Contact networks
A contact network represents potentially infectious interactions between individuals or groups. It replaces the assumption that every susceptible individual mixes equally with every infectious individual.
Nodes and edges
In an individual-level contact network, each node represents a person. An edge indicates a type of contact through which transmission could occur.
The edge definition must match the pathogen. A contact relevant for an airborne infection may differ from one relevant for a sexually transmitted or vector-borne infection.
Adjacency matrix
For a binary contact network, let
\[A_{ij}=\begin{cases}1,&\text{if }i\text{ and }j\text{ have a modelled infectious contact},\\0,&\text{otherwise.}\end{cases}\]For undirected contact,
\[A_{ij}=A_{ji}.\]Directed transmission opportunities can instead be represented by a non-symmetric matrix.
Degree
For an undirected network, the degree of node \(i\) is
\[\boxed{k_i=\sum_j A_{ij}}.\]It counts the number of direct contacts represented by the network.
A high-degree individual has more potential transmission connections, but actual epidemic importance also depends on the states and positions of those neighbours.
Epidemic states on the network
Suppose each node has an epidemic state
\[X_i(t)\in\{S,I,R\}.\]The network structure \(A\) specifies possible contacts, while \(X_i(t)\) specifies the current disease state.
Thus topology and epidemic state are separate parts of the model.
Infectious neighbours
Let
\[I_j(t)=\begin{cases}1,&\text{if node }j\text{ is infectious at time }t,\\0,&\text{otherwise.}\end{cases}\]The number of infectious neighbours of node \(i\) is
\[\boxed{m_i(t)=\sum_j A_{ij}I_j(t)}.\]This quantity changes through time even on a static network because neighbours change epidemic state.
Infection rate from infectious neighbours
Suppose transmission across each susceptible–infectious edge occurs at rate \(\tau\). If node \(i\) is susceptible and has \(m_i\) infectious neighbours, its total infection rate is
\[\boxed{\lambda_i(t)=\tau m_i(t)=\tau\sum_jA_{ij}I_j(t)}.\]Over a sufficiently short interval \(\Delta t\),
\[P(i\text{ becomes infected in }\Delta t)\approx\lambda_i(t)\Delta t.\]This is the network analogue of constructing infection pressure from infectious contacts.
Exact infection probability over a finite interval
If the infection rate \(\lambda_i\) remains constant over an interval of length \(\Delta t\), the exponential waiting-time model gives
\[\boxed{P(\text{infection during }\Delta t)=1-e^{-\lambda_i\Delta t}}.\]For small \(\Delta t\),
\[1-e^{-\lambda_i\Delta t}\approx\lambda_i\Delta t.\]Recovery
If each infectious node recovers at rate \(\gamma\), then an infectious individual has mean infectious duration
\[\frac1\gamma\]under the exponential-duration assumption.
Network structure affects transmission opportunities, while recovery can be specified independently at each node.
Weighted contact networks
Contacts can differ in intensity. Let \(w_{ij}\ge0\) measure the strength of contact between \(i\) and \(j\). A simple weighted infection rate is
\[\boxed{\lambda_i(t)=\tau\sum_jw_{ij}I_j(t)}.\]The weight may represent duration, frequency or another exposure measure, provided its biological interpretation is stated.
Edge-specific transmission
Different contacts may have different transmission rates \(\tau_{ij}\). Then
\[\boxed{\lambda_i(t)=\sum_jA_{ij}\tau_{ij}I_j(t)}.\]This allows household, workplace or other contacts to carry different transmission intensities.
Static networks
A static contact network assumes
\[A(t)=A.\]The contacts remain fixed during the modelled period, although infection states continue to change.
This can be a useful approximation for relatively stable relationships.
Temporal contact networks
If contacts change with time, use
\[A=A(t).\]The infection rate becomes
\[\lambda_i(t)=\tau\sum_jA_{ij}(t)I_j(t).\]Now transmission requires not only a network connection but a connection at the relevant time.
Why contact timing matters
Suppose \(A\) contacts \(B\), and later \(B\) contacts \(C\). Infection can potentially follow the time-respecting route
\[A\to B\to C.\]If those contacts occur in the reverse temporal order, the same aggregated static edges exist, but transmission from \(A\) through \(B\) to \(C\) may not be possible during that period.
Therefore aggregating temporal contacts can create transmission routes that never existed in the correct time order.
Repeated contacts
Repeated interactions with the same neighbour are not equivalent to contacts with many different neighbours. Depending on the pathogen, repeated exposure may increase transmission probability while adding no new route to other parts of the network.
This distinction is lost if every contact count is interpreted simply as degree.
Clustering
Contacts often form triangles: two neighbours of a person may also contact one another.
For a node \(i\) with degree \(k_i\ge2\), a local clustering coefficient for a simple undirected graph is
\[\boxed{C_i=\frac{2e_i}{k_i(k_i-1)}},\]where \(e_i\) is the number of edges among the neighbours of \(i\).
High clustering can create repeated exposure within tightly connected groups rather than continually opening new transmission routes.
Assortative mixing
Contact patterns may depend on characteristics such as age, occupation or degree. Assortative mixing means similar nodes are more likely to connect to one another than expected under random mixing.
Age-assortative contact, for example, can strongly shape epidemic spread between age groups.
Degree correlations
Networks can also be assortative by degree: high-degree nodes may preferentially connect to other high-degree nodes.
Alternatively, high-degree nodes may connect mainly to low-degree nodes. These patterns can affect epidemic reach and intervention priorities.
Households and groups
Some contacts naturally occur in groups such as households, schools and workplaces. Representing every pairwise edge may be sufficient for some questions, but group-based or multilayer models can preserve additional structure.
Homogeneous mixing versus contact networks
A homogeneous-mixing epidemic model effectively averages over who contacts whom. A network model retains individual or group-level contact structure.
Homogeneous mixing can be appropriate when contacts are numerous and rapidly changing. Explicit networks become more useful when heterogeneity, clustering or persistent relationships strongly affect transmission.
Contact network versus transmission network
A contact edge means transmission is possible; it does not mean transmission actually occurred.
The realised transmission network is generally a subset of the epidemiological contact opportunities and depends on infection states, timing and stochastic transmission events.
Network data
Contact networks can be estimated from surveys, diaries, proximity sensors, movement data or other measurements. Each method observes a different proxy for epidemiologically relevant contact.
Missing contacts, reporting bias and the chosen time window can alter estimated network structure.
Interventions can change the network
Isolation removes or suppresses contacts associated with infectious individuals. School or workplace closure can remove entire groups of edges. Behavioural changes may reduce edge weights or contact duration.
Vaccination may leave the contact network unchanged while changing susceptibility or infectiousness at selected nodes.
What degree alone cannot tell us
Degree is important, but two nodes with equal degree can occupy very different network positions. One may connect otherwise separated communities while another is embedded within a tightly connected group.
This motivates later measures of centrality and network intervention.
Transition to degree distributions
The next lesson moves from individual degrees \(k_i\) to the distribution of degree across the whole population. This gives a first mathematical description of contact heterogeneity.