The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH3063: Nonlinear ODEs & Applications Semester 1, 2022
Web Page: http://www.maths.usyd.edu.au/u/UG/SM/MATH3063/
Lecturer: Nathan Duignan
Due Sun 08 May 2022, 11:59PM. To be submitted as a pdf online via gradescope on Canvas.
In lectures, the standard SIR model for epidemics was analysed through phase plane analysis. Ultimately,
the phase plane analysis allowed us to understand when an epidemic would occur for a disease, and hence, what
policy could be implemented to avoid this. The standard SIR model assumes that the rate of infection is simply
βIS, for some positive constant β. However, in general, the rate of infection may be given by g(I)S, for some
function g(I) satisfying g(0) = 0 and g(I) > 0 whenever I > 0. The function g is known as the “infection force”
of the disease. It essentially measures how probable it is to be infected when there are I people infected.
In the 1978 paper [1], Capasso and Serio proposed an infection force for the SIR model aimed at under-
standing psychological effects of a disease. The modification of the SIR model is given as,
S˙ = g(I)S
I˙ = g(I)S γI
R˙ = γI,
(1)
where g(I) = βIe αI , and α, β, γ are positive constants.
1. (a) Draw the compartmental model for (1).
(b) Plot the infection force g(I) for I ≥ 0 showing any maxima or minima. Use this plot to argue
why the model (1) proposed by Capasso and Serio does indeed factor in psychological effects of the
disease.
2. (a) Explain mathematically why N = S + I +R is a conserved quantity.
(b) Use the conservation of N to reduce the 3D system to a 2D system of ODEs in S, I. Find a scaling
of S, I, t to scaled variables x, y, τ , respectively, which brings this 2D system into the form
dx
dτ
= R0xye ay
dy
dτ
= R0xye
ay y,
with R0, a positive constants. Give an expression for R0, a in terms of the original constants
β,N, γ, α.
(c) If R(0) = 0, explain why the allowed initial conditions (x0, y0) must lie on the line x+ y = 1.
(d) Perform a phase plane analysis; find the fixed points, characterise their stability, find the nullclines,
sketch the phase portrait in the biologically relevant domain.
(e) From the phase plane analysis, show that an epidemic will occur if
eay0
1 y0 < R0, that is, the infected
population will grow to a peak before the disease goes extinct. Show that no epidemic occurs if
eay0
1 y0 > R0, that is, the infected population will simply decline from y0 until the disease goes
extinct.
3. (a) Compare the results of the phase plane analysis for the Capasso-Serio psychological model to the
standard SIR model analysed in lectures. In particular, compare whether the diseases will be en-
demic, the conditions for an epidemic, and when the expected peak will occur in the case of an
epidemic. Argue whether the differences and similarities are expected or not from a biological per-
spective.
(b) What human psychology could influence the value of the constant α How could the population be
influenced to ensure an epidemic is avoided
Copyright c 2022 The University of Sydney 1
References
[1] Capasso, V. & Serio, G., A Generalization of the Kermack-McKendrick Deterministic Epidemic Model,
Mathematical Biosciences, Volume 42, Issues 1-2, 1978.
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