Table 2.
The variables we need to calculate the epidemic dynamics. In all of these is a test individual: randomly chosen from the population and modified so that it cannot infect others, although it can become infected.
Figure 1.
Flow diagram showing the flux of individuals between the different compartments.
Because we have an explicit expression for , if we know
we do not need to explicitly determine the flux from
to
.
Figure 2.
Flow diagram for the flux of partners through different states.
The top three boxes ,
, and
make up
and represent the different states the partner can be in if it has not transmitted. The lower box
is the probability the partner has transmitted.
Table 1.
The degree distribution used in simulations in [14].
Figure 3.
Results of simulations for ,
, and
individuals.
The solid curve gives our prediction for the final sizes of epidemic in a large population. Colors are log scale giving probability of that particular epidemic size. Each simulation is for a new network generated using the from table 1, with
and
. We randomly select a proportion
of the population to initially infect and compare final size with the prediction of theory. The number of simulations for each
for
,
, and
was
,
, and
respectively. To show that this is sufficient to resolve the distribution, for
there were
,
and
simulations performed respectively for each
. This only slightly improves the tails of the distribution. Note that when the initial number (not proportion) of infections is small, a large fraction of simulations end without an epidemic.
Figure 4.
Epidemic final sizes in population of individuals with half having degree
and half with degree
.
The disease parameters are ,
. Results of simulations having initial infections chosen with probability proportional to square of degree (left) or inverse square of degree (right). For each initial number of infections,
simulations were performed, each with a different network. For the range
,
simulations were performed to give insight into how well resolved the distribution is. Note that for small numbers of initial infections, epidemics are less likely when the lower degree individuals are chosen.
Figure 5.
A comparison of the observed and predicted number of infections from simulations.
Left: 5% initially infected, chosen randomly from the population. Right: 5% initially infected, chosen with probability proportional to squared degree. Each simulation curve represents a single simulation of the given size. As population size increases, the results converge to the theoretical prediction.
Figure 6.
Epidemics begin at with
of the population infected. Left: epidemic curve without interventions, and with each intervention introduced at time
. Right: horizontal axis is
, showing final effectiveness if interventions introduced at different times
Figure 7.
Bifurcation diagram with ,
, and
as given.
The figure on right zooms in on the bifurcation point. Disease parameters are and
. In each all members of the population have degree either
or
, with the proportions chosen so that
takes the values on the horizontal axis. The dashed curves denote unstable equilibria and the solid curves stable equilibria. Approximate curves (dotted) come from equation (5). Only the equilibria with
are biologically meaningful.