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In this section several commonly used epidemic models which incorporate some form of heterogeneity are shown to have a bipartite graph based representation.

Household model

The widely studied household model, described briefly above, with a within house- hold infection rateη(n) =nλH, is readily represented as a bipartite graph epidemic.

When the number of households isnh this has a bipartite graph representation with

an adjacency matrix of sizenp×(nh+ 1) where np is the sum of all the household sizes. With a fixed household sizem thennp =nhm.

The bipartite graph epidemic of equation 2.6.1 is obtained by settingλ1 =λG

andλj+1=λH for 1≤j≤mand the adjacency matrix isai,1= 1 for alli≤npand

ai,j+1 = 1 for Nj−1 < i≤Nj , where N0 = 0 andNj is the sum of household sizes

1. . . j. For example for 4 houses of sizes 2,3,3,4 the adjacency matrix and associated

infection rates is shown in table 2.6.1.

λG λH λH λH λH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Table 2.6.1: bipartite representation of a household model

Multi-type model

The frequently used multi-type model mentioned in section 2.5.3 with infection rates

ψi,j between an infective in group iand a susceptible in groupj, can be considered as a bipartite graph epidemic model subject to conditions onψi,j.

Theorem 4. If Ψ = (ψi,j1≤i, j≤m) is symmetric and ψi,i ≥Pj6=iψi,j for all i

then a multi-type epidemic model with m types and infection rates Ψhas an equiv- alent representation as a bipartite graph epidemic model withm(m+ 1)/2 groups. Proof. By construction, set λk = ψi,i −Pj6=iψi,j for k = 1. . . m and assign the

elements of ψi,j where i < j to λk for k = m+ 1. . . m(m+ 1)/2. Construct an adjacency matrix with columnsk= 1. . . mfor the within type infections each being an indicator vector for typekand the remainingm(m−1)/2 columns for the between type infections being a ’logical or’ of columnsiand j.

The condition will usually apply if the groups are geographically separate but may not if the groups are split by ages or if varying susceptibility and infectiousness is modelled by a product form forψi,j. For example for 3 types of sizes 3,2,4

ψ1,1−ψ1,2−ψ1,3 ψ2,2−ψ2,1−ψ2,3 ψ3,3−ψ3,1−ψ3,2 ψ1,2 ψ1,3 ψ2,3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .

Bipartite graph epidemics with asymmetric infection rates A frequently studied model has both susceptibility and infectiousness varying between groups so thatψi,j =cidj whereciis the infectiousness anddj the susceptibility of individuals

in groupsi and j. An extension to the bipartite graph epidemic defined in section 2.6.3 could be considered: where the infection ratesλkk ∈ V are replaced by two

sets of ratesλc_k and λd_k and equations 2.6.1 and 2.6.2 are replaced by

ηj(t) = X k∈V ajkλdk X l∈I(t) alkλckIk(t) (2.6.3)

as the rate of infections at timetfor each susceptible individualj∈ S(t). This would give an immediate correspondence between this extended model and the important subset of multitype models, but at the expense of extra complexity and is not considered further here.

Other models

Random graph Although any graph can be represented in a bipartite form using the clique decomposition, the usual model with a constant infection rate along each edge does not in general have a bipartite graph epidemic representation. The exceptions include the set of graphs formed from the lower projection of a bipartite graph with an adjacency matrix that contains no repeated rows with more than one 1. This set is composed of graphs composed of cliques with the overlap between cliques containing at most one vertex.

Spatial models A frequently used spatial model for epidemics is to have the infec- tion rate between two individuals depend inversely on a spatial kernel, a very similar set of infection rates can be obtained by combining a bipartite graph epidemic, with overlapping spatial tilings.

The simplest example is to choose a small numbermof tilings, 3 for example and construct the first tiling with vertices at (im, jm) for i, j ∈_Z and subsequent ones at (im+ 1, jm+ 1) (im+ 2, jm+ 2) etcetera. Now take the spatial locations, suitably scaled, and for each tiling and each individual determine which square contains the location, each square on each tiling corresponds to a group/column. Any pair of rows/individuals will be in the same square for 0,1,2 or 3 tilings and so have 0,1,2 or 3 columns in common. The infection rates of the spatial kernel model and the proposed bipartite graph model will be approximately proportional. Increasingm will bring the models closer but increases the computational burden.

A three level model A model incorporating households, schools and workplaces is considered by Britton et al. (2011) which can also be represented as a bipartite graph epidemic model. Their example has 500 households of size 4, where the 2 adults in a house each attend one of 40 work places and the 2 children attend the same school of size 100. A straightforward extension of the household representa- tion above is used to represent this with 551 columns 500+40+10 + 1 for a global infection possibility. This model goes a long way to capturing the most obvious heterogeneities in urban life and has been used to simulate examples from their model.