Stochastic heterogeneous SIR model with infection-age dependent infectivity on large random graphs

We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and is represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. To each individual is associated a random infectivity function of its infection age, which is allowed to depend upon the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, and reflects the heterogeneities in individual characteristics and social connectivity.

💡 Research Summary

The paper introduces a novel stochastic individual‑based SIR epidemic model that simultaneously incorporates (i) heterogeneous individual attributes—such as spatial location, age, or social behavior—and (ii) infection‑age‑dependent infectivity functions. Each individual is represented by a vertex in a random graph; the vertex carries a characteristic (X_i) taking values in a Polish space (\mathcal X). The probability that an edge connects two vertices (i) and (j) is given by a general kernel (W(X_i,X_j)), which may be random but converges, under a suitable scaling, to a deterministic graphon as the population size (N) tends to infinity. This scaling allows the graph to be dense (order (N^2) edges) or intermediate (order (N^{1+\alpha}), (0<\alpha\le 1)), thereby covering a broad class of realistic contact networks.

In addition to the network heterogeneity, each individual carries a random infectivity function (\lambda_i(a)=\bar\lambda(X_i,Y_i,a)) that depends on the infection age (a) and on an auxiliary random variable (Y_i) independent of the initial sigma‑field. The infectivity functions are uniformly bounded by a constant (\lambda^*) but are otherwise arbitrary, allowing for non‑exponential infectious periods and for individual‑specific transmission profiles. The infection duration (\eta_i) is defined as the supremum of ages for which (\lambda_i(a)>0).

The epidemic dynamics are described by three measure‑valued processes:

• (\mu^S_t) on (\mathcal X) for the susceptible population,
• (\mu^I_t) on (\mathcal X\times\mathbb R_+) for infected individuals together with their infection ages,
• (\mu^R_t) on (\mathcal X) for recovered individuals.

These processes evolve in the Skorokhod space (D(\mathbb R_+, \mathcal M(\cdot))) and are driven by the aggregate force of infection \