1.3 Math Models

1.3 What/Why/How

What: A math model is a DE that describes something in engineering, physics, chemistry, …

Why: Much cheaper and easier to change than collecting data.

How: Depends on the application

Bio

Lots of different examples.  Lets look at one in detail.

Suppose we have a bunch of students trapped at an isolated college campus and someone brings a disease onto campus.  Lets try to describe how the disease might spread.

SIR Model

Categories

Susceptible:	 "Not Sick yet"

Infected: 	 "Sick"

Recovered: 	 "Better or Dead"

Behavior

Susceptible:	 "Not Sick yet"

Still going to class. Can catch the disease from someone sick.

Infected: 	 "Sick"

Still going to some classes.  Can give the disease to someone sick, gets better in a few days.

Recovered: 	 "Better or Dead"

Going to classes.  Has immunity can not get sick again.

ODE Model

Susceptible:	 "Not Sick yet"

s'[t]=-a s[t] i[t]
s[0]=

Infected: 	 "Sick"

Still going to some classes.  Can give the disease to someone sick gets better in a few days.

i'[t]=a s[t] i[t]-b i[t]
i[0]=

Recovered: 	 "Better or Dead"

r'[t]=b i[t]
r[0]=0

Lots of extra stuff you could add in.  Two real constants a and b.

ODE Simulation

Working out reasonable parameter values is OK.  here is an example plotwith a twelve day plague.

Clear[a,b,Po, s,r,i,t,SIRModel]
SIRModel[a_,b_,Pop_]:={
s'[t]== - a s[t] i[t],
i'[t]==a s[t] i[t] - b i[t],
r'[t]==b i[t],
s[0]==Pop - 1,
i[0]==1,
r[0]==0}

TMax=12
SIRSol={s[t],r[t],i[t]}/.NDSolve[
SIRModel[0.001,0.1,5000],
{s[t],r[t],i[t]},
{t, 0, TMax}
][[1]]
Plot[
SIRSol,
{t, 0, TMax},
PlotStyle->{Red, Green, Blue}
]