Graph-Based SIR Contagion Model

• The paper introduces a graph-based SIR model that precisely maps contagion dynamics on locally tree-like networks to a system of ODEs.
• It derives explicit formulas linking network degree distributions and epidemic parameters to final outbreak sizes, highlighting discrepancies with mean-field approximations.
• Numerical validations confirm that the ODE framework reliably predicts outbreak dynamics in both SIR and SEIR models, underscoring its practical forecasting value.

A graph-based SIR contagion model describes the infection dynamics of Susceptible–Infected–Recovered (SIR) processes on random graphs whose limiting local structure is tree-like, including Erdős–Rényi, random regular, and configuration model networks. In the large-graph regime, this formulation yields systems of ordinary differential equations (ODEs) that govern the limiting empirical distribution of susceptible, infected, and recovered nodes. For both SIR and SEIR models with time-varying or constant infection and recovery rates, these ODEs provide exact descriptions of outbreak dynamics and final epidemic sizes, elucidating the influence of network degree distributions and model parameters on macroscopic outcomes. Major findings include explicit formulas for the final outbreak fraction and rigorous comparisons with mean-field theory.

1. Network Ensembles and Local Tree-Likeness

Two primary network models underpin the limiting theory:

• Erdős–Rényi $\mathrm{ER}(n, c/n)$: Each pair of $n$ nodes is connected independently with probability $c/n$, yielding asymptotic mean degree $c$.
• Configuration Model $\mathrm{CM}_n(\zeta)$: Given degree distribution $\zeta$ with finite third moment, random graphs are constructed by random matching of half-edges.

In either ensemble, as $n \to \infty$, the local neighborhood of a typical node converges in distribution to a rooted Galton–Watson tree with offspring law $\zeta$. Neighbors of non-root nodes follow the size-biased law

$$\zeta^{(1)}(k) = \frac{(k+1)\,\zeta(k+1)}{\sum_j j\,\zeta(j)}$$

for $k \geq 0$. This structure, known as local weak convergence, is the basis for exact ODE closure on these networks (Cocomello et al., 2023).

2. ODE System for Limiting SIR Dynamics

Let $\beta_t > 0$ and $\rho_t > 0$ be time-dependent infection and recovery rates. Define the generating function for degree law $\zeta$,

$M_\zeta(x) = \sum_{k=0}^\infty \zeta(k) e^{k x},$

and its logarithmic derivative

$$\Phi(z) = \frac{M'_\zeta(-z)}{M_\zeta(-z)} = \frac{\sum_k k\,\zeta(k) e^{-kz}}{\sum_k \zeta(k) e^{-kz}}.$$

The macroscopic SIR dynamics are described by coupled ODEs for the quantities $f_S(t)$ (surviving susceptible branch), $f_I(t)$ (infectious branch), and $F_I(t)$ (cumulative infection hazard): $$\begin{cases} \dot f_S(t) = f_S(t) f_I(t) \beta_t \big(1 - \Phi(F_I(t))\big),\ \dot f_I(t) = f_S(t) f_I(t) \beta_t \Phi(F_I(t)) - f_I(t)\big[\rho_t + \beta_t - \beta_t f_I(t)\big],\ \dot F_I(t) = \beta_t f_I(t), \end{cases}$$ with initial conditions $f_S(0) = s_0$, $f_I(0) = 1-s_0$, $F_I(0) = 0$.

The limiting fraction of susceptible nodes is

$s^{(\infty)}(t) = s_0\, M_\zeta\big(-F_I(t)\big),$

and the infectious fraction is

$$i^{(\infty)}(t) = e^{-\int_0^t \rho_u du} \Big[(1-s_0) + s_0 \int_0^t M'_\zeta\big(-F_I(u)\big) e^{\int_0^u \rho_v dv} \beta_u f_I(u) du\Big].$$

Recovered fraction is given by $$r^{(\infty)}(t) = 1 - s^{(\infty)}(t) - i^{(\infty)}(t)$$. These formulas hold as $n \to \infty$ uniformly in $t$ in probability for $G_n$ (Cocomello et al., 2023).

3. Probabilistic Derivation and Conditional Independence Structure

The ODE system arises from the hydrodynamic limit, employing a coupling between the finite-$n$ process and an infinite unimodular Galton–Watson tree. The key tool is a conditional-independence (broadcast) property: for the SIR process started from i.i.d. $\{S, I\}$-labels on the root, the infection status of subtrees rooted at each neighbor of the root, conditional on the root being susceptible at time $t$, are independent and identically distributed. Filtering arguments and symmetry reduce the high-dimensional stochastic process to a closed Markov system on three summary statistics, enabling the ODE closure (Cocomello et al., 2023).

4. Final Outbreak Size and Explicit Fixed-Point Equation

For constant rates $\rho_t/\beta_t \equiv r$, the limit $F_I(\infty)$ (total cumulative infection hazard) determines the final size. The limiting fraction of susceptibles satisfies

$$s^{(\infty)}(\infty) = s_0 M_\zeta\left(-F_I(\infty)\right),$$

where $F_I(\infty)$ solves the strictly monotone fixed-point equation: $$\Psi_r(F) = F + \ln M_\zeta(-F) - \ln\big[1 + r(1 - e^{-F})\big] + \ln s_0 = 0.$$ Consequently,

$$1 - s_0 M_\zeta(-F_\infty) = r\big(1 - e^{-F_\infty}\big),$$

with $F_\infty = F_I(\infty)$. This equation has a unique positive solution and provides an explicit, asymptotically exact link between degree distribution, epidemic parameters, and final epidemic size (Cocomello et al., 2023).

5. Comparison with Mean-Field SIR and the Impact of Network Structure

The classical mean-field SIR ODE (corresponding to fully mixed or $\kappa$-regular graphs) predicts final susceptible fraction $\sigma_\kappa$ via

$$s_0 e^{\frac{\kappa}{\rho}(\sigma_\kappa - 1)} - \sigma_\kappa = 0.$$

It has been rigorously established that in the sparse, locally tree-like regime, the mean-field formula overestimates the outbreak size: $$1 - \sigma_\kappa > 1 - s^{(\infty)}(\infty), \quad \forall \kappa.$$ This discrepancy quantifies the reduction in epidemic size due to network heterogeneity and finite mean degree, as well as the limitation of mean-field approximations on sparse networks (Cocomello et al., 2023).

6. Extensions: SEIR Dynamics and Time-Varying Rates

The methodology extends directly to SEIR models, incorporating an exposed latent state $E$ and corresponding transition rates. When $\rho_t/\beta_t$ is constant, the final fraction infected in SEIR coincides with that from SIR, regardless of the latent-to-infectious transition rate $\lambda_t$. When $\rho_t/\beta_t$ varies, SIR and SEIR models diverge in their limiting outbreak sizes, even with identical initial conditions. Simulations confirm that with time-varying infection or recovery rates, the timing and magnitude of outbreaks are strongly affected by both network structure and compartmental delays (Cocomello et al., 2023).

7. Numerical Validation and Practical Implications

Numerical integration of the ODE system on moderate-sized ($n \sim 100$–$300$) ER and random regular graphs matches closely the Monte Carlo simulation results for both SIR and SEIR processes. These results confirm the efficacy of the limiting ODE description even outside the strict $n \to \infty$ regime. Further, the framework demonstrates how the phase and amplitude of seasonal or periodic variation in infection rates can nontrivially affect final outbreak size, emphasizing the critical role played by non-Markovian and heterogeneous effects in realistic epidemic forecasting (Cocomello et al., 2023).