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.
Mathematical models play a crucial role in understanding and managing the spread of infectious diseases. These models offer insight into how various parameters influence the dynamics of an epidemic, which can be instrumental in forecasting outbreaks and designing effective control and mitigation strategies.
The base epidemic model is the Susceptible-Infected-Recovered (SIR) compartmental model. The deterministic version of this model is characterized by a system of differential equations that captures the evolution of the proportion of susceptible, infected, and recovered individuals in the population over time. The same differential equations also appear in the stochastic version of the model, as a large population limit established through a functional law of large numbers (FLLN). In the present paper, we establish such a FLLN for a considerably more general version of the base SIR model.
The basic version of the SIR model relies on simplifying assumptions that are typically not satisfied in practice. One key assumption is homogeneous mixing, which implies that every individual in the population is equally likely to come into contact with every other individual. If we represent each individual in the population as a vertex in a graph and each social connection through which the epidemic can spread as an edge, then the assumption of homogeneous mixing is equivalent to assuming that every edge is present in the graph. However, this assumption overlooks several crucial aspects of real-world social networks during a pandemic. In reality, the network is heterogeneous, reflecting that some individuals have more social connections than others. It is also dynamic, as connections can appear and disappear over time. Moreover, these dynamics are co-evolutionary: the individual epidemiological states (i.e., susceptible, infected, or recovered) are influenced by the structure of the network (since the epidemic spreads through existing connections) and the network itself evolves in response to the epidemic, as individuals may choose to alter their connections to prevent further spread of the disease.
At a more precise level, a realistic model should incorporate two types of co-evolution:
(a) First, the dependence of the network on the individual epidemiological states can be local.
For example, susceptible individuals may choose to break ties with other individuals they suspect to be infected. (b) Second, there can be global feedback mechanisms. For instance, quarantine measures may be introduced once the proportion of infected individuals exceeds a certain threshold.
Ideally, we would work with a model in which the underlying graph is heterogeneous (i.e., not complete), is dynamic (i.e., evolves over time), and exhibits co-evolution (i.e., the network influences the epidemic dynamics and is, in turn, shaped by them -both at local and global scales).
In this paper we introduce a stochastic SIR model that explicitly captures the realistic network features discussed above. We analyze this model in the dense regime in which the number of edges scales roughly as n 2 , with n denoting the number of individuals. Our first goal is to establish a FLLN for the proportion of susceptible, infected, and recovered individuals over time. While our model is inherently more complex, this result is in a similar spirit as the FLLN for the base SIR model. In view of the general co-evolutionary dynamics incorporated in the model, we are also interested in how the network evolves during the pandemic. Our second objective is therefore to establish a FLLN for the dynamic network. The proposed model is capable of reproducing complex real-world phenomena, such as the emergence of multiple infection peaks -patterns that cannot be captured in SIR models without incorporating the co-evolutionary feedback between individual states and network structure. We use our results to gain insight into the evolution of the epidemic and the state of the network as the epidemic evolves.
Our results should be viewed within the context of SIR models on random graphs. We begin by providing a brief overview of this subarea. Broadly, the literature can be categorized into four main groups:
˝A FLLN for the SIR model on a sparse static configuration model was proposed in [37]. This result was subsequently rigorously established in [8], [10], [19], and [26] under progressively weaker assumptions on the distribution of the vertex degrees in the configuration model. A similar result for an SIR model on a sparse static stochastic block model appeared more recently in [13].
˝FLLNs for SIR models on dense static random graphs have been considered in [20,28,34]. In these papers, the random graph through which the epidemic spreads is constructed by sampling from a reference graphon. The results in [34] apply when the process exhibits non-Markovian dynamics but hold only when then number of edges scale as n 2 , whereas the results in [20,28] consider Markovian dyna
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