MA 450 - Fall 2021 (Instructor: Haider)
Test #3 [45 PTS]
Submit via Moodle no later than Friday Nov. 19 at 11:59pm
Show all steps in your derivation; all work must be your own; no collaboration
1(a) [6 PTS]. Consider the following model for spread of an infectious disease among subpopulations
that are susceptible S(t), infected I(t) and recovered R(t), where all constants are positive:
dS
dt
= −aSI (1)
dI
dt
= −bI + aSI (2)
dR
dt
= bI (3)
subject to the initial conditions
S(0) = S0, I(0) = I0, R(0) = 0 (4)
Indicate whether the following statements are TRUE or FALSE.
(i) The sum of the population of susceptible and infected people decreases over time.
(ii) The population of infected people decreases via interactions with susceptible people.
(iii) The population of susceptible people decreases via interactions with infected people.
(iv) The population of infected people increases via interactions with susceptible people.
(v) The population of recovered people increases via interactions with susceptible people.
(vi) If I0 = 0, then I(t) = 0 for all time.
1(b) Consider the reversible reaction:
3A+B ↔ C
(i) [3 PTS]. Denoting the forward reaction rate as r1 and the reverse reaction rate as r−1, determine
expressions for r1 and r−1 in terms of the concentrations A,B,C and the associated rate constants k1
and k−1.
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(ii) [4 PTS]. Write an initial value problem that governs the time evolution of concentrations of A(t), B(t)
and C(t)
(iii) [2 PTS]. Find the largest set of independent conservation laws for the system in (b), i.e. no subset
of the conservation laws that you identify should be linearly dependent.
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2. [15 PTS] Michaelis-Menten kinetics provide a model for evolution of concentrations of a substrate
S(t), an enzyme catalyzing the reaction E(t), an intermediate complex C(t) and a product P (t). This
system is represented by the reaction equations:
S + E ↔ C
C → P + E
Apply the Law of Mass Action to the system of reactions above to write an initial value problem that
models Michaelis-Menten reaction kinetics when the system initially contains only the substrate and the
enzyme. By identifying independent conservation laws satisfied by the species, reduce the model to a
system of two ODEs in the variables E(t) and P (t).
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3. Consider using asymptotic series expansions to construct a composite solution to the differential
equation:
y′′ + 2y′ + y2 = 0, 0 < x < 1, where: << 1 (5)
subject to the boundary conditions:
y(0) = 0, y(1) = 1 (6)
(a) [4 PTS] Determine the outer solution of (5), assuming that the boundary layer is located at x = 0.
(b) [8 PTS] By introducing a boundary layer coordinate and balancing, determine a boundary layer
solution. Show all cases in your balancing argument. Enforce the appropriate boundary conditions for
the boundary layer solution.
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(c) [3 PTS] Using your solutions in (a) and (b) perform matching and construct a composite expansion
that approximates the solution of (5)-(6).
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