Murray J.D. Mathematical Biology: I. An Introduction
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306 9. Perturbed and Coupled Oscillators and Black Holes
and the system of equations, (9.45) and (9.46), becomes a partial differential equation
system.
The algebra in the rest of this section is also rather involved. The end result, namely,
equation (9.69) is required in the following Section 9.10, where an important result for
coupled oscillators is derived.
If we now substitute (9.50) with (9.51) into (9.45) and (9.46) and equate powers of
ε we get the following hierarchy of equations.
O(1) :
∂
0
A
1
∂t
−Φ(
0
θ
1
)
0
A
1
= kU(
0
θ
1
,
0
θ
2
) + λΦ(
0
θ
1
),
∂
0
θ
1
∂t
= 1,
∂
0
A
2
∂t
−Φ(
0
θ
2
)
0
A
2
= kU(
0
θ
2
,
0
θ
1
),
∂
0
θ
2
∂t
= 1.
(9.52)
O(ε) :
∂
1
A
1
∂t
−Φ(
0
θ
1
)
1
A
1
={Φ
(
0
θ
1
)
0
A
1
+k
∂U
∂θ
1
(
0
θ
1
,
0
θ
2
) +λφ
(
0
θ
1
)}
1
θ
1
+k
∂U
∂θ
2
(
0
θ
1
,
0
θ
2
)
1
θ
2
+Ψ
1
(
0
A,
0
θ
θ
θ) −
∂
0
A
1
∂τ
,
∂
1
θ
1
∂t
=[Ω(
0
θ
1
) − Γ(
0
θ
1
)]
0
A
1
+λγ (
0
θ
1
) + kr(
0
θ
1
)
+kV(
0
θ
1
,
0
θ
2
) −
∂
0
θ
1
∂τ
,
∂
1
A
2
∂t
−Φ(
0
θ
2
)
1
A
2
={Φ
(
0
θ
2
)
0
A
2
+k
∂U
∂θ
2
(
0
θ
2
,
0
θ
1
)}
1
θ
2
+k
∂U
∂θ
1
(
0
θ
2
,
0
θ
1
)
1
θ
1
+Ψ
2
(
0
A,
0
θ
θ
θ) −
∂
0
A
2
∂τ
,
∂
1
θ
2
∂t
=[Ω(
0
θ
2
) − Γ(
0
θ
2
)]
0
A
2
+kr(
0
θ
2
) +kV(
0
θ
2
,
0
θ
1
) −
∂
0
θ
2
∂τ
.
(9.53)
If we integrate the 2nd and 4th of (9.52) we get
0
θ
i
= t +ψ
i
(τ), i = 1, 2, (9.54)
where, at this stage the ψ
i
(t) are arbitrary functions of τ. If we now substitute these into
the 1st and 3rd equations of (9.52) we get
9.9 Singular Perturbation Analysis: Two-Time Expansion 307
∂
0
A
1
∂t
−Φ(t +ψ
1
)
0
A
1
= kU(t + ψ
1
, t + ψ
2
) +λφ(t + ψ
1
),
∂
0
A
2
∂t
−Φ(t +ψ
2
)
0
A
2
= kU(t + ψ
2
, t + ψ
1
).
(9.55)
To get the required solutions let us digress again, and consider the less cluttered
equations:
dx
ds
−Φ(s)x = f (s),
dy
ds
−Φ(s)y = U(s, s + χ),
(9.56)
where χ ≡ ψ
2
−ψ
1
. Remember that the functions Φ, φ and U are all T -periodic. The
complementary or homogeneous solution of each is
exp [v(s)],v(s) =
s
0
Φ(σ)dσ.
From (9.47) we have v(s)<0, which is necessary for the stability of the uncoupled
limit cycle oscillators. So, the complementary solutions will decay, eventually, to zero.
We can now show that each of (9.56) has a unique T -periodic solution. To do this
consider, say, the first of (9.56). The exact solution is
x(s) = x(0) exp
s
0
Φ(σ)dσ
+
s
0
exp
s
α
Φ(σ)dσ
φ(α)dα. (9.57)
The equation for x is unchanged if we replace s by s + T . Since the above solution for
x(s) is periodic
x(0) = x(T ) = x(0) exp
T
0
Φ(σ)dσ
+
T
0
exp
T
α
Φ(σ)dσ
φ(α)dα,
which is an equation for the inital value x(0), substitution of which into (9.57) gives
the unique T -periodic solution of the first of (9.56). Similarly the second of (9.56) has
a unique T -periodic solution. Denote these periodic solutions by
x = p(s), y = ρ(s,χ). (9.58)
In terms of these solutions (9.58) the general solutions of (9.55) are
0
A
1
= kρ(t + ψ
1
,χ)+ λp(t + ψ
1
) +h
1
(τ) exp [v(t +ψ
1
)],
0
A
2
= kρ(t + ψ
2
, −χ) + h
2
(τ) exp [v(t +ψ
2
)],
(9.59)
where the h
1
and h
2
are arbitrary functions of τ. If we now substitute these into the
θ-equations in (9.53) we get
308 9. Perturbed and Coupled Oscillators and Black Holes
∂
1
θ
1
∂t
={kρ(t +ψ
1
,χ)+ λp(t + ψ
1
)}{Ω(t + ψ
1
) −Γ(t + ψ
1
)}+λγ (t +ψ
1
)
+kr(t +ψ
1
) + kV(t +ψ
1
, t +ψ
2
) −
dψ
1
dτ
+h
1
(τ) exp [v(t +ψ
1
)]{Ω(t + ψ
1
) −Γ(t + ψ
1
)},
∂
1
θ
2
∂t
= kρ(t + ψ
2
, −χ){Ω(t +ψ
2
) −Γ(t + ψ
2
)}
+kr(t +ψ
2
) + kV(t +ψ
2
, t +ψ
1
)
−
dψ
2
dτ
+h
2
(τ) exp [v(t +ψ
2
)]{Ω(t + ψ
2
) −Γ(t + ψ
2
)}.
(9.60)
If f (t) is a T -periodic function we can write
f (t) = µ + ω(t), µ =
1
T
T
0
f (s) ds, (9.61)
where ω(t) is T -periodic and has zero mean. If we now use this fact in (9.60) we
get
∂
1
θ
1
∂t
= µ
1
(χ) +ω
1
(t,τ),
∂
1
θ
2
∂t
= µ
2
(χ) +ω
2
(t,τ),
(9.62)
where
µ
1
= H(χ) +λβ −
dψ
1
dτ
,
µ
2
= H(−χ)−
dψ
2
dτ
,
(9.63)
with
β =
1
T
T
0
{p(s)[Ω(s) − Γ(s)]+γ(s)}ds,
H(χ) =
1
T
T
0
{kρ(s,χ)[Ω(s) −Γ(s)]+kr(s) +kV(s, s + χ)}ds.
(9.64)
The functions ω
1
(t,τ) and ω
2
(t,τ) are made up of exponentially decaying terms and
periodic terms of zero mean, so on integrating (9.62) we get
1
θ
1
= µ
1
(χ)t + W
1
(t,τ),
1
θ
2
= µ
2
(χ)t + W
2
(t,τ),
(9.65)
9.9 Singular Perturbation Analysis: Two-Time Expansion 309
where W
1
and W
2
are bounded functions. If we now substitute these solutions into the
third of (9.53) and use (9.54) and (9.57), the equation for
1
A
2
becomes
∂
1
A
2
∂t
−Φ(t +ψ
2
)
1
A
2
={S
1
(t,τ)µ
1
+ S
2
(t,τ)µ
2
}t + B(t,τ),
S
1
= k
∂U
∂θ
1
(t + ψ
2
, t +ψ
1
),
S
2
= k
∂U
∂θ
2
(t + ψ
2
, t +ψ
1
) + kρ(t + ψ
2
, −χ)Φ
(t +ψ
2
),
(9.66)
where B(t,τ)is another function which consists of an exponentially decaying term and
a T -periodic part.
In the usual asymptotic singular perturbation way we now require the O(ε) part
of the amplitude, that is,
1
A
2
, and its time derivative ∂
1
A
2
/∂t, to be bounded for all
time—this ensures that the series solution (9.50) is uniformly valid for all time. From
(9.66) this requires that
{S
1
(t,τ)µ
1
+ S
2
(t,τ)µ
2
}t
must be bounded for all time. However, since S
1
and S
2
are T -periodic functions (and
so cannot tend to zero as t →∞) the only way we can get boundedness is if
S
1
(t,τ)µ
1
+ S
2
(t,τ)µ
2
≡ 0. (9.67)
In general S
1
and S
2
are two different periodic functions, so the only way to ensure
(9.67) is if µ
1
and µ
2
are both zero. That is, from (9.63), we require
µ
1
= H(χ) +λβ −
dψ
1
dτ
= 0,µ
2
= H(−χ) −
dψ
2
dτ
= 0
and so
dψ
1
dτ
= H(χ) + λβ,
dψ
2
dτ
= H(−χ). (9.68)
Recalling that χ = ψ
2
−ψ
1
, if we subtract the last two equations we get the following
single equation for χ,
dχ
dτ
= P(χ) − λβ,
where P(χ) = H(−χ) − H(χ), χ = ψ
2
−ψ
1
.
(9.69)
The dependent variable χ is the phase shift due to the coupling: the ordinary differential
equation (9.69) governs the time evolution of this phase shift. The derivation of this
equation is the main purpose of the singular perturbation analysis in Sections 9.8 and
9.9. It is also the equation which we shall use to advantage not only here but also in
Chapter 12.
310 9. Perturbed and Coupled Oscillators and Black Holes
9.10 Analysis of the Phase Shift Equation and Application to
Coupled Belousov–Zhabotinskii Reactions
The functions H(χ) and H(−χ) are T -periodic functions from their definition (9.64).
So, P(χ) in the phase shift equation (9.69) is also a T -periodic function. If χ = 0,
P(0) = H(0) − H(0) = 0. From the form of H(χ ) in (9.64) its derivative at χ = 0is
H
(0) =
1
T
T
0
k
∂ρ
∂χ
(s, 0)[Ω(s) − Γ(s)]+
∂V
∂θ
2
(s, s)
ds. (9.70)
The function ρ(s,χ)is the T -periodic solution of the second of (9.56); namely,
∂ρ
∂s
−Φ(s)ρ = U(s, s + χ),
which on differentiating with respect to χ and setting χ = 0gives
∂
∂s
∂ρ
∂χ
(s,0)
−Φ(s)
∂ρ
∂χ
(s,0)
=
∂U
∂θ
2
(s,s)
.
From the definition of U in (9.44) we see that [∂U/∂θ
2
]
(s,s)
= 0 and so the only periodic
solution of the last equation is [∂ρ/∂χ]
(s,0)
= 0. Using (9.41) which gives the definition
of V ,weget[∂ V/∂θ
2
]
(s,s)
= 1. With these values (9.70) gives
H
(0) =
1
T
T
0
kds= k.
So P
(0) =−H
(0) − H
(0) =−2k. Figure 9.16(a) illustrates a typical P(χ ).
Let us now consider the time evolution equation (9.69) for the phase shift χ =
ψ
2
−ψ
1
. Phase locking is when the phase difference is constant for all time, that is, when
χ = χ
0
is constant. Phase locked solutions are given by (9.69) on setting dχ/dτ = 0,
namely, solutions χ
0
of
Figure 9.16. (a) Schematic form of the T -periodic function P(χ) in the phase shift equation (9.69). (b)
Determination of steady state solutions χ
0
from the intersection of P(χ) = λβ.
9.10 Phase Shift Equation Analysis and Application 311
P(χ) − λβ = 0. (9.71)
The linear stability of χ
0
is given by linearising (9.69) about it, which gives
d(χ −χ
0
)
dτ
≈ P
(χ
0
)(χ −χ
0
),
and so
χ
0
is
stable
unstable
if P
(χ
0
)
< 0
> 0
. (9.72)
For example, if the coupled oscillators are identical, λ = 0 from (9.16), and from Fig-
ure 9.16(a) χ = 0 is a solution of P(χ) = 0 and its derivative P
(0)<0. So χ = 0is
stable. This means that coupling synchronizes identical oscillators.
Suppose now that the coupled oscillators are not identical; that is, λ = 0andε = 0
in (9.16). In this case steady states χ
0
of (9.71) will depend on whether the horizontal
line λβ intersects the curve of P(χ) in Figure 9.16(a). So there are at least two steady
state solutions in 0 ≤ χ
0
≤ T if
min P(χ) = P(χ
m
)<λβ<P(χ
M
) = max P(χ). (9.73)
Referring to Figure 9.16(b), we see that two typical solutions, χ
1
and χ
2
, are respec-
tively unstable and stable from (9.72), since by inspection P
(χ
1
)>0andP
(χ
2
)<0.
In this situation the coupled oscillator system will evolve to stable limit cycle oscilla-
tions with a constant phase shift χ
2
between the two oscillators after a long time, by
which we mean τ large, which in turn means εt large. Whether or not there are more
than two steady state solutions for the phase shift χ depends on the form of P(χ);the
example in Figure 9.16 is illustrative of the simplest situation.
As long as λβ lies between the maximum and minimum of P(χ), that is, (9.73)
is satisfied, the steady state solutions χ
0
, of (9.71) depend continuously on λβ .Thisis
clear from Figure 9.16(b) if the line P = λβ is moved continuously between the upper
and lower bounds P(χ
m
) and P(χ
M
). For example, as λβ increases the stable steady
state phase shift between the oscillators decreases.
Suppose now that λβ is such that the two solutions coalesce, either at P(χ
m
) or
P(χ
M
), and we ask what happens when λβ is such that these solutions no longer exist.
To examine this situation let us be specific and consider the case where λβ is slightly
less than the critical value, (λβ)
c
= P(χ
m
). With λβ just less than P(χ
m
) there is no
intersection of P(χ) and the line λβ .Let
λβ = (λβ)
c
−δ
2
= P(χ
m
) −δ
2
, 0 <δ
2
1. (9.74)
Now write the phase difference equation (9.69) in the form
dχ
dτ
={P(χ) − P(χ
m
)}+{P(χ
m
) − λβ }
={P(χ) − P(χ
m
)}+δ
2
.
(9.75)
312 9. Perturbed and Coupled Oscillators and Black Holes
Figure 9.17. (a) Situation for the χ-equation (9.75) where there is no steady state solution; here 0 <δ
2
1.
(b) Schematic solution, during which there are long periods of very little change in χ interspersed with rapid
variations.
Figure 9.17(a) illustrates dχ/dτ as a function of χ for δ
2
> 0. This is the same curve
as in Figure 9.16(a) simply moved up a distance −P(χ
m
) + δ
2
, which in our situation
is sufficient to make (9.75) have no steady state solutions for χ; that is, the curve of
dχ/dτ as a function of χ does not cross the dχ/dτ = 0axis.
The solution problem posed by (9.75) is another singular perturbation one when
0 <δ 1 and can be dealt with by standard singular perturbation techniques (see
Murray 1984). However it is not necessary to carry out the asymptotic analysis to see
what is going on with the solutions. First note that when δ = 0 (that is, λβ = P(χ
m
))
there are solutions χ = χ
m
+ nT for n = 0, 1, 2,... since P(χ) is a T -periodic
function. To be specific, let us start with χ ≈ χ
m
in Figure 9.17(a). From (9.75) and
from the graph in Figure 9.17, dχ/dτ = O(δ
2
)>0 which implies that χ ≈ χ
m
+δ
2
τ
and so for times τ = O(1), χ does not vary much from χ
m
. However, for all τ>0,
dχ/dτ>0andsoχ slowly increases. For τ sufficiently large so that χ starts to diverge
significantly from χ
m
, P(χ) − P(χ
m
) is no longer approximately zero, in which case
dχ/dτ is O(1) and χ changes measurably in times τ = O(1) and χ → χ
m
+T .When
χ is close to χ
m
+ T , once again P(χ ) − P(χ
m
) ≈ 0anddχ/dτ = O(δ
2
) again.
The qualitative picture is now clear. The solution stays in the vicinity of the solutions
χ = χ
m
+nT for a long time τ = O(1/δ
2
), then changes in times τ = O(1) to the next
solution (with δ = 0), where again it stays for a long time. This process repeats itself in a
quasi-periodic but quite different way from the T -periodic limit cycle behaviour we got
before. The solution is illustrated in Figure 9.17(b). The rapidly changing regions are
the singular regions while the roughly constant regions are the nonsingular parts of the
solutions. This behaviour is known as rhythm splitting.So,asλβ → (λβ)
c
= P(χ
m
)
the solution for χ bifurcates from phase synchronization to rhythm splitting.
Now that we know the qualitative behaviour of χ = ψ
2
−ψ
1
we can determine ψ
1
and ψ
2
from (9.68); namely,
dψ
1
dτ
= H(χ) + λβ,
dψ
2
dτ
= H(−χ). (9.76)
The solutions x
i
and y
i
are then given by
x
i
= X (t +ψ
i
(τ)), y
i
= Y (t + ψ
i
(τ)), i = 1, 2. (9.77)
Exercise 313
Figure 9.18. (a) The frequency dθ/dt of one of the oscillators when λβ is slightly less than the minimum
P(χ
m
) and rhythm splitting occurs. (b) The time variation in x
i
when phase locking just bifurcates to rhythm
splitting.
The frequency of the oscillations is given to O(ε) by
dθ
i
dt
= 1 +ε
dψ
i
dτ
=
1 +ε[H(χ ) +λβ ], i = 1
1 +ε H(−χ), i = 2
. (9.78)
Figure 9.18(a) illustrates the rhythm splitting solution. The small bumps are the rapid
variations in χ in times O(1) while the long flat regions correspond to the slowly vary-
ing solutions where the phase difference ψ
2
− ψ
1
(= χ) is approximately constant.
Figure 9.18(b) illustrates a typical solution x
i
as a function of t corresponding to this
rhythm splitting. Note the sudden change in frequency which occurs when the phase
difference χ goes through its region of rapid change from one solution χ
m
to the next
χ
m
+ T .
This bifurcation phenomenon exhibited by coupled oscillators, where there is an
abrupt change from phase locking to rhythm splitting, is exactly what was demonstrated
experimentally by Marek and Stuchl (1975), as described in Section 9.5. That is, they
first observed phase locking. Then when the parameters were changed so that the au-
tonomous limit cycle frequencies were sufficiently different they observed that the phase
difference was slowly varying for long periods of time but punctuated by short periods
of rapid fluctuations. This is the rhythm splitting phenomenon we have discussed above.
Exercises
1 Consider the limit cycle oscillator
dr
dt
= r(1 −r),
dθ
dt
= 1
314 9. Perturbed and Coupled Oscillators and Black Holes
to be perturbed by an impulse I parallel to the x-axis (refer to Figure 9.4). Determine
the resulting phase resetting curves and discuss the possible existence of black holes.
2 Discuss the bifurcation situation for coupled oscillators, using equation (9.73) for
the phase difference χ, when the interaction parameters λβ are just greater than
P(χ
M
), the maximum P(χ) in Figure 9.16(a). Sketch the equivalent functions to
those illustrated in Figures 9.17 and 9.18 and discuss the practical implications.
10. Dynamics of Infectious Diseases:
Epidemic Models and AIDS
10.1 Historical Aside on Epidemics
The history of epidemics is an ever fascinating area; the 14th century Black Death is just
the most famous epidemic historically (see Chapter 13, Volume II, which deals with the
spatial spread of epidemics, for a brief history of it). In Europe, which had a population
of around 85 million at the time, about a third of the population died.
One epidemic which has exercised classical scholars for a very long time is the
Plague of Athens (430–428
BC) described in great detail by Thucydides including the
symptoms and disease progression. He also gave some exact figures such as that 1050 of
4000 soldiers on an expedition died of the disease. The disease described so minutely
by Thucydides, even to the fact that dogs who ate the dead bodies also suffered, has
been the source of numerous articles over some hundreds of years with cases being
made (with great conviction and defended vehemently) for an incredible range of dis-
eases such as bubonic plague, measles, Malta fever, smallpox, scarlet fever, typhus, ty-
phoid fever and many others. The symptoms described by Thucydides are (i) heat in the
head, (ii) inflammation of the eyes, (iii) suffusion with blood of the tongue and throat,
(iv) foetid breath, (v) hoarseness with violent coughing, (vi) vomiting of bile, (vii) retch-
ing and convulsions, (viii) pustular and ulcerating eruptions of the skin, (ix) total body
hyperaesthesia and restlessness, (x) irresistible desire for water to assuage thirst and im-
mersion therein to alleviate body heat, (xi) terminal exhaustion apparently produced by
diarrhoea, (xii) loss of toes, fingers and genitalia, (xiii) destruction of eyes and, (xiv) if
recovery occurs, amnesia, the latter no doubt a blessing. Based on the symptoms none of
the above suggestions seems to fit the Athens disease. Whatever it was it was certainly
very nasty. An interesting review article on the Athens plague is given by Poole and
Holladay (1979). They conclude that it has either become extinct or has been modified
over the millennia. Since then other articles have appeared with yet other possiblities.
One of the interesting aspects of Thucydides’ account is that there is no mention of
person-to-person contagion which we now accept so freely with diseases. It was only in
the 19th century that it was beginning to be discussed. Evil exhalations from the earth,
aerial miasmata and so on were generally accepted. The latter explanation for some dis-
eases, or rather illnesses, is not as ridiculous as it might at first appear when you think of
the number of people, with the same epidemiclike medical problems, who live on con-
taminated ground or in regions where the water is iodine-deficient resulting in goitres
to mention just two examples. Many South-East Asians can be forgiven for believing