Infection models on dense dynamic random graphs

Abstract

We consider Susceptible-Infected-Recovered (SIR) models on dense dynamic random graphs, in which the joint dynamics of vertices and edges are co-evolutionary, i.e., they influence each other bidirectionally. In particular, edges appear and disappear over time depending on the states of the two connected vertices, on how long they have been infected, and on the total density of susceptible and infected vertices. Our main results establish functional laws of large numbers for the densities of susceptible, infected, and recovered vertices, jointly with the underlying evolving random graphs in the graphon space. Our results are supported by simulations, which characterize the limiting size of the epidemics, i.e., the limiting density of susceptible vertices, and how the peak of the epidemics depends on the rate of the evolution of the underlying graph. The proofs of our main results rely on the careful construction of a mimicking process, obtained by approximating the two-way feedback interaction between vertex and edge dynamics with a mean-field type interaction, acting only as one-way feedback, that remains sufficiently close to the original co-evolutionary process. To treat the more general setting in which edge dynamics are affected by the proportions of susceptible and infected individuals, we introduce a methodological extension of existing techniques. We thus show that our model exhibits multiple epidemic peaks -- a phenomenon observed in real-world epidemics -- which can emerge in models that incorporate mutual feedback between vertex and edge dynamics.

Figures (5)

• Figure 2.1: Illustration of the relation between graph, adjacency matrix, and graphon.
• Figure 2.2: Two graphons with different labellings (left and middle panels), plus the limiting graphon when the labelling of the middle panel has been used (right panel).
• Figure 3.1: Each solid curve represents the numerical solution of the system of PDEs for $\pi_{{\rm S}{\rm S}}=p_0$, $\lambda=4$, $q_0=0.1$, and varying $p_0,\pi_{{\rm S}{\rm I}},\gamma$.
• Figure 3.2: The grey trajectories in the top row, from left to right, correspond to simulations of 100 stochastic SIR trajectories of the fraction of infected individuals for $n=200,500,1000$, respectively. The solid blue curves are the numerical solution of the system of PDEs with the same parameters. Observe that the 'cloud' of simulated trajectories shrinks with $n$, reflecting the convergence to the deterministic limiting path. The second row displays the empirical graphons corresponding to the black trajectory ($n=200$) when $t=0.69$ (first peak), $t=1.4$ (valley between the first two peaks) and $t=1.71$ (second peak); a dot represents an edge. The labels of the vertices are updated dynamically so that they are ordered lexicographically, i.e., the vertices with state ${\rm S}$ have lower labels than the vertices with state ${\rm I}$, which in turn have lower labels than the vertices with state ${\rm R}$, and then by increasing type. The third row displays the corresponding FLLN. The bottom row displays $\bar{F}_{{200}}(t;x)$ (dashed line) corresponding to the black trajectory and $\bar{F}(t;x)$ (solid line) when $t=0.69,1.4,1.71$ for $x\in[0,1]$.
• Figure 3.3: An illustration of $\phi(t)$ for $t\in[0,5]$: the black and the blue lines correspond to the empirical black trajectory and the smooth blue curve, respectively, of the top left panel in Figure \ref{['fig:doublepeak']}.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• Lemma 5.1
• proof
• Lemma 5.2
• proof
• Lemma A.1
• proof