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Edited by: Francesca Fulminante, University of Bristol, United Kingdom

Reviewed by: Paul Bogdan, University of Southern California, United States; Franco Ruzzenenti, University of Groningen, Netherlands

This article was submitted to Social Networks, a section of the journal Frontiers in Human Dynamics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this article, the author studies epidemic diffusion in a spatial compartmental model, where individuals are initially connected in a social or geographical network. As the virus spreads in the network, the structure of interactions between people may endogenously change over time, due to quarantining measures and/or spatial-distancing (SD) policies. The author explores

In the last two years, the still ongoing diffusion of the Coronavirus disease 2019 (COVID-19) pandemic has spurred a large body of scientific contributions, attempting to explore how compartmental models (Keeling and Rohani,

Most of this work has been focusing on models in which the mixing process between people in different states or compartments does not depend on the social or geographical space where they are embedded in. However, some previous literature has shown that the (complex) structure of networks describing the way agents can meet and possibly get infected may affect the dynamics of the epidemic diffusion and its long-run properties (Keeling and Eames,

Motivated by these observations, the paper introduces a generalized spatial susceptible, exposed, infected, recovered, dead (SEIRD) model that, besides the standard four compartments (susceptible, exposed, infected, recovered, dead), also considers an additional “quarantined” state, i.e., a susceptible, exposed, infected, quarantined, recovered, dead (SEIQRD) model (Peng et al., ^{1}

More specifically, the author plays with a finite population of agents (i.e., nodes) initially placed on four different families of interaction structures: (a) regular 2-dimensional lattices with Moore neighborhoods; (b) small-world lattice (Watts and Strogatz,

The author begins by describing a simple model where no SD policies are enforced. Consider a population _{0} = (_{0}), where _{0} is the initial edge list, defined as the set of pairs (_{0} if and only if there exists an edge between _{0}—which, as we will see below, is going to evolve through time as the epidemics spreads—can be considered as describing social or geographical links through which people normally meet friends or neighbors.

At time _{it}, where _{it} = {_{t}} and _{t} is the current edge list. In each time period, transitions between compartments (i.e., states) occur through a parallel updating mechanism according to the following rules:

An agent in state

An agent in state ^{k} if s/he meets _{it}|.^{2}

An agent in the state _{t}). Agents in state ^{3}

An agent in state _{t}), recovers (in state _{t}), or stays quarantined otherwise. Recovered agents are assumed to be immunized and re-establish connections that they used to have in _{0} (provided that neighbors are still alive and are not quarantined).

A flow-chart description of model dynamics is provided in the

The initial network _{0} is assumed to belong to one out of the following graph families:

Regular 2-dimensional boundary-less lattices endowed with the Chebyshev distance (^{LA} ≥ 1 and degrees

Small-worlds lattice (Watts and Strogatz, ^{SW} > 0 and expected average degree ^{SW} ≥ 1 is the interaction radius on the initial ring (

Erdös-Renyi random graphs (Erdos and Renyi, ^{ER} > 0 and expected average degree

Scale-free networks with linear preferential-attachment (Barabasi and Albert, ^{SF} ≥ 1 new nodes, generating an expected average degree

These four graph families have been chosen as they represent the simplest and most widely used network structures employed in the literature. Nevertheless, additional, more complex graph families can be employed to describe the initial social or geographical setup, e.g., graphs displaying self-similarity and multi-fractal patterns (Song et al.,

To summarize network topology, the author focuses, besides average degree, on three statistics that have been found to influence, in general, the spread of epidemics on graphs (Lloyd and Valeika, _{k}), global clustering coefficient (^{LA} ∈ {1, 2, 3, 4} and thus _{k}, _{k} and

All simulations refer to a population of

The epidemic parameters of the model are calibrated using data at the national level for Italy, made available by “Dipartimento della Protezione Civile,” see _{t} = _{t} = _{t} =

As to initial network structures, the author experiments with average degrees ^{LA} ∈ {1, 2, 3, 4}. Therefore, it follows that ^{LA} ∈ {4, 12, 24, 40}, ^{SF} ∈ {4, 12, 24, 40}. Refer to

For each choice of model parameters, the author independently runs

In order to get insights about within-simulation model behavior, the author keeps track of several _{k}, _{0}) of the epidemics. Finally, the author also explores the spatial correlation coefficients of compartments (SCCC), calculating, for each state {

To summarize the aggregate behavior of the model (i.e., across runs), the following set of additional statistics are computed: (i) peak-time of infections (PTI), defined as the first day in which the share of infected people reach its overall maximum; (ii) the shares of agents in states {_{k},

Monte Carlo averages of all the above summarizing statistics will then be compared across initial networks families, initial average degrees, and epidemiological setups.

The author begins studying the dynamic behavior of disease spreading across the four network families, focusing on the “Mid Impact” epidemic scenario with

Within-simulation evolution of agent shares in the six compartments over time. Initial

In these two networks, epidemic diffusion reaches a higher peak of infections than in the case of

As the epidemic process develops over time, the share of

To get a better feel about this coevolutionary process, _{k} and _{k} first increases due to the injection of _{k} follows the same time pattern of

Within-simulation evolution of network metrics, re-scaled to match the [0, 1] interval. _{k});

In the

The author now investigates the behavior of the model when the initial average degree increases in the range of {8, 24, 40, 80}, keeping fixed the epidemic scenario to the “Mid Impact” one. If agents initially have, on average, more neighbors they can meet more infective people. Therefore, the probability to become

For example, as shown in

Furthermore, in all networks, the fraction of infected people at PTI immediately jumps up when

Next, the author explores what happens in the model when alternative epidemic scenarios are assumed (cf. Section 2.3). For the sake of comparison, the author keeps fixed

Comparing the behavior of the model across three epidemic scenarios: Bad vs. Mid vs. Good (see Section 2.3).

Spatial distancing is implemented in the model in a very stylized way (cf. Achterberg et al., _{t}(^{⋆} ∈ (0, 1). The SD policy aims at making more difficult face-to-face meetings between neighbors and can be enforced with increasing strengths. Of course, its _{t}(^{*}, thus avoiding stop-and-go patterns. In the following simulations, the author considers three SD setups: (a) strong: (^{⋆}, θ) = (0.02, 0.7); (b) intermediate: (^{⋆}, θ) = (0.04, 0.5); (c) mild: (^{⋆}, θ) = (0.06, 0.3), whilst keeping fixed throughout

Effects of spatial distancing (SD). Comparing the behavior of the model across three SD setups: Strong vs. Intermediate vs. Bad (see Section 3.4).

This is due to how network structures evolve during a typical run, as shown in

In this article, the author studied a generalized spatial SEIQRD model to explore the impact of alternative social network structures on the diffusion of the COVID-19 disease. The introduction of quarantined agents generates a coevolving process between epidemic spreading and network structure, ultimately shaping steady-state outcomes and the speed of diffusion. In order to make sounder comparisons across different graph structures, the author kept their initial average degree fixed. Therefore, the average degree plays the role of a re-scaling parameter, which is linked to the expected values of the other topological indicators considered (cf.

In the simplest framework, without SD policies and a given benchmark choice of initial average degrees and epidemic parameters, the initial network structure does not affect the final shares of susceptible, dead, and recovered people, but it strongly impacts the timing and the speed of diffusion. In ER and, in particular, in SF networks, more agents become exposed earlier and diffusion takes place quicker and more strongly than in the LA and SW cases. This is linked with how network structure coevolves across time with the shares of

In particular, since the epidemics initially diffuses quicker in ER and SF networks, one typically observes more

The effect of SD policies depends on the strength of the model with which they are enforced, as well as whether they are temporary or permanent. In particular, whereas permanent SD policies allow for better results than temporary ones irrespective of network structure, permanent (and strong) SD measures are more effective in LA and SW structures, whereas temporary (and strong) SD policies should be preferred if interactions occur through ER or SF graphs. This is again due to the interplay between network structure and compartment shares in the evolution of the epidemics. Indeed, switching on and off SD policies may hit the system when the topological properties of its network structure are very different, depending on the initial graph family describing social interactions (cf.

More generally, results suggest that, in order to predict how epidemic phenomena evolve in networked populations, it is not enough to focus on the properties of initial interaction structures. In fact, if the epidemic diffusion requires quarantining people, and possibly enforcing SD policies, the coevolution of network structures and compartment shares strongly shape the way in which the virus spreads into the population, especially in terms of its speed. On the one hand, the average and standard deviation of the degree distribution, as well as clustering, of initial networks are, together with epidemic parameters, important determinants of the subsequent diffusion patterns. On the other hand, the topology of social interaction structures evolves over time, due to the rise and fall of

The foregoing analysis can be improved in several directions. To begin with, alternative parameterizations for the epidemic process, more in line with evidence from the ongoing second wave, could be tested. Furthermore, it would be interesting to assess the extent to which results are robust to increasing population size, additional network structures (e.g., core-periphery graphs), and different values for the share of agents that become initially exposed. In this last respect, one could also play with alternative assumptions as to the mechanism governing the way in which exposures initially occur, e.g., allowing for the emergence of spatially-clustered exposed agents, instead of just supposing that a randomly-chosen share of people gets infected. One can also perform a deeper analysis to better understand how the topology of network structures influences epidemic diffusion, for example, asking whether centrality indicators such as k-coreness measures (Kitsak et al.,

The model, as simple as it is, can be extended to explore additional issues related to epidemic spreading in networked populations. For example, the presence of a non-zero share of susceptible agents at the end of the diffusion process, especially when the initial graph looks like a random graph or a scale-free network (cf.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

GF: wrote the code, performed simulations, and wrote this article.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

The Supplementary Material for this article can be found online at:

^{1}see

^{2}In other words, π is the probability of being infected by at least one infective neighbor in a random sequence of meetings.

^{3}Since I do not distinguish between mild and severe symptoms in the development of the illness, there is not any difference in the model between being quarantined at home or at the hospital.