Math 4600, Homework 3
1. Consider the SIR-model from class, with permanent immunity and recovery rate γ = 10
a. A city of 1500 individuals experiences an epidemic, resulting in a total of 1282 documented cases of
infection before the infection ceased. Assuming that the infection began with 1 infectious individual,
estimate the parameter β.
b. Assume that you counsel leaders of a city containing 1000 susceptible individuals. Determine whether
the infection will spread throughout this city, given your estimate of β in part (a) (explain why or
why not). If it does, how many individuals would have to be immunized to prevent an epidemic?
Why might this value of β not accurately describe disease transmission in your city of 1000?
c. (computing) Plot a simulation of the infection for the 1000 person case in the phase plane in R.
Next, separately, sketch the time course of the same solution by hand (be as specific as you can, e.g.
mark the maximum of I and steady state value of S correctly, even if you don’t know exactly at
what time the maximum occurs). Produce the same graph in the case where the minimum number
of immunizations have been made to avoid an epidemic (part b) . Compare the development of the
epidemic in the two cases (discuss the maximum number of infected people, duration, total population
affected).
2. Imagine an epidemic in a population with the following categories of people: susceptible (S), infected (I),
recovered (R), and healers (H). Assume that:
- healers cannot be infected
- there is permanent immunity
- the disease is spread on contact, the rate of transmission is α
- to recover from the disease an infected person needs to come in contact with a healer. Rate of contacts
with healers is β
- a healthy person (susceptible or recovered) can become a healer. This happens at the rates of γS for
susceptibles and γR for recovered people.
- deaths and births can be ignored.
a. Draw a state diagram describing interactions between these different classes of people.
b. Write a system of differential equations describibg the course of the epidemic
c. Reduce the system to 3 variables, find the steady states.
3. Below is an SIR model with births (in addition to deaths). S(t) is the number of susceptible individuals,
and I(t) is the number of the infected individuals. It is assumed that susceptible individuals can produce
an uninfected offspring with rate b, that they become infected when they come in contact with infected
people at rate β, and that the infected individuals die at rate k. This is described by the following
equations:
dS
=bS − βIS
dt
dI
=βIS − kI
dt
a. (computing) Solve this system in R with b = 3, β = 3, k = 3. Your code should be analogous to
what you used in the previous homework. Try initial conditions S(0) = 1, and I(0) equal to either .1,
.5 or 1.3. Look at the solutions in the phase plane. Where do you think the steady states are, what
types are they?
b. Find the steady states analytically
c. Describe in words your solutions from a) in terms of number of susceptible and infected individuals.
Why does it make sense from an epidemiological perspective?
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4. (computing)The goal of this exercise is to verify the validity of the Michaelis-Menten term in an enzyme
reaction. Suppose molecules S,E,C,P obey the following the reaction dynamics:
S+E C →P +E
a. Write down a system of 4 differential equations describing this reaction.
b. In R, numerically simulate this system with the parameter values e0 = 1, k− = 120, k+ = 100, k2 = 2,
and initial conditions S(0) = 100, C(0) = 0, E(0) = e0 , P (0) = 0. Simulate this system until S
gets close to 0. Produce a graph of the production rate of P ( dP
dt ) vs. S(t). To this plot, add the
Michaelis-Menten term vs. S. Where do the curves match? Where don’t they match? Why?
c. Compare the accuracy of the M.M. approximation when k− = 12, k+ = 10. Comment on why the
matching of the M.M. term differs.
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