differential equation project

Names:

MA 226 Project 2

Due April 25

Consier the classical SIR model for the spread of an epidemic. A population is divided into three groups - the

susceptible, infected, and recovered (resistant in lecture) individuals. For a particular strain of flu, let’s define the

variables:

• S(t) is the fraction of the population at time t that can catch the disease.

• I(t) is the fraction of the population at time t that has the disease and can spread it to the susceptibles.

• R(t) is the fraction of the population at time t that has recovered from the disease and can not catch it again.

and state the assumptions:

• Once a person has had this disease, their immune system prevents them from catching it again.

• The flu spreads fairly quickly, so time is in days. Also, there is a low enough mortality rate that we can assume the total population is unchanged at 1, meaning S + I + R = 1 for all time. The only changes

between the 3 parts of the population is due to the flu.

• The rate that susceptible and infected people interact is proportional both the number of susceptibles and infecteds.

• Some fraction of these interactions lead to a susceptible becoming infected.

• Infected individuals recover at a rate proportional to the number of infecteds.

Putting all of this together yields the model

dS

dt = �↵SI

dI

dt = ↵SI � �I

dR

dt = �I

If S + I + R captures the entire population, then we need only keep track of S(t) and I(t), and we can consider

only the planar system

dS

dt = �↵SI

dI

dt = ↵SI � �I

9) There was an outbreak of the Coda virus in Essen, and it spilled over into neighboring cities. The Coda virus is

unique in that it drives the infecteds to work together to increase the number of infecteds. We can modify the SIR

model to include the assumption that infecteds actively infect susceptibles by replacing I with p I in the

interaction term. We obtain the system

dS

dt = �↵S

p I

dI

dt = ↵S

p I � �I

a) Calculate the equilibrium points of this model.

b) Find the region of the phase plane where dI/dt > 0.

c) Coda victim = zombie. If someone infected with the Coda virus is a zombie, what is a recovered in this

context? What does the term ��I mean, contextually? d) Fix ↵ = 0.2 and � = 0.1 and use technology to sketch the phase portrait. What does the model predict for

the spread of zombies? Do di↵erent initial conditions result in di↵erent qualitative outcomes?

10) The military has responded to the zombie threat and has distributed grenade belts to the susceptibles (i.e.,

surviving humans). This means that susceptibles destroy as many zombies as they can. We can assume that

zombies become “recovereds” at a rate proportional to the size of the surviving human population. We obtain

dS

dt = �↵S

p I

dI

dt = ↵S

p I � �S

a) Calculate the equilibrium points of this model.

b) Find the region of the phase plane where dI/dt > 0.

c) If the surviving humans had better equipment or better teamwork, how would you adjust the model?

d) Fix ↵ = 0.2 and � = 0.1 and use technology to sketch the phase portrait. What does the model predict for

the spread of zombies? Do di↵erent initial conditions result in di↵erent qualitative outcomes?

11) Take your favorite zombie film, television show, or literature. Select the zombie model and parameter values

that fit your choice, and pick an initial condition that fits your choice. State the assumptions that informed your

choices. Use technology to sketch the phase portrait and solution curves, and explain what is happening in the

context of your choice.

Think of it as a technical plot summary for an excerpt from your show/movie/book. Try to have fun with this.

If you have never seen a zombie film, you’ll want to pick parameter values in the context of human and zombie

behavior. For example, humans make the worst decisions in The Walking Dead, so ↵ may be high. Milla Jovovich

murders zombies at an absurd Legolas-level rate in Resident Evil, so � may be high. The rabid zombies in 28

Days Later are super fast, so ↵ may be high, or if you wanted to experiment, the interaction term could be SI1/3.

Will Smith found a cure for zombies or vampires or whatever they were in that mess of an adaptation, so perhaps

recovereds rejoin susceptibles.