Computational Biology 1 (FFR110/FIM740)
Problem set 3
for March 9, 2017
1 Stochastic dynamics in large but finite populations. In this task you
will contrast the deterministic disease dynamics (valid in the limit of infinite
populations) to the corresponding disease dynamics in large but finite populations. The analysis will be made for the SIS model introduced in the lecture
notes. In what follows, S denotes the population size of susceptibles, and I
denotes the population size of infectives. The deterministic dynamics under the
model is given by
α
dI
=
SI − βI
dt
S+I
α
dS
=−
SI + βI
dt
S+I
(1)
(2)
Here, the parameters α and β are positive constants.
1a. For which values of the parameters of the model (1)-(2) will the infection
sustain ad infinitum? Support your result by finding the steady states of the
system (1)-(2) and performing their linear stability analysis. (0.5p)
1b. Contrast the deterministic model (1)-(2) to the corresponding stochastic
model. Assume that the population consists of N individuals (where N is constant in time). An individual in the population is either infected or susceptible.
Assume that in a short time interval, the number of infected individuals changes
by +1 or −1 because of
n−1
new infection : n − 1 → n at rate bn−1 = α 1 −
(n − 1) ,
N
recovery : n + 1 → n at rate dn+1 = β(n + 1) .
Write down the Master equation for the probability ρn (t) to observe n infected
individuals at time t in a finite population consisting of N individuals. Derive
Eq. (1) in the limit of N → ∞. Assume that the parameter values are such
that the infection under the deterministic model persists ad infinitum. Does
this result apply to the stochastic model as well? Discuss in which way the
stochastic model differs from the deterministic one. Relate your discussion to
the quasi-steady state. (0.5p)
1c. Set the parameters so that the number of infected individuals in the quasisteady state is positive. Simulate the number of infected individuals under the
stochastic model. Perform the simulation as follows. Start from a chosen initial number of infected individuals, and simulate the processes of infection and
recovery according to the rates given above. Make sure to choose appropriate
parameter values so that the quasi-steady state can be reached in a reasonable
Computational Biology 1 (FFR110/FIM740)
amount of time. Repeat the simulation many times (starting from the same
initial condition). Plot the average number of infected individuals as a function
of time. Plot also the time dependence of the average number of infected individuals conditional on that the disease has not gone extinct at a given time.
You should see that the two curves agree at short times, but differ at long times.
Why? Compare with the results of your analysis of the deterministic model.
(1p)
1d. In the limit of large but finite values of N , the average time to the extinction of the disease Text is approximated by
log(Text ) ∼ N S(α, β) + . . . .
(3)
Find an approximate expression for S(α, β) (see the procedure outlined in the
lecture notes). Compute the average time needed for the disease to go extinct
using your simulations. Initialise the population in the quasi-steady state and
trace the number of infected individuals until the disease dies out. Repeat
the computation many times and average the result over the different runs.
Determine the range of validity of Eq. (3). (1p)
1e. By means of simulations compute the quasi-steady state distribution of
the number of infected individuals. In the limit of large but finite values of N
compare the findings with the predictions of Eq. (1). In the lecture notes an
approximate expression for the quasi-steady state distribution is derived from
the Master equation. Compare this expression to the results of your simulations. Does the theory correctly predict deviations from a Gaussian? Does
it correctly describe the far left tail of the distribution (corresponding to few
infected individuals)? (1p)
2. Synchronisation. In the lectures the Kuramoto model was introduced. It
describes the dynamics of the phases θi of N coupled oscillators (i = 1, ..., N ):
N
KX
dθi
= ωi +
sin(θj − θi )
dt
N j=1
The frequencies ωi are random, drawn from a symmetric distribution g(ω) with
one maximum. Assume that g(ω) = (γ/π)[ω 2 + γ 2 ]−1 .
2a. In the lecture notes it was shown that the degree of synchronisation in this
model is, within a mean-field approximation, described by the order parameter
r which satisfies the self-consistent equation
Z
π/2
1=K
−π/2
dθ cos2 (θ)g(Kr sin θ) .
(4)
Computational Biology 1 (FFR110/FIM740)
By expanding this equation in the vicinity of the bifurcation (onset of synchronisation), the bifurcation value Kc was found (see the lecture notes). It was
further shown that, in the vicinity of the bifurcation, the order parameter is
√
given by r = C µ where 0 < µ = (K − Kc )/Kc 1. Determine the coefficient
C. (0.5p)
2b. Perform numerical simulations of the Kuramoto model (4) for a value of
K below Kc , and for two values of K above Kc . Of the latter two, let one value
be very close to Kc . For each simulation, initialise the phases of the oscillators
to random numbers between −π/2 and π/2. Plot how the order parameter
depends on time. Compare the results of your simulations to the predictions
from 2a. It was argued in the lecture notes that the meanfield theory is expected
to work well when the number of oscillators N is sufficiently large. Check this by
simulating the Kuramoto model for several different values of N (for example,
choose N = 20, 100, 300). Discuss: how well does the mean-field theory work?
How do your results depend on the choice of N ? (1.5p)