Epidemics on networks
Network epidemic models place infection and recovery dynamics on an explicit contact structure. Transmission is possible only through epidemiologically relevant connections rather than through homogeneous population-wide mixing.
States on the nodes
For an SIR network model, each node \(i\) has state
\[X_i(t)\in\{S,I,R\}.\]A susceptible node can become infectious through infectious neighbours, while an infectious node eventually recovers.
The network specifies which transmissions are possible; the epidemic process specifies when state changes occur.
Continuous-time infection process
Let \(A\) be the adjacency matrix and let
\[I_j(t)=\begin{cases}1,&X_j(t)=I,\\0,&\text{otherwise.}\end{cases}\]If transmission along each susceptible–infectious edge occurs at rate \(\tau\), the infection rate of susceptible node \(i\) is
\[\boxed{\lambda_i(t)=\tau\sum_jA_{ij}I_j(t)}.\]If \(m_i(t)\) is the number of infectious neighbours, then simply
\[\lambda_i(t)=\tau m_i(t).\]Recovery process
Suppose each infectious node recovers at rate \(\gamma\). Then
\[I\longrightarrow R\qquad\text{at rate }\gamma.\]Under the exponential waiting-time assumption, the mean infectious duration is
\[\boxed{E[T_I]=\frac1\gamma}.\]The network epidemic as a CTMC
For a fixed finite graph, a continuous-time stochastic SIR epidemic can be represented as a continuous-time Markov chain whose state records the disease state of every node.
With \(N\) nodes and three possible SIR states per node, the full state space can contain as many as
\[3^N\]configurations.
This rapid growth is one reason why exact node-level probability equations become computationally difficult for large networks.
Small-time event probabilities
For sufficiently small \(\Delta t\), a susceptible node with infection rate \(\lambda_i\) has
\[P(S_i\to I_i\text{ during }\Delta t)\approx\lambda_i\Delta t.\]An infectious node has
\[P(I_i\to R_i\text{ during }\Delta t)\approx\gamma\Delta t.\]These are continuous-time rate approximations, not fixed per-step probabilities.
Competing transmission and recovery
Consider one infectious node connected to one susceptible neighbour. Transmission across that edge occurs at rate \(\tau\), while recovery occurs at rate \(\gamma\).
There are therefore two competing exponential events. The probability that transmission occurs before recovery is
\[\boxed{T=\frac{\tau}{\tau+\gamma}}.\]This quantity is called the edge transmissibility for this Markovian SIR model.
General transmissibility
If the infectious period is a random variable \(D\) and transmission along an edge follows a Poisson process with rate \(\tau\), then
\[\boxed{T=1-E[e^{-\tau D}]}.\]For \(D\sim\operatorname{Exp}(\gamma)\), this reduces to
\[T=\frac{\tau}{\tau+\gamma}.\]Thus the infectious-period distribution can affect transmissibility even when its mean is unchanged.
Early epidemic branching
At the start of an epidemic, most neighbours are susceptible. On a large locally tree-like network, early spread can therefore be approximated by a branching process.
If an infection is reached by following an edge, the expected number of remaining edges is
\[\frac{\langle k^2\rangle-\langle k\rangle}{\langle k\rangle}.\]Multiplying by transmissibility gives the approximate edge-based reproduction quantity
\[\boxed{R_*=T\frac{\langle k^2\rangle-\langle k\rangle}{\langle k\rangle}}.\]Network epidemic threshold
Under the configuration-model and independent-transmission assumptions, a large SIR outbreak becomes possible when
\[\boxed{T\frac{\langle k^2\rangle-\langle k\rangle}{\langle k\rangle}>1}.\]Equivalently, the critical transmissibility is
\[\boxed{T_c=\frac{\langle k\rangle}{\langle k^2\rangle-\langle k\rangle}}.\]Large degree variance can therefore lower the threshold in this class of networks.
Poisson-network special case
For a Poisson degree distribution with mean \(z\),
\[\langle k^2\rangle-\langle k\rangle=z^2.\]Hence
\[R_*=Tz,\]and the threshold becomes
\[\boxed{Tz>1}.\]Regular-network special case
If every node has degree \(k\), an infected node reached along one edge has \(k-1\) remaining edges. Thus
\[\boxed{R_*=T(k-1)}.\]The subtraction of one occurs because one edge was used to reach the infected node.
Why heterogeneous degree matters
High-degree nodes are more likely to become infected through randomly followed edges because they have more edges through which they can be reached.
Once infected, they may also have many remaining susceptible neighbours. This size-biased sampling explains why \(\langle k^2\rangle\) enters network epidemic thresholds.
Final epidemic size and percolation
For SIR epidemics on large locally tree-like networks with independent edge transmissibility \(T\), final epidemic size can be related to bond percolation.
Each contact edge is regarded as open with probability \(T\). A major outbreak corresponds to the infection seed reaching a giant connected component of the resulting transmission network.
This provides a structural interpretation of the final outbreak rather than a time-dependent description of the epidemic curve.
Generating-function formulation
Let
\[G_0(x)=\sum_kP(k)x^k\]be the degree generating function and
\[G_1(x)=\frac{G_0'(x)}{G_0'(1)}\]the excess-degree generating function.
If \(u\) is the probability that following a randomly chosen edge does not eventually lead into the giant transmission cluster, then
\[\boxed{u=1-T+TG_1(u)}.\]The corresponding fraction of nodes in the giant transmission component is
\[\boxed{S=1-G_0(u)}.\]These equations apply under the usual locally tree-like configuration-model assumptions.
SIS dynamics on networks
In an SIS model, recovery returns an infectious node to susceptibility:
\[S\to I\to S.\]Unlike SIR dynamics, infection can revisit the same node and edges can participate repeatedly in transmission.
Therefore the simple final-size percolation mapping used for SIR does not directly describe SIS endemic dynamics.
Spectral threshold idea
For several SIS approximations on a fixed network, epidemic persistence is related to the largest eigenvalue \(\rho(A)\) of the adjacency matrix.
A common first-order node-based mean-field threshold has the form
\[\boxed{\frac{\tau}{\gamma}\rho(A)>1}.\]This shows mathematically how network topology can enter through a spectral quantity.
Node-based mean-field equations
Let \(p_i(t)\) approximate the probability that node \(i\) is infectious in an SIS model. A common independence approximation gives
\[\boxed{\frac{dp_i}{dt}=\tau(1-p_i)\sum_jA_{ij}p_j-\gamma p_i}.\]The term \((1-p_i)\) represents susceptibility, while the network sum approximates infectious pressure from neighbours.
The approximation neglects statistical dependence between connected node states.
Why independence can fail
Neighbouring epidemic states become correlated because transmission itself acts along edges. If one node infects another, their states are not independent samples from the population.
This motivates pair-based models that track quantities such as susceptible–infectious edges.
Pair variables
Let \([S]\) and \([I]\) denote expected numbers of susceptible and infectious nodes, while \([SI]\) denotes the expected number of susceptible–infectious pairs under a stated counting convention.
In a simple continuous-time SIR pair formulation, population-level infection occurs through \(SI\) edges, so schematically
\[\frac{d[S]}{dt}=-\tau[SI].\]Equations for pairs then involve triples such as \([SSI]\) and \([ISI]\), producing a hierarchy.
Moment or pair closure
To obtain a finite system, triples are approximated using lower-order quantities. On a homogeneous locally tree-like network, a closure may take a form proportional to
\[[ABC]\approx\frac{k-1}{k}\frac{[AB][BC]}{[B]},\]with precise factors depending on ordered-versus-unordered counting conventions.
Closure converts an otherwise unclosed hierarchy into an approximate system of ODEs.
Clustering
Triangles violate the locally tree-like assumption. Two neighbours of an infectious individual may also be connected to each other, so transmission paths overlap.
High clustering can therefore change both epidemic growth and final size relative to a random network with the same degree distribution.
Communities
Dense connections within groups and sparse connections between groups can create rapid local outbreaks followed by slower spread between communities.
A small number of bridge edges may then have disproportionate influence over population-wide transmission.
Temporal networks
If contacts vary through time, write
\[A=A(t).\]The infection rate becomes
\[\lambda_i(t)=\tau\sum_jA_{ij}(t)I_j(t).\]Static aggregation can incorrectly create paths whose edges never occurred in a transmission-compatible temporal order.
Weighted and heterogeneous transmission
If edge \((i,j)\) has transmission rate \(\tau_{ij}\), then
\[\boxed{\lambda_i(t)=\sum_jA_{ij}\tau_{ij}I_j(t)}.\]This allows household, workplace and other interaction types to have different transmission intensities.
Network interventions
Network structure creates intervention options unavailable in homogeneous models. Contacts can be removed, high-risk edges reduced, particular nodes vaccinated, or bridges between communities targeted.
However, identifying the best targets requires more than degree alone and is developed later in the network-intervention lesson.
Network epidemics versus homogeneous mixing
A homogeneous SIR model uses aggregate infection pressure such as \(\beta SI/N\). A network model replaces this average interaction assumption with explicit susceptible–infectious edges.
If contacts are sufficiently numerous, rapidly changing and weakly structured, a homogeneous approximation may be adequate. Persistent heterogeneous contacts make explicit network models more informative.
What network models add
Network models can represent who becomes exposed, local depletion of susceptible neighbours, heterogeneity in contact number, clustering, communities, temporal ordering and targeted interventions.
The price is greater mathematical and computational complexity.
Transition to gene networks
Epidemic networks use edges to represent transmission opportunities. The next lesson changes the biological interpretation: nodes become genes or regulatory components and directed edges describe regulatory influence.