Table 1.
Models investigated in this article.
Figure 1.
Sample generation of a Configuration Model network.
The steps to generate a Configuration Model network with . (Top left) Each node/individual is independently assigned either
or
stubs with equal probability. (Successive plots) Pairs of stubs are randomly chosen and joined into edges until no stubs remain (bottom right). In the limit of a large population, about half of the individuals have
partners, about half have just
partner, and a random partner of a given individual is three times as likely to have
partners versus just
. The number of triangles in the network is small; there is little clustering. For this
we have
.
Figure 2.
Flow diagram for Configuration Model networks.
Given a test individual and a random partner
, the probability
has not transmitted to
is
. We divide
into three compartments based on the status of
:
where the subscript gives the status of
. The flow diagram on the left gives the flux between these subcompartments within
and the flux from
to
(which comes specifically from the
subcompartment of
). The flow diagram on the right shows the flux of individuals through the
,
, and
compartments. To find
, we must find
, which is
. We find
. Because the flux into
and
are proportional, we can find
. Thus
, and we are able to find a differential equation for
in terms of
. To find
,
, and
, we note that the probability
is in each state is equal to the proportion of the population in each state, so the susceptible proportion is equal to the probability that
is susceptible, which is
. We determine
by
and
by
.
Table 2.
Variables and equations for the basic EBCM approach.
Figure 3.
Flow diagram for a network with directed and nondirected edges. We consider the two edge types separately. The evolution of edges is similar to figure 2. We can assign different infection rates for each edge type.
Figure 4.
Results for the directed networks described in section 2.2.1 using ,
, and
. We choose
to correspond to
cumulative incidence. Theory (dashed) correctly predicts dynamics of simulations in a population of
individuals (solid).
Figure 5.
Heterogeneous infectiousness/susceptibility model.
We separate nodes by type , but assume that
has no effect on connectivity. Both infectiousness and susceptibility may depend on
. We must consider edges between each pair of types
and
separately. The evolution of edges is similar to before.
Figure 6.
Heterogeneous infectiousness/susceptibility example.
Epidemics spreading in a population for which half have received a leaky vaccine described in section 2.2.2. Vaccinated individuals are half as infectious and half as susceptible. We choose to correspond to
% cumulative incidence. Simulations in a population of
individuals (solid) and theory (dashed) are in good agreement.
Figure 7.
Assortative mixing by type model.
We separate nodes by type. We assume that type may influence infectiousness and susceptibility as well as connections. For simplicity, we assume a finite number of groups. The resulting system is similar to our system for correlated infectiousness and susceptibility.
Figure 8.
Assortative mixing by type example.
Comparison of theory and simulated results for mixing with demographic groups described in section 2.2.3. We also show the predicted levels of infection in each subgroup. Simulations in a population of individuals (solid) and theory (dashed) are in good agreement. The difference between the top and bottom result from changing the correlations of within and between-group mixing. We choose
to correspond to
% cumulative incidence.
Figure 9.
Multiple modes of transmission model.
Flow diagram showing the flux of edges for the -th contact type for a disease which has multiple modes of transmission.
Figure 10.
Multiple modes of transmission example.
Disease spread in a population with three different types of partnerships, each with a different degree distribution described in 2.2.4. Simulations in a population of individuals (solid) and theory (dashed) are in good agreement. We choose
to correspond to
% cumulative incidence.
Figure 11.
Multiple infectious stages model.
Flow diagrams for a disease with several infected stages. When a disease progresses through several states (or has an infectious period that is not exponentially distributed) it is convenient to use a stage-progression model to represent the state of an edge.
Figure 12.
Multiple infectious stages example.
The spread of the disease described in 2.2.5 with three infectious stages. Simulations in a population of individuals (solid) and theory (dashed) are in good agreement. We choose
to correspond to
% cumulative incidence.
Figure 13.
Fixed-degree serosorting model.
Flow diagram showing the interplay involved in serosorting with fixed-degree. We do not consider a recovered class, which simplifies the equations. The framework can be adapted to include a recovered class. The variables represent the total proportion of stubs involved in edges between the two types and the
variables are the proportion of dormant stubs belonging to nodes of each type. The
variables are as before. For the right hand side, we are able to determine most of the variables analytically, so we only need the fluxes into and out of
. We expect that the edge breaking and rejoining rates
will depend on values of
and
.
Figure 14.
Variable-degree serosorting model.
Flow diagram showing the interplay involved in serosorting. We do not consider a recovered class, which simplifies the equations significantly. The framework can be adapted to include a recovered class. The variables give the proportion of contacts that would be formed with susceptible or infected individuals assuming that their behavior is not altered by disease. The
variables are the probability that a current contact of the test node is with an individual of given type, under the assumption that the test node always behaves as if susceptible, and does not transmit to its partners. We expect that the edge breaking and rejoining rates
will depend on values of
and
. Note that
need not equal
.