Mathematical Biology
Homework Assignment 4
2024–25
Please submit solutions to the following two questions as
Homework Assignment 4
by 16:00pm on Monday, November 11, 2024.
I. Consider the fast-slow system
where ε is a small parameter.
(a) State the slow and fast systems, as well as the corresponding reduced and layer problems.
(b) Determine the critical manifold M0 for Equation (1), and verify its normal hy-perbolicity.
(c) Describe the reduced flow on the manifold M0, as well as the layer flow off it.
(d) Sketch the phase portrait of (1) in the singular limit, i.e. for ε = 0.
(e) Discuss the applicability of Fenichel’s theorems to Equation (1).
(f) Rewrite (1) as an equation for y(x), and find the exact solution under the initial condition that y(0) = 2/1.
II. Consider an SIR model in which a fraction θ of infectives is isolated in a perfectly quarantined class Q. Under the assumption that individuals make a contacts in unit time, of which a fraction I/(N − Q) are, infective, the governing system of equations is given by
with initial condition (S,I, Q, R)(0) = (S0,I0, 0, 0) for S0 + I0 = N constant. Here, S, I, Q, and R denote the populations of susceptible, infected, quarantined, and recovered individuals, respectively.
(a) Show that Equation (2) can be written in non-dimensionalised form. as
with (u, v, w, z)(0) = (u0, v0, 0, 0), where
as well as
(b) Explain why Equation (3) is naturally defined on the simplex
and why it hence suffices to consider the (u, v, w)-subsystem
only.
(c) Deduce that, by item (b), the equilibria for Equation (6) are located on the seg-ment of the positive u-axis.
(d) Show that the basic reproductive number R0 for (6) reads R0 = 1+δ/βu0, where u0 is again the (non-dimensionalised) initial population of susceptibles.
(e) Give an epidemiological interpretation of the effect of quarantine in the model.
(Hint: rewrite R0 from item (d) in terms of the original, dimensional parameters in (2), assuming that the population is fully susceptible initially, with u0 = 1.)