NAME:
MATH 5110
Midterm
Do all ve problems (20 points each). Write readable answers on the test, but feel free to use or hand
in additional paper if necessary. Open book and notes, no calculators, no cheating.
1. A computer virus has been rampaging through a network. Each infected computer sends out r infected
emails per week. Of these, a fraction p have a chance of infecting the target. Only uninfected targets
can be infected. A fraction of computers are cleared of the virus each week. Let V represent the
number of infected computers in week t, and N be the total number of computers (a constant). A
discrete-time dynamical system for V is
t
t
Vt+1 = rpVt (1 ?
Vt
) ? Vt:
N
a. Explain the terms in this system.
b. Find the equilibria and their stability. What bifurcations occur as a function of p?
2. The endish designers of the computer virus have included a feature to make some viruses sleepers.
Ignoring the complexities in problem 1, suppose the following happens each week. Every active virus
makes r active ones and r sleepers (all succeed in infecting their targets) and then self-destructs. A
fraction of the sleepers remain asleep and the rest become active (none are cleared).
a. Write a matrix equation describing this population of viruses.
b. What is the condition on r and for the virus to spread? Does it make sense?
3. Problem 1 ignores the fact that the number of computers changes over time. Extend that problem to
include the assumption that the number of computers (N ) increases by S (1 ? NV 0 ) computers each
t
t
week.
a. Write a two-dimensional dynamical system describing these assumptions.
b. Find the equilibrium. Why is there only one?
c. Show that some parameter values exist for which the equilibrium is unstable.
d. (Extra credit) Draw a bifurcation diagram as a function of S .
4. Problem 1 treats all computers as being equally susceptible to attack.
a. Suppose that computers dier due to the level P of paranoia of the owner. The probability of
infection is 1 +1P 2 and the probability density function for paranoia is g(P ) = 2P for 0 P 1.
What is the average probability of a successful attack?
b. Assume that computers that had been previously attacked are less susceptible to future attack
due to a temporary increase in paranoia. Sketch a model that extends problem 1 to include
this phenomenon. For simplicity, you might want to include only two levels of paranoia, such
as P = 0 and P = 1.
5. Consider again the situation in problem 1, but suppose that the probability p of successful attack is
given by p = 1 +1P 2 (as in problem 4). All members of the population must choose the same level
of paranoia P . and work together to nd the optimal P . For simplicity, set = 0 and r = 2.
a. Find the positive equilibrium V as a function of P . What level of paranoia is required to
eliminate the virus?
b. Suppose the payo to society is (0:5V + (N ? V ))(1 ? P ). Explain this functional form.
c. Find the level of paranoia that maximizes the payo to society. What lesson can be drawn from
this result?