of the population from day $k$ to day $k+1.$ Note that if some number of individuals move from

one class to another, you have to remove them from the class they were in originally $($the columns of your

matrixM should all sum to $1$

THIS IS MY CURRENT CODE ONLY A$5$ IS CORRECT

$\%$ Given parameters

recovery$\_$rate $=1/1000$;

infection$\_$rate $=1/200$;

reinfection$\_$rate $=1/10000$;

$\%$ Scenario $1$: p $=0$

p $=0$;

$\%$ Transition matrix

M $=[$

$1-$ infection$\_$rate, $0,$ reinfection$\_$rate $+$ p;

infection$\_$rate, $1-$ recovery$\_$rate, $0$;

$0,$ recovery$\_$rate, $1-$ reinfection$\_$rate $-$ p

$]$;

$\%$ Save the matrix as A$5$

A$5=$ M;

$\%$ Initialize the state vector

x$0=[0\mathrm{.}9$; $0\mathrm{.}09$; $0\mathrm{.}01]$;

$\%$ Simulation for steady state

for day $=1$:$100000$

x$0=$ M $*$ x$0$;

if day $==1$

D$0=$ day;

F$0=$ x$0(2)$; $\%$ Fraction of population infected

elseif abs$($x$0(2)-$ F$0)<mtext\; id="EditorElement\_478417"></mtext><mn\; id="EditorElement\_478418">1</mn>$e$-8$

break;

end

end

F$0=$max$(0,$min$(1,$F$0))$;

$\%$ Save the row vector A$6=[$D$0,$ F$0]$

A$6=[$D$0,$ F$0]$;

$\%$ Scenario $2$: p $=2/1000$

p$1=2/1000$;

$\%$ Update the transition matrix

M $=[$

$1-$ infection$\_$rate $-$ p$1,0,$ reinfection$\_$rate $+$ p$1$;

infection$\_$rate, $1-$ recovery$\_$rate, $0$;

p$1,$ recovery$\_$rate, $1-$ reinfection$\_$rate $-$ p$1$

$]$;

$\%$ Save the matrix as A$7$

A$7=$ M;

$\%$ Initialize the state vector

x$0=[0\mathrm{.}9$; $0\mathrm{.}09$; $0\mathrm{.}01]$;

$\%$ Simulation for steady state

for day $=1$:$100000$

x$0=$ M $*$ x$0$;

if day $==1$

D$1=$ day;

F$1=$ x$0(2)$; $\%$ Fraction of population infected

elseif abs$($x$0(2)-$ F$1)<mtext\; id="EditorElement\_479024"></mtext><mn\; id="EditorElement\_479025">1</mn>$e$-8$

break;

end

end

$\%$ Save the row vector A$8=[$D$1,$ F$1]$

A$8=[$D$1,$ F$1]$;