Adaptive Network Modeling of Social Distancing Interventions
AAdaptive Network Modeling of SocialDistancing Interventions
Carl Corcoran *1 and John Michael Clark Department of Mathematics, University of California, Davis, Davis, CA, USA Department of Mathematics, Oklahoma State University, Stillwater, OK, USA
February 16, 2021
Abstract
The COVID-19 pandemic has proved to be one of the most disruptive pub-lic health emergencies in recent memory. Among non-pharmaceutical in-terventions, social distancing and lockdown measures are some of the mostcommon tools employed by governments around the world to combat thedisease. While mathematical models of COVID-19 are ubiquitous, fewhave leveraged network theory in a general way to explain the mechanicsof social distancing. In this paper, we build on existing network models forheterogeneous, clustered networks with random link activation/deletiondynamics to put forth realistic mechanisms of social distancing using piece-wise constant activation/deletion rates. We find our models are capable ofrich qualitative behavior, and offer meaningful insight with relatively fewintervention parameters. In particular, we find that the severity of socialdistancing interventions and when they begin have more impact than howlong it takes for the interventions to take full effect.
The global COVID-19 pandemic has upended modern life an placed an enor-mous epidemiological, economic, and social burden on the world’s resources.The gravity of events has brought the need for epidemiological modeling intosharp focus. As the pandemic spread around the world in the absence of avaccine, non-pharmaceutical interventions including social distancing, quar-antine, and lockdown measures proliferated, and bringing these interventionsinto modeling efforts has remained paramount.In recent years, network-based models of epidemic spread have become anincreasingly popular paradigm (Pastor-Satorras et al, 2015; Kiss et al, 2017), and * Corresponding Author. Email: [email protected] a r X i v : . [ m a t h . D S ] F e b etwork science generally has been recognized for its potential to contributesolutions to the current crisis (Eubank et al, 2020). Most network models repre-sent individuals as nodes in a network, and their contacts as edges connectingthe nodes. Moreover, many models assume that the network is static–that theedges between nodes don’t change over time–and thus the epidemic spreadsfrom node to node across these edges. Among static network models, pairwisemodels (Keeling, 1999; Eames and Keeling, 2002) are both frequently used andwell-studied. Pairwise models track not only the number of nodes in a givenstate, but pairs, triples, and higher order motifs as well (Fig. 1). An advantageof pairwise models in that in their full form, they exactly model (in expecta-tion) the continuous time Markov chain formulation of epidemic spread on anetwork (Taylor et al, 2012).Pairwise models have been successfully applied to a number of disease nat-ural histories and different network types. Two important network featuresthat play a role in the theory of pairwise models are degree heterogeneity andclustering. The degree of a node in a network is the number of edges to whichit is connected, and the degree distribution is the probability distribution ofselecting a random node with a given degree. The degree distribution plays afundamental role in many network models, and is particularly powerful whendescribed as a probability generating function. The clustering coefficient isthe ratio of triangles to connected triples in the network. While clustering inan important component of network structure, it has not widely been incor-porated to pairwise models. We acknowledge two major benefits of degreeheterogeneous, clustered models. First, including both or either as modelingconsideration affects epidemic dynamics in a nontrivial way (House and Keel-ing, 2011; Keeling, 1999) and second, both have been shown to be features ofrealistic contact networks (Read et al, 2008). (a) (b) (c) Figure 1: Diagrams of network structures whose evolution is modeled by thepairwise model: (a) node in state A , (b) pair in state A − B , (c) triple in state A − B − C .Though static networks model some forms of complexity well, an impor-tant aspect of real-world contact networks is that some connections changein response to disease dynamics or public health measures. By relaxing thestatic network assumption, dynamic or “adaptive” network models (Gross andSayama, 2009) can capture both the dynamics of the network and the epi-2emic dynamics on the network. A number of models have been recentlyproposed for that describe a variety of network dynamic processes. Grosset al (2006) introduced a model of edge rewiring, where susceptible individ-uals break connections with infectious individuals and reconnect to suscep-tible individuals at random. Another model for network dynamics is randomlink addition/deletion (Kiss et al, 2012) where individuals break and form newcontacts and constant rates. Their approach is notable for its intuitiveness as asimple dynamic model, and also its use of probability generating functions as atool to describe network dynamics. A related model is link addition/deletionon a fixed network (Tunc et al, 2013; Shkarayev et al, 2014), where individu-als can temporarily deactivate contacts with infectious individuals, and reacti-vate them when their contact is not infectious. While much of the focus of theadaptive network literature has been involved in analyzing the resulting dy-namical systems, particularly for SIS-type diseases, some works have focusedon the role of network dynamics in controlling or mitigating epidemic spread(Youssef and Scoglio, 2013; S´elley et al, 2015).Network models in general offer a compromise between two other com-mon modeling techniques: compartment models and agent-based simulations.They are able to capture more complex contact structure than simple com-partment models, while offering analytical tractability that many agent-basedsimulations lack. Despite this, models of non-pharmaceutical interventionshave tended to favor simulation or compartment models (Ahmed et al, 2018;Davey et al, 2008). In the early stages of the COVID-19 pandemic, complexindividual-based simulations offered major insights about the effectiveness ofnon-pharmaceutical interventions (Ferguson et al, 2020). However, the highcomputational cost can make investigating the impacts of intervention policieswith a large number of parameters a challenging endeavor. Network models,especially those with a relatively small number of equations, can offer broadinsights at reduced cost. While some models of social distancing have in-corporated contact network structure as a major consideration (Valdez et al,2012; Glass et al, 2006), differential equation network models of such inter-ventions are uncommon. Adaptive network models in particular can offer anew perspective on questions surrounding social distancing and other non-pharmaceutical interventions made pressing by the COVID-19 pandemic.In this paper, we develop simple, novel mechanisms to incorporate so-cial distancing into a network model of epidemic spread, using COVID-19 asthe central case study to investigate the impact of a range of interventions.First, we develop a pairwise SEIR model with random link activation/deletiondynamics–that is edges are added and deleted at constant rates independentof the epidemic dynamics on the network. Furthermore, the model incorpo-rates degree heterogeneity and clustering, which offers increased realism oversimpler network or compartment models. To apply the model, we use bipar-tite mixing networks to generate large heterogeneous, clustered contact net-works coupled with disease dynamics given by epidemiological parametersestimated for COVID-19. Next, we develop two mechanisms of social distanc-ing using piecewise constant link activation and deletion rates. The first is a3ingle intervention event, where the average number of contacts decreases, isheld constant, and then recovers; the second allows for multiple interventionswhich restart depending on the prevalence of the disease. While we investigatethe implications of these policies for COVID-19 on a specific type of hetero-geneous, clustered network, both the adaptive network model and the socialdistancing schemes are more generally applicable to a variety of networks andepidemiological parameters. Finally, we consider the public health implica-tions of the latter model, finding that certain intervention parameters are moreimportant than others in achieving an effective reduction in overall infections. To begin construction of the full model, we consider SEIR dynamics on a staticnetwork. Pairwise equations for an SEIR epidemic can be found in Keelinget al (1997) and Rand (1999). Model variables include the expected numberof susceptible, exposed, infectious, and recovered nodes ( [ S ] , [ E ] , [ I ] and [ R ] respectively) as well as the expected number of pairs in each state. For ex-ample, [ SS ] is the expected number of connected pairs of susceptible nodes,while [ SI ] is the expected number of connected pairs of susceptible and in-fectious nodes. The expected number of connected triples is also considered( [ SSI ] , [ ESI ] , [ ISI ] ), though differential equations for these variables are notwritten. The full SEIR pairwise model is ˙[ S ] = − β [ SI ] , (1) ˙[ E ] = β [ SI ] − η [ E ] , (2) ˙[ I ] = η [ E ] − γ [ I ] , (3) ˙[ SS ] = − β [ SSI ] , (4) ˙[ SE ] = β [ SSI ] − β [ ESI ] − η [ SE ] , (5) ˙[ SI ] = η [ SE ] − β [ SI ] − β [ ISI ] − γ [ SI ] , (6) ˙[ EE ] = 2 β [ ESI ] − η [ EE ] , (7) ˙[ EI ] = β [ ISI ] + β [ SI ] + η [ EE ] − ( γ + η )[ EI ] , (8) ˙[ II ] = 2 η [ EI ] − γ [ II ] , (9)where β is the transmission rate, γ is the recovery rate, and η is the rate atwhich exposed individuals become infectious. The nodes and edges also obeyconservation equations N = [ S ] + [ E ] + [ I ] + [ R ] (10)and (cid:104) k (cid:105) N = [ SS ] + [ EE ] + [ II ] + [ RR ]+ 2 ([ SE ] + [ SI ] + [ SR ] + [ EI ] + [ ER ] + [ IR ]) (11)4here N is the number of nodes and (cid:104) k (cid:105) is the average degree of the network.We note that with the conservation equations, we do not need terms of the type [ AR ] to determine the evolution of [ S ] , [ E ] , [ I ] , and [ R ] . The full model requires dynamical equations for triples of the form [ ASI ] and higher order motifs as well, leading to a system that is prohibitively largefor computations. To make the model tractable, we approximate the expectednumber of triples [ ASI ] in terms of pairs and individual nodes, thus closing thesystem (1)-(9). An approximation of this kind is referred to as a triple closure.For triples of the type A − S − I, House and Keeling (2011) give a triple closureapproximation as [ ASI ] ≈ [ AS ][ SI ] (cid:80) k ( k − k )[ S k ]( (cid:80) k k [ S k ]) (cid:18) − φ + φ N (cid:104) k (cid:105) [ AI ]( (cid:80) k k [ A k ]) ( (cid:80) k k [ I k ]) (cid:19) (12)where [ A k ] is the expected number of nodes in state A with degree k. Usingthe network degree distribution probability generating function and introduc-ing new dynamical variables, they develop an SIR model for heterogeneous,clustered networks. In Appendix A, we derive an analogous heterogeneous,clustered SEIR model complete with link activation and deletion. However,the model complexity induced by (12) is not necessary to accurately capturethe combined epidemic and network dynamics, and thus a simpler triple clo-sure will suffice.A simple yet useful assumption is that degree and state are independent,and thus [ A k ] = p k [ A ] where p k is the proportion of nodes with degree k. Withthis assumption, the resulting triple closure becomes: [ ASI ] ≈ (cid:104) k − k (cid:105)(cid:104) k (cid:105) [ AS ][ SI ][ S ] (cid:18) − φ + φ N (cid:104) k (cid:105) [ AI ][ A ][ I ] (cid:19) , (13)where (cid:104) k (cid:105) = (cid:80) N − k =0 kp k and (cid:104) k − k (cid:105) = (cid:80) N − k =0 ( k − k ) p k . We note that if ahomogeneous degree distribution is assumed, the closure reduces to clusteredclosure from Keeling (1999).With the static model closed, we now incorporate the effects of networkdynamics. Kiss et al (2012) introduced a simple model of network dynamics,termed random link activation/deletion (RLAD). In this model, independentof epidemic dynamics nonexistent edges are added to the network (or acti-vated) at a constant rate α and existing edges are removed from the network(or deleted) at a constant rate ω . Ignoring epidemic spread and looking at theeffects of activation/deletion only, the equation for edges of type [ AA ] is [ ˙ AA ] = α ([ A ]([ A ] − − [ AA ]) − ω [ AA ] (14)and for type [ AB ] we have [ ˙ AB ] = α ([ A ][ B ] − [ AB ]) − ω [ AB ] . (15)Next, we have to consider the effect of activation/deletion on the now time-dependent network quantities: degree distribution moment terms (cid:104) k (cid:105) ( t ) , (cid:104) k − (cid:105) ( t ) and the clustering coefficient φ ( t ) . Following the example of Kiss et al(2012), dynamical equations for the first two can be easily derived by findingthe partial differential equation for the degree distribution generating function g ( x, t ) = N − (cid:88) k =0 p k ( t ) x k . (16)The Kolmogorov equations describe the evolution of p k ( t ) , the proportion ofdegree k nodes at time t :˙ p k = α ( N − k ) p k − − ( α ( N − − k ) + ωk ) p k + ω ( k + 1) p k +1 . (17)With some straightforward algebra, we derive a partial differential equationfor the degree distribution generating function: ∂g∂t = ( x − (cid:18) α ( N − g − ( αx + ω ) ∂g∂x (cid:19) . (18)The network quantities (cid:104) k (cid:105) and (cid:104) k − k (cid:105) can be computed from the generatingfunction as (cid:104) k (cid:105) = g x (1 , t ) and (cid:104) k − k (cid:105) = g xx (1 , t ) . Then, from (18) we derivethe dynamical equations: (cid:104) ˙ k (cid:105) = α ( N − − ( α + ω ) (cid:104) k (cid:105) , (19) (cid:104) ˙ k − k (cid:105) = 2 α ( N − (cid:104) k (cid:105) − α + ω ) (cid:104) k − k (cid:105) . (20)The clustering coefficient is defined as the ratio of triangles to connected triplesin the network. To compute ˙ φ , we start with the Kolmogorov equations for q k ( t ) , the probability that there are k triangles in the network at time t :˙ q k = α ( L − k − q k − − ( α ( L − k ) + 3 ωk ) q k + 3 ω ( k + 1) q k +1 (21)where L = N (cid:104) k − k (cid:105) / is the number of connected triples. From this we derivethe differential equation for the expected number of triangles (cid:104) T (cid:105) as (cid:104) ˙ T (cid:105) = αL − α + ω ) (cid:104) T (cid:105) , (22)and compute the equation for the clustering coefficient φ ( t ) :˙ φ = 3 α − (cid:18) α + ω + 2 α ( N − (cid:104) k (cid:105)(cid:104) k − k (cid:105) (cid:19) φ. (23)Finally, we have a full set of equations for a pairwise SEIR for a heterogeneous,6lustered network with random link activation and deletion: ˙[ S ] = − β [ SI ] , (24) ˙[ E ] = β [ SI ] − η [ E ] , (25) ˙[ I ] = η [ E ] − γ [ I ] , (26) ˙[ SS ] = − β [ SSI ] + α [ S ]([ S ] − − ( α + ω )[ SS ] , (27) ˙[ SE ] = β [ SSI ] − β [ ESI ] − η [ SE ] + α [ S ][ E ] − ( α + ω )[ SE ] , (28) ˙[ SI ] = η [ SE ] − β [ SI ] − β [ ISI ] − γ [ SI ] + α [ S ][ I ] − ( α + ω )[ SI ] , (29) ˙[ EE ] = 2 β [ ESI ] − η [ EE ] + α [ E ]([ E ] − − ( α + ω )[ EE ] , (30) ˙[ EI ] = β [ ISI ] + β [ SI ] + η [ EE ] − ( γ + η )[ EI ]+ α [ E ][ S ] − ( α + ω )[ EI ] , (31) ˙[ II ] = 2 η [ EI ] − γ [ II ] + α [ I ]([ I ] − − ( α + ω )[ II ] , (32) (cid:104) ˙ k (cid:105) = α ( N − − ( α + ω ) (cid:104) k (cid:105) , (33) (cid:104) ˙ k − k (cid:105) = 2 α ( N − (cid:104) k (cid:105) − α + ω ) (cid:104) k − k (cid:105) , (34) ˙ φ = 3 α − (cid:18) α + ω + 2 α ( N − (cid:104) k (cid:105)(cid:104) k − k (cid:105) (cid:19) φ, (35)where [ SSI ] = (cid:104) k − k (cid:105)(cid:104) k (cid:105) [ SS ][ SI ][ S ] (cid:18) − φ + φ N (cid:104) k (cid:105) [ SI ][ S ][ I ] (cid:19) , (36) [ ESI ] = (cid:104) k − k (cid:105)(cid:104) k (cid:105) [ SE ][ SI ][ S ] (cid:18) − φ + φ N (cid:104) k (cid:105) [ EI ][ E ][ I ] (cid:19) , (37) [ ISI ] = (cid:104) k − k (cid:105)(cid:104) k (cid:105) [ SI ] [ S ] (cid:18) − φ + φ N (cid:104) k (cid:105) [ II ][ I ] (cid:19) . (38)To demonstrate the validity of this model, we test it against numerical sim-ulations (Fig. 2) on a heterogeneous, clustered network—the construction ofwhich is described in Section 2.1. Clearly, the model (24)-(35) is in excellentagreement with the simulations. The goal of this paper is to investigate social distancing policies through ran-dom link activation/deletion dynamics, which are controlled by the activationand deletion rates α and ω. Moreover, in building intervention schemes in Sec-tion 3 new parameters are introduced. In order to consistently compare theefficacy of intervention schemes, network and epidemiological parameters areheld the same across schemes. As such, we restrict our attention to a particu-lar heterogeneous, clustered network and epidemiological parameters that areplausible for COVID-19. For completeness, other network types and epidemi-ological parameters are considered in the Appendix B.7 a) (b)
Figure 2: Comparison of the model to simulation. 100 trials were run on aunipartite contact network generated from a bipartite network with Poissondegree distributions and N = 500 , M = 125 , λ = 4 . Initial conditions are [ E ] =[ I ] = 10 , [ S ] = N − , [ R ] = 0 . Epidemiological and network parametersare R = 2 . , η = 1 / , γ = 1 / , α ≈ . × − , ω = 3 . × − . Individualsimulations are shown in light gray with the mean in black. Model results are(a) [ I ]( t ) , red circles, (b) [ R ]( t ) , green circles.A consistent challenge of network models is constructing realistic contactnetworks. In particular, degree heterogeneity and significant clustering are ob-served in real world social networks (Read et al, 2008). To construct such a con-tact network, we consider a bipartite mixing network (Eubank et al, 2004) with N individuals and M mixing locations (Fig. 3a). Two individuals are in contactif they both connect to the same mixing location, so we form a contact networkas the unipartite projection of the bipartite mixing network (Fig. 3b). To in-troduce degree heterogeneity, we construct a bipartite mixing network whereboth individuals and mixing locations have Poisson degree distributions. Theaverage individual degree λ and average mixing location degree µ are relatedby N λ = M µ, (39)so only
N, M, and λ are needed to characterize this network. Using generatingfunction techniques (Newman et al, 2001), we compute8 a)(b) (c) Figure 3: Example contact network (b) and its degree distribution (c) gener-ated from a bipartite mixing network (a). Degree distributions for the indi-viduals and mixing locations are Poisson (as described in Section 2.1) with N = 200 , M = 50 , and λ = 2 . 9 k (cid:105) = NM λ , (40) (cid:104) k − k (cid:105) = (cid:18) NM (cid:19) λ ( λ + 1) , (41) φ = 1 λ + 1 , (42)for the unipartite contact network, which exhibits both degree heterogeneity(Fig. 3c) and clustering. Unless otherwise specified, the networks in this articleare generated using N = 10 , , M = 2 , , and λ = 4 . We acknowledgethat though we use a bipartite mixing network to generate a heterogeneous,clustered unipartite contact network, our network dynamics are limited to thecontact network. Mobility networks (Chang et al, 2021) have been used to greateffect for COVID-19, and suggest a fruitful path forward for bipartite networkdynamics.Numerous recent studies have estimated important epidemiological quan-tities for the spread of Sars-CoV-2, including the length of the incubation pe-riod, the length of the infectious period, and the basic reproduction number R . We choose the plausible estimates in line with recent studies: averageincubation period of 5 days (Linton et al, 2020; Zhang et al, 2020), averageinfectious period of 10 days (You et al, 2020), and R = 2 . (Li et al, 2020;Anastassopoulou et al, 2020). To incorporate these into the model, we note that /η and /γ are the average lengths of the incubation and infectious periodsrespectively, and thus η = 0 . , γ = 0 . . We do not derive R for the model (24)-(35), but instead consider the basic reproduction number for a heterogeneous,clustered population from Miller (2009), which is given as the series R = (cid:104) k − k (cid:105)(cid:104) k (cid:105) ββ + γ − φ (cid:104) k − k (cid:105)(cid:104) k (cid:105) (cid:18) ββ + γ (cid:19) + . . . (43)Ignoring higher order terms, we can compute β from (43) when R = 2 . Withthese parameters, we plausibly model the spread of COVID-19 through a mod-erately sized heterogeneous, clustered population in the following sections,while introducing various social distancing interventions to mitigate or con-trol the epidemic.
Social distancing and lockdown measures have been used to curb the spreadof infectious diseases throughout history, and are some of the most ubiquitousnon-pharmaceutical interventions in the current COVID-19 pandemic. Manycompartment-based models that incorporate social distancing do so in a phe-nomenological manner through the transmission rate, but adaptive networkmodels present an opportunity to describe a social distancing mechanism in10 imple InterventionPrevalence-Dependent Intervention
Infections increase through threshold Intervention Implemented Intervention in Place Intervention Relaxed
Impact
Infections decrease through thresholdIntervention Implemented Intervention in Place Intervention Relaxed
Impact
Infections increase through threshold Infections increase through threshold
Figure 4: Schematic of the Simple and Prevalence-Dependent Interventions.Both interventions are triggered by a threshold condition, and proceed throughthe described intervention until the epidemic ends and the impacts of the in-terventions can be evaluated.a fundamental way. A simple model of such interventions can be naturallycharacterized by the link activation/deletion process. During periods of socialdistancing and lockdown, individual contacts break; during periods of relax-ation of the measures, individual contacts form. In this section, we develop twosocial distancing schemes (Fig. 4). Both social distancing schemes begin whenthe prevalence [ I ]( t ) reaches some specified threshold level. For the simpleintervention scheme, contacts break as the intervention is implemented, thencontacts stay fixed as the intervention is in place, and finally contacts form untilthey reach their pre-intervention levels. The prevalence-dependent scheme un-folds similarly, but with two notable differences. First, after the intervention,contacts do not start forming again until the prevalence has dropped belowthe threshold. Second, any time the prevalence reaches the threshold again,the intervention restarts. This allows for multiple implementations of a socialdistancing intervention throughout the course of the epidemic.Critically, we do not treat these schemes as a mere modeling exercise, butare interested in the impact of each intervention scheme at the end of the epi-demic. We develop two simple metrics to evaluate the effectiveness of thesimple and prevalence-dependent interventions. First, we consider each inter-vention’s ability to reduce the cumulative number of infections, known as thefinal size of the epidemic. Second, we also consider how many infections oc-cur above the threshold value for prevalence. These two measures reflect twodifferent yet crucial public health goals, and do not necessarily agree on whichinterventions are the most effective. Both must be considered to get a complete11icture of an intervention’s impact. In this section, we derive these two metricsmathematically, and describe the simple and prevalence-dependent interven-tions while assessing their overall effects. The first measure of intervention effectiveness we introduce is the
Relative Changein the Final Size (RCFS). The final size of an epidemic is the cumulative numberof infections that occur over the course of the epidemic. In terms of the model,the final size can be found as the limiting value of the recovered individuals [ R ] : lim t →∞ [ R ]( t ) = R ∞ . We compare the final size of the epidemic with no intervention R ∞ to the fi-nal size where an intervention has been implemented R int ∞ . We then define theRCFS as RCFS = R int ∞ − R ∞ R ∞ . (44)An effective intervention will lead to a decrease in final size, so an RCFS near 0is unsuccessful, while an RCFS near − is extraordinarily successful. However,it is important to note that for brief, intense intervention schemes, it is possi-ble that the final size actually increases. In this case, the network parameterschange quickly, before significant disease spread, so the epidemic unfolds on afundamentally different static network.While the relative change in the final size provides an overall measure of theeffectiveness of interventions, reducing cumulative infections alone is not theonly public health goal that an intervention scheme might seek to accomplish.In some schemes, a large number of infections occur above the threshold de-spite a large reduction in the final size of the epidemic. This can be particularlypernicious if the threshold represents some fixed resource such as healthcarecapacity, where a large number of infections above the threshold could lead tohigher mortality and other negative outcomes. To account for this, we computethe Cumulative Infections Above Threshold (CIAT). Let t , t , ... be the sequence oftimes when [ I ] = qN. Assuming [ ˙ I ] (cid:54) = 0 at any time in the sequence, the conti-nuity of [ I ]( t ) implies that the prevalence is above the threshold on the intervals [ t i − , t i ] for i = 1 , , , ... Thus, the CIAT may be defined asCIAT = (cid:88) i (cid:90) t i t i − [ I ]( t ) − qN dt. (45)We note that the units of CIAT are person-time–for a metric with units of pop-ulation, we compute the Average Infections Above Threshold (AIAT):AIAT = CIAT (cid:80) i t i − t i − (46)12sing the relation [ ˙ R ] = γ [ I ] , equation (46) becomesAIAT = (cid:80) i [ R ]( t i ) − [ R ]( t i − ) γ (cid:80) i t i − t i − − qN, (47)which is convenient for computations. For a simple model of social distancing, we consider a scheme that unfoldsin three successive phases, each with variable length. The effects of the inter-vention scheme on the contact network are characterized through the averagenumber of contacts (cid:104) k (cid:105) ( t ) . The intensity of the intervention can be thought ofas how severely the average number of contacts are reduced, so we introduce aseverity parameter p ∈ [0 , . The top panel of Fig. 5 shows how the (cid:104) k (cid:105) changesFigure 5: Simple Intervention. Once the intervention begins, edges are deletedat rate ω ∗ for L I days until the average number of contacts (cid:104) k (cid:105) drops to p (cid:104) k (cid:105) . For the next L H days, no changes are made to the nework. Then, edges areadded at rate α ∗ for L R days, until the average number of contacts (cid:104) k (cid:105) increasesto back to (cid:104) k (cid:105) over time as the result of the intervention. In the first phase, as social distanc-ing measures are put into place, the average number of contacts decreases fromits pre-intervention level (cid:104) k (cid:105) to p (cid:104) k (cid:105) . In the second phase, with the measuresfully in place, the average number of contacts remains constant at p (cid:104) k (cid:105) . In the13hird phase, social distancing measures are relaxed and the average number ofcontacts increases to its pre-intervention level (cid:104) k (cid:105) .To achieve this effect in the evolution of the average number of contacts,we consider link activation rate α ( t ) and deletion rate ω ( t ) functions that arepiecewise constant. These rate functions can be seen in the bottom two panelsof Fig. 5. Since contacts are only broken in the first phase, ω ( t ) = ω ∗ in thefirst phase and otherwise. Since contacts are only formed in the third phase, α ( t ) = α ∗ in the third phase and otherwise. As the dynamical equation for (cid:104) k (cid:105) (33) is a first-order linear ODE, the resulting curve for (cid:104) k (cid:105) ( t ) will be piecewiseexponential, and the values of α ∗ and ω ∗ are easily computed for a given p. Other than p, four other parameters characterize the simple intervention: thelengths of the three phases L I , L H , and L R , and the threshold proportion of thepopulation q ∈ [0 , to initiate the intervention. The full simple interventionscheme can be described as follows:• No intervention: the epidemic spreads unabated until [ I ] increases through qN ( α = ω = 0 ).• Intervention Phase (length L I ): intervention occurs, edges are removedat a constant rate ( α = 0 , ω = ω ∗ ).• Holding Phase (length L H ): intervention holds, edges are neither re-moved nor added ( α = ω = 0 ).• Relaxation Phase (length L R ): interventions are relaxed, edges are addedat a constant rate ( α = α ∗ , ω = 0) .As this scheme requires five “intervention” parameters, p, q, L I , L H , and L R ,exploring the full impact of the interventions is difficult. To better see the ef-fects, we consider an example scheme where we fix two parameter values ineach and allow the other three to vary. To focus on the impact of the severityparameter p and the lengths of the intervention and relaxation phases L I and L R , we set L H = 15 and q = 0 . for the remainder of this section. Thus,the intervention begins when infections reach one percent of the population,and the holding phase is fixed at 15 days for all interventions. The other threeparameters are allowed to vary. This allows for both abrupt and gradual im-plementations of interventions and relaxation of measures, and different levelsof intervention intensity. Fig. 6 shows the prevalence of some example inter-vention schemes, showing rich qualitative behavior. To assess the effectivenessof the simple intervention we plot the RCFS and the AIAT for a large numberof parameter combinations. We allow the lengths of both the intervention andrelaxation periods L I and L R to vary from to days, and consider threedifferent intensities p = 0 . , . , . . The results are shown in Fig. 7.A significant common feature of the plots in Fig. 7a is a qualitative bound-ary (solid white) that divides ( L I , L R ) space into two distinct classes of theresulting infection curve (for p = 0 . , this occurs outside the boundaries ofthe plot). To the right of the boundary, infection curves are characterized by a14 a) (b) (c) Figure 6: Example infection curves [ I ]( t ) for the simple intervention with q =0 . . The other intervention parameters are (a) p = 0 . , L I = 30 , L R = 90 ,(b) p = 0 . , L I = 60 , L R = 60 , and (c) p = 0 . , L I = 15 , L R = 150 . Solidorange curves are [ I ]( t ) under the intervention, while dashed orange curvesare [ I ]( t ) without any intervention. Gray dashed lines denote the starts of theintervention, holding, and relaxation periods.single “uniform spike,” defined by an prevalence curve [ I ]( t ) with two inflec-tion points and a single local maximum (Fig. 8a). To the left of the boundary,infection curves take the form of either a single “non-uniform spike” (Fig. 8b),with more than two inflection points but only one local maximum, or multiplespikes (Fig. 8c), with more than two inflection points and multiple local max-ima. For p = 0 . and p = 0 . , only multiple spikes occur to the left of theboundary. For small L I , the first spike is small and the second spike is large,and occasionally the final size of the epidemic surpasses the static case due tonetwork alterations. As L I approaches the qualitative boundary, the secondspike becomes shorter and occurs later until negligible. This phenomenon canalso be seen in Fig. 7b: as L I increases, the AIAT decreases until the secondspike drops below the threshold qN, at which point the AIAT increases as thefirst spike grows taller. For p = 0 . on the other hand, both nonuniform spikesand multiple spikes are possible to the left of the boundary. Multiple spikes oc-cur in the region of ( L I , L R ) space enclosed by the dashed white curve, whilea single nonuniform spike occurs elsewhere left of the qualitative boundary.A few other observations warrant comment. First, the length of the in-tervention L I appears to be more important in determining epidemic’s finalsize compared to L R . This is intuitive, as the most significant changes to net-work structure occur during the intervention phase. Second, as p increases,the qualitative boundary shifts generally left. This means that for less severeinterventions, single uniform spikes will occur for smaller L I values. This ob-servation carries weight for repeated interventions, explored in Section 3.3, assingle uniform spikes are heavily penalized by the AIAT. Third, nonuniformspikes occur for p = 0 . , but not for p = 0 . or p = 0 . . We hypothesize thatthere may exist some threshold p ∗ where nonuniform spikes don’t occur below p ∗ , but do above p ∗ . a)(b) Figure 7: Plots of the RCFS (a) and AIAT (b) for the L H = 15 and q = 0 . .For intensities p = 0 . , . , and . , the intervention period and relaxationperiod lengths L I and L R vary from to days. In (a), the solid white curvedenotes the qualitative boundary, to the right of which uniform spikes occur.The dashed white line in the third panel denotes the boundary of the regionwhere two spikes occur. (a) (b) (c) Figure 8: Types of infection curves with the simple intervention: (a) uniformspike, (b) non-uniform spike, (c) multiple spikes. Black dots denote inflectionpoints. 16 .3 Prevalence-Dependent Intervention
While the simple intervention scheme provides a simple yet general model ofsocial distancing, its implementation lacks a degree of realism. Interventionsare put into place only once, and the epidemic continues, often with infectionsspiking after measures begin to relax. In reality, we would expect public healthmeasures to be responsive to rising prevalence. Moreover, continued inter-ventions might be triggered by some indicator, such as case numbers, deaths,hospital capacity, etc... In this section, we adapt the intervention scheme fromSection 3.2 so that it may be reimplemented when a prevalence-based con-dition is satisfied, forming the prevalence-dependent intervention. We beginFigure 9: Prevalence-Dependent Intervention. The intervention begins when [ I ] = qN, and edges are deleted at a constant rate ω ∗ until (cid:104) k (cid:105) decreases to p (cid:104) k (cid:105) , at which point there is no change to the network until [ I ] drops below thethreshold qN. Then, edges are added at a constant rate α ∗ until (cid:104) k (cid:105) returns to (cid:104) k (cid:105) or [ I ] increases through the threshold qN , at which point the interventionbegins again.with two more realistic assumptions about how a public health response mightunfold. First, interventions are reimplemented any time the prevalence in-creases through some threshold. Second, the relaxation phase of an interven-tion doesn’t begin until the prevalence has dropped below the threshold. Weincorporate these assumptions into a new prevalence-dependent interventionscheme. The scheme is determined by four parameters: q, p, L I , and L R . Asbefore, interventions begin when [ I ] reaches qN , p is the severity of the inter-17ention, and L I and L R are now the maximum lengths of the intervention andrelaxation periods, which determine ω ∗ and α ∗ as in Section 3.2. We can definethe new scheme as follows:• As [ I ] increases through qN, a new intervention is implemented.• Intervention Phase: Once an intervention is implemented, edges are deletedat rate ω = ω ∗ until (cid:104) k (cid:105) = p (cid:104) k (cid:105) . • Holding Phase: At the end of the intervention period, a holding periodbegins ( α = ω = 0 ) until the prevalence has dropped below the threshold qN. If the prevalence drops below the threshold during the interventionperiod, the holding period has length 0.• Relaxation Phase: Edges are added at rate α = α ∗ until (cid:104) k (cid:105) = (cid:104) k (cid:105) , or anew intervention is implemented.It worth noting that compared to the simple intervention in Section 3.2, the in-tervention, holding, and relaxation phases can all be of variable length. For in-stance, if the average number of contacts (cid:104) k (cid:105) has not rebounded to (cid:104) k (cid:105) by thetime a new implementation begins, the resulting relaxation period is shorterthan L R . Moreover, in the subsequent intervention phase, edges delete until (cid:104) k (cid:105) = p (cid:104) k (cid:105) and the phase is shorter than L I . In sum, while ω ∗ and α ∗ arefixed, the average number of contacts is never less than p (cid:104) k (cid:105) and the effectivelengths of different intervention and relaxation phases may vary. An exam-ple implementation of the prevalence-dependent scheme is shown in Fig. 9,which shows both holding periods of nonzero length as well as interventionand relaxation periods that are shorter than L I and L R respectively. (a) (b) (c) Figure 10: Example infection curves [ I ]( t ) for the prevalence-dependent inter-vention. Parameters shown are (a) q = 0 . , p = 0 . , L I = 60 , L R = 60 , (b) q = 0 . , p = 0 . , L I = 15 , L R = 60 , (c) q = 0 . , p = 0 . , L I = 30 , L R = 120 .Solid orange curves are [ I ]( t ) under the intervention, while dashed orangecurves are [ I ]( t ) without any intervention. Dashed gray lines denote timeswhen [ I ] = qN . 18igure 11: Relative change in final size (RCFS) for the prevalence-dependentintervention. Each plot represents a choice of p and q, with L I and L R on theaxes, ranging from for .A notable feature of the prevalence-dependent intervention is its ability togenerate infection curves with multiple spikes as the epidemic progresses. Ex-amples of this behavior are shown in Fig. 10. To fully explore the intervention,we again consider the RCFS for a variety of parameter combinations. Fig. 11shows the RCFS for different thresholds ( q = 0 . , . , . ) and intensities( p = 0 . , . , . ) as L I and L R both vary from to days. Though notshown, as with the simple intervention each case has a qualitative boundary, tothe right of which infection curves are single, uniform spikes. The most signif-icant departure from the simple intervention though is to the left of the quali-tative boundary. In the simple case, infection curves from this region took theform of either two spikes or a single nonuniform spike. With the prevalence-dependent intervention, the infection curve behavior is richer.The region is characterized by “waves” in the RCFS, particularly for lowervalues of p . The boundaries of these waves can be described by the numberof spikes that occur over the course of the epidemic. Holding L R fixed andincreasing L I through one of these contours helps explain the behavior of theinfection curve in this region (Fig. 12). At the crest, the final spike peaks just19 a) (b) (c) Figure 12: Progression of the infection curve [ I ]( t ) as L I increases, showingthe shrinking of the final spike and the penultimate spike dropping below thethreshold qN . Parameters are q = 0 . , p = 0 . , L R = 90 and L I = 70 (a), (b), (c). Solid orange curves are [ I ]( t ) under the intervention, while dashedorange curves are [ I ]( t ) without any intervention. Dashed gray lines denotetimes when [ I ] = qN .below the threshold qN (Fig. 12a). As L I increases, the final spike occurs laterand peaks lower (Fig. 12b) and the RCFS decreases until the spike vanishes.Then, the penultimate spike becomes the new final spike, peaking just belowthe threshold (Fig. 12c) and the RCFS jumps up as a new wave crests. Thisunderscores a potential limitation of a threshold-based intervention: if a spikedoes not reach the threshold and no intervention occurs, the spike occurs overa longer period of time and more infections accumulate than if the spike hadtriggered an intervention. A practical implication of this observation is thatno spike in infections should go unaddressed by interventions if the goal is toreduce the number of cumulative infections. We also consider the AIAT for thesame parameter combinations (Fig. 13), though the conclusions by this metricare less complex. For any combination of p and q, increasing L I leads to a largerAIAT. This suggests that when considering interventions with the same RCFS,more abrupt interventions (smaller L I ) are preferable. However, an interestingobservation is that the AIAT increases rapidly as the epidemic changes fromthree to two spikes.While Figs. 11 and 13 show the overall behavior of the prevalence-dependentintervention, by considering fixed values of L I and L R and allowing p and q to vary, we get a more pointed perspective on the effectiveness of this typeof intervention. Fig. 14 shows increasingly gradual interventions from left toright with plots of the RCFS and AIAT as p and q vary on the axes. Notably,regardless of L I , low values of p and q are able to produce interventions thatboth greatly decrease the final size of the epidemic, and the average infectionsabove threshold. This suggests that for sufficiently low thresholds ( q ) and suf-ficiently severe decreases in contacts ( p ), the length over which the decreasein contacts occurs ( L I ) does not play an important role in the effectiveness ofinterventions. However, as q or p increases, L I has a more pronounced impact.20igure 13: Average infections above threshold (AIAT) for the prevalence-dependent intervention. Each plot represents a choice of p and q, with L I and L R on the axes, ranging from for .In particular, for low values of p and large values of q, a longer, more gradualintervention can lead to more average infections above threshold. Moreover, astark change in both effectiveness metrics occurs for large values of p, (around p = 0 . for L I = 15 and L I = 30 ). This suggests that if an intervention doesn’treduce average contacts sufficiently, a highly effective intervention isn’t possi-ble, regardless of the other parameter values. In this paper, we have developed a new SEIR model on a network with ran-dom link activation/deletion dynamics. Using piecewise constant activationand deletion rate functions, we propose two simple mechanisms for social dis-tancing interventions. The simple intervention models a single interventionevent, where contacts are decreased over a period of time, stay constant, andthen return to pre-intervention levels. The prevalence-dependent interventionexpands the simple case to more complex scenarios, where interventions can21 a)(b)
Figure 14: Plots of the RCFS (a) and AIAT (b) for the prevalence-dependent in-tervention with L I = 15 , , and L R = 90 as p varies from 0 to 1 and q variesfrom to . . Notably, both measures indicate highly-effective interventionsfor small values of p and q .be reintroduced in the face of rising prevalence. Using the unipartite projec-tion of a bipartite network, and epidemiological parameters representative ofCOVID-19, we examine the effectiveness of a wide range of potential socialdistancing policies on relatively large heterogeneous, clustered networks.Both intervention schemes are shown to capture a wide variety of behaviorsin the prevalence “curve,” which has received considerable attention in bothacademic studies and public health messaging. The simple intervention mani-fests curves with one or two spikes, while the curves for prevalence-dependentintervention can have many more. Moreover, the behavior of the prevalencecurve is consistent across a number of parameters and can be described quali-tatively with success. This is despite the simplicity of social distancing mech-anism introduced by the piecewise constant activation and deletion rates α ( t ) and ω ( t ) , which take on values α ∗ or ω ∗ respectively, or zero. We have notconsidered the cases where the values of α ∗ and ω ∗ may change over time, orwhere α ( t ) and ω ( t ) are not piecewise constant. As such, our model has naturalextensions that may capture an even richer variety of qualitative behaviors.Furthermore, the mechanisms proposed in this paper offer insights intowhat makes for a successful intervention. We have used two metrics as sim-plified public health goals to evaluate the effectiveness of interventions: therelative change in final size (RCFS) and the average infections above thresh-old (AIAT). For the more realistic prevalence-dependent intervention scheme,we find that the most effective interventions come when the threshold num-ber of infections is low and the intervention severely decreases average con-22acts. When these conditions are met, the relative change in the final size isgreatly decreased and the length over which the intervention is implementedhas little impact on the effectiveness. However, even small increases in thethreshold value can greatly impact the effectiveness of interventions. As well,if interventions do not sufficiently reduce contacts (around fifty percent), theyare rendered significantly less effective by both measures.While this is a first foray into the use of adaptive networks to model so-cial distancing for an SEIR disease, we acknowledge some limitations of ourmodel. First, there is a trade-off between complexity of the disease natural his-tory model and the number of equations of the pairwise model; age-structuredmodels or other more complex compartmental models are popular for COVID-19, but added compartments require tracking an increasing number of edgetypes. However, even simple extensions (such as the inclusion of an asymp-tomatic infectious state) present interesting opportunities. Second, while therandom link activation/deletion process is simple to implement, it has someunrealistic features. In particular, in the t → ∞ limit, one can show from the de-gree distribution generating function that the resulting network approaches anErd˝os-R´enyi random graph, with vanishing clustering and an approximatelyPoisson degree distribution. One manifestation of this property is a rapidlydeclining clustering coefficient over time. While the piecewise constant activa-tion and deletion rates mitigate this to an extent, the network resulting fromthese social distancing policies is fundamentally different than the initial net-work state. To overcome this limitation, future investigations might involvenew processes for network dynamics, such as activation/deletion on a fixednetwork or network dynamics on an underlying bipartite mixing network. Acknowledgements
This paper is a continuation of a project that began at the “Dynamics and datain the COVID-19 pandemic” workshop hosted by the American Institute ofMathematics. The authors would like to thank Stephen Schecter, Hans Kaper,and the rest of the workshop staff for their guidance. We also thank Alan Hast-ings for his insightful comments.
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Adaptive SEIR with Complex Closure
In this appendix, we develop an adaptive network SEIR pairwise model forheterogeneous, clustered networks. The model is analogous to the SIR modelfor heterogeneous, clustered networks in House and Keeling (2011) with ran-dom link activation/deletion dynamics included.The generic triple closure (12) proposed by House and Keeling (2011) canbe further developed by introducing a new variable θ ( t ) , the proportion ofedges that have not transmitted the infection. With the observation that [ S k ] = N p k θ k , (12) becomes [ ASI ] ≈ [ AS ][ SI ] g (cid:48)(cid:48) ( θ ) N ( g (cid:48) ( θ )) (cid:18) (1 − φ ) + φg (cid:48) (1) N [ AI ]( (cid:80) k k [ A k ]) ( (cid:80) k k [ I k ]) (cid:19) (A.1)Moreover, we can express (cid:80) k k [ S k ] = N θg (cid:48) ( θ ) , and we introduce auxiliaryvariables Y = (cid:80) k k [ E k ] , Z = (cid:80) k k [ I k ] , and θ . Observing that (cid:80) k k [ A k ] =[ AS ] + [ AE ] + [ AI ] + [ AR ] , it follows that the dynamical equations for Y and Z (without network dynamics) are ˙ Y = β θg (cid:48)(cid:48) ( θ ) g (cid:48) ( θ ) [ SI ] − ηY (A.2) ˙ Z = ηY − γZ (A.3)Now we incorporate the effects of link activation and deletion. Notably, theprobability generating function for the degree distribution is now time- depen-dent, taking the form g ( x, t ) = N − (cid:88) k =0 p k ( t ) x k . (A.4)As a consequence, the ordinary derivatives of g in (A.1)-(A.3) become partialderivatives with respect to x. From (14) and (15), we can derive network- dy-namical versions of (A.2) and (A.3): ˙ Y = β θg xx ( θ, t ) g x ( θ, t ) [ SI ] − ( η + α + ω ) Y + α ( N − E ] (A.5) ˙ Z = ηY − ( γ + α + ω ) Z + α ( N − I ] (A.6)Next, the non-epidemiological network quantities in this model are entirely de-termined by the degree distribution probability generating function g ( x, t ) andthe clustering coefficient φ ( t ) . We can express (23) in terms of the generatingfunction g ( x, t ) as ˙ φ = 3 α − (cid:18) α + ω + 2 α ( N − g x (1 , t ) g xx (1 , t ) (cid:19) φ. (A.7)Equations (A.1),(A.5), and (A.7) require g ( x, t ) and its derivatives explicitly,27hich can be found by solving (18) using the method of characteristics: g ( x, t ) = g (cid:18) ω + αx + ω ( x − e − ( α + ω ) t ω + αx − α ( x − e − ( α + ω ) t (cid:19) (cid:18) ω + αx − α ( x − e − ( α + ω ) t α + ω (cid:19) N − (A.8)where g ( x ) = g ( x, . Finally, we derive the evolution equation for θ ( t ) bydifferentiating ˙[ S ] = N g ( θ ( t ) , t ) and solving for ˙ θ :˙ θ = − β [ SI ] N g x ( θ, t ) − (1 − θ ) (cid:18) αθ + ω − α ( N − g ( θ, t ) g x ( θ, t ) (cid:19) . (A.9)Thus, we arrive at the pairwise SEIR for a heterogeneous, clustered networkwith random link activation and deletion: [ S ] = N g ( θ, t ) (A.10) ˙[ E ] = β [ SI ] − η [ E ] , (A.11) ˙[ I ] = η [ E ] − γ [ I ] , (A.12) ˙[ SS ] = − β [ SSI ] + α [ S ]([ S ] − − ( α + ω )[ SS ] , (A.13) ˙[ SE ] = β [ SSI ] − β [ ESI ] − η [ SE ] + α [ S ][ E ] − ( α + ω )[ SE ] , (A.14) ˙[ SI ] = η [ SE ] − β [ SI ] − β [ ISI ] − γ [ SI ] + α [ S ][ I ] − ( α + ω )[ SI ] , (A.15) ˙[ EE ] = 2 β [ ESI ] − η [ EE ] + α [ E ]([ E ] − − ( α + ω )[ EE ] , (A.16) ˙[ EI ] = β [ ISI ] + β [ SI ] + η [ EE ] − ( γ + η )[ EI ] + α [ E ][ S ] − ( α + ω )[ EI ] , (A.17) ˙[ II ] = 2 η [ EI ] − γ [ II ] + α [ I ]([ I ] − − ( α + ω )[ II ] , (A.18) ˙ Y = β θg xx ( θ, t ) g x ( θ, t ) [ SI ] − ( η + α + ω ) Y + α ( N − E ] , (A.19) ˙ Z = ηY − ( γ + α + ω ) Z + α ( N − I ] , (A.20) ˙ θ = − β [ SI ] N g x ( θ, t ) − (1 − θ ) (cid:18) αθ + ω − α ( N − g ( θ, t ) g x ( θ, t ) (cid:19) , (A.21) ˙ φ = 3 α − (cid:18) α + ω + 2 α ( N − g x (1 , t ) g xx (1 , t ) (cid:19) , φ (A.22)where [ SSI ] = [ SS ][ SI ] g xx ( θ, t ) N ( g x ( θ )) (cid:18) (1 − φ ) + φg (cid:48) (1) [ SI ] θg x ( θ, t ) Z (cid:19) , (A.23) [ ESI ] = [ SE ][ SI ] g xx ( θ, t ) N ( g x ( θ, t )) (cid:18) (1 − φ ) + φg (cid:48) (1) N [ EI ] Y Z (cid:19) , (A.24) [ ISI ] = [ SI ] g xx ( θ, t ) N ( g x ( θ, t )) (cid:18) (1 − φ ) + φg (cid:48) (1) [ II ] θg x ( θ, t ) Y (cid:19) . (A.25)28 Additional Networks and EpidemiologicalParameters
In this appendix, we consider the prevalence-dependent intervention on twoalternative heterogeneous, clustered networks with our COVID-19 parameters R = 2 . , η = 0 . and γ = 0 . . We also consider the case where R has in-creased to on the same networks, as well as the original unipartite projectioncontact network from the main text. The two networks considered are a Watts-Strogatz “small world” network and a power law network with clustering.Both networks consist of N = 10 , nodes, as with the contact network in themain text. For the small world network, (cid:104) k (cid:105) ≈ , (cid:104) k − k (cid:105) ≈ , φ ≈ . ;for the the power law network with clustering, the relevant initial network pa-rameters are (cid:104) k (cid:105) ≈ , (cid:104) k − k (cid:105) ≈ , φ ≈ . . For each network, we report four sets of figures. First, we consider therelative change in final size (RCFS) and that average infections above thresh-old (AIAT) for the prevalence-dependent intervention with q = 0 . and p =0 . , . , . . Each plot represents a combination of these two parameters,and L I and L R vary on the axes from to . Second, we fix L R = 90 and L I = 15 , , and allow p and q to vary on the axes, with p ranging from to and q ranging from to . . (a)(b) Figure B.1: Plots of the RCFS for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed q = 0 . .29 a)(b) Figure B.2: Plots of the AIAT for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed q = 0 . . (a)(b) Figure B.3: Plots of the RCFS for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed L R = 90 .30 a)(b) Figure B.4: Plots of the AIAT for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed L R = 90 .31 a)(b)(c) Figure B.5: Plots of the RCFS for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed q = 0 . .32 a)(b)(c) Figure B.6: Plots of the AIAT for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed q = 0 . .33 a)(b)(c) Figure B.7: Plots of the RCFS for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed L R = 90= 90
In this appendix, we consider the prevalence-dependent intervention on twoalternative heterogeneous, clustered networks with our COVID-19 parameters R = 2 . , η = 0 . and γ = 0 . . We also consider the case where R has in-creased to on the same networks, as well as the original unipartite projectioncontact network from the main text. The two networks considered are a Watts-Strogatz “small world” network and a power law network with clustering.Both networks consist of N = 10 , nodes, as with the contact network in themain text. For the small world network, (cid:104) k (cid:105) ≈ , (cid:104) k − k (cid:105) ≈ , φ ≈ . ;for the the power law network with clustering, the relevant initial network pa-rameters are (cid:104) k (cid:105) ≈ , (cid:104) k − k (cid:105) ≈ , φ ≈ . . For each network, we report four sets of figures. First, we consider therelative change in final size (RCFS) and that average infections above thresh-old (AIAT) for the prevalence-dependent intervention with q = 0 . and p =0 . , . , . . Each plot represents a combination of these two parameters,and L I and L R vary on the axes from to . Second, we fix L R = 90 and L I = 15 , , and allow p and q to vary on the axes, with p ranging from to and q ranging from to . . (a)(b) Figure B.1: Plots of the RCFS for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed q = 0 . .29 a)(b) Figure B.2: Plots of the AIAT for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed q = 0 . . (a)(b) Figure B.3: Plots of the RCFS for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed L R = 90 .30 a)(b) Figure B.4: Plots of the AIAT for the prevalence-dependent intervention onthe (a) small world network and (b) power law network with clustering for R = 2 . , η = 0 . , γ = 0 . and a fixed L R = 90 .31 a)(b)(c) Figure B.5: Plots of the RCFS for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed q = 0 . .32 a)(b)(c) Figure B.6: Plots of the AIAT for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed q = 0 . .33 a)(b)(c) Figure B.7: Plots of the RCFS for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed L R = 90= 90 .34 a)(b)(c) Figure B.8: Plots of the AIAT for the prevalence-dependent intervention on the(a) unipartite projection network, (b) small world network and (c) power lawnetwork with clustering for R = 5 , η = 0 . , γ = 0 . and a fixed L R = 90= 90