Pandemic modelling with network structure and behaviour

Pandemic modelling with network structure and behaviour 1

Pandemic modelling with network structure and behaviour: Lockdown rules and the Lucas critique

Non-pharmaceutical interventions (NPIs) are the fundamental public health policies available in an epidemic (Bricongne and Meunier 2021), before the advent of reliable vaccination measures – and even after, in some cases. It is therefore of first order importance to measure the relative effectiveness of different such NPIs in diffusion models of infections (Baldwin 2020). Building on our previous work on the spatial-SIR model (Bisin and Moro 2021), in Bisin and Moro (2022) we present a model that accounts for a demographic and a network structure of the social contacts driving the diffusion of the epidemics. In this model, population and location heterogeneity interact with the spatial dimension of contacts;1 we also account for agents’ and firms’ behavioural responses to the diffusion of the epidemics and to policy interventions.2

Specifically, in our simulations, agents belong to three demographic types: young (students/workers), not employed, and old. Each day, the types potentially visit three locations in a structured network – the city, school/work locations, and home – which differ by contact density. Depending on the spread of the epidemic, contagion-risk averse agents may reduce their social interactions, and firms may instruct their workers to operate remotely according to a rule that depends on the current level of infections in the city.

We calibrate the model to reproduce several stylised facts about the diffusion of SARS-CoV-2 epidemics. Agent types and family composition match the US family size distribution by age. The three locations’ relative densities match the data on the distribution of contacts reported by Mossong et al. (2008). Behavioural responses are as calibrated by Engle et al. (2021). Transition probabilities between states match various SARS-CoV-2 parameters from epidemiological studies, notably Ferguson et al. (2020).3

Several important factors are relevant to understanding the heterogeneous rates of diffusion of the epidemic, highlighting that local herd immunities may form not only geographically (our focus in Bisin and Moro 2021) but also along socio-demographic dimensions depending on the network structure. For instance, in the absence of policy interventions, the city originates more infections than school/work and school/work more than home, since home is the smallest, though densest, location and the city is largest, though the least dense. Herd immunity therefore is achieved earlier at home and school/work locations than in the city. Similarly, the epidemic is primarily concentrated on the young because they are exposed at school/work, where there are more contacts than at home. Our analysis also illustrates the relevance of the indirect effects induced by agents’ and firms’ behavioural responses across locations and demographic types. For instance, the young’s behavioural responses, preventing exposure in school/work locations, have a sizable substantial effect on the old in the city, where most of the old get infected.

We use this model to study the effects of several public health policy interventions, such as lockdown rules, where the public health authorities set thresholds levels of active infection cases determining when social interactions are restricted and when they are allowed. Lockdown rules interact in rather subtle ways with the dynamics of herd immunity. When the lockdown is placed too early with respect to the spread of the epidemic, the fraction of infected is too small, herd immunity is too far out, and a second wave of infections at reopening reaches a peak higher than the first one (see Figure 1, Panel A). If reopening is delayed until a lower fraction of active cases than the one that triggered the lockdown is reached – a strategy we labelled cautious reopening – herd behaviour is more advanced upon reopening, substantially dampening the second wave (see Figure 1, Panel B).

Figure 1 Lockdown rules

Pandemic modelling with network structure and behaviour 2             

Note: The figure illustrates the dynamics of active cases (Asymptomatics (A)+Symptomatics(Y)) as a fraction of the total population under different lockdown policies. Under a lockdown, agents are prevented from visiting the city, and 50% of students/workers from reaching their school/work locations. Lockdown-once (left panel): lockdowns are triggered by reaching the indicated fraction of active cases; reopening never occurs. Cautious reopening (right panel): after lockdowns are triggered, reopening occurs after reaching the indicated lower level of infections. The horizontal lines indicate the proportion of infected triggering a lockdown or a reopening. Steady state outcomes and infections by location are available in Bisin and Moro (2022). 

We notice that the flattening of the infection curve is substantial, and so is the reduction of the total number of infections (dead plus recovered individuals) at the steady state. Importantly, the lockdown has rather large effects on the old (not reported in the table). While either lockdown reduces the fraction of dead at the steady-state of only one-tenth of 1%, this reduction is four to five times as big for the old.

We also simulate the ability of the government to exploit granular information on the epidemics by imposing lockdowns at the neighbourhood level based on the local progression of the infection. Neighbourhood-level lockdown policies can flatten the curve of infections and generate fewer infected at steady state, if compared to a city-wide lockdown (see Figure 2, Panel A). However, the effectiveness of these localised lockdowns crucially depends on how segregated neighbourhoods are, in particular on the level of interaction across neighbourhoods caused by agents visiting school/work locations.

Figure 2 Neighbourhood-specific lockdowns

Pandemic modelling with network structure and behaviour 3             

Note: The figure illustrates the dynamics of active cases (Asymptomatics (A)+Symptomatics(Y)) as a fraction of the total population under a cautious reopening policy (5-2% rule) applied at the neighbourhood level. The green solid line (baseline) corresponds to a policy applied city-wide. In the left panel we simulate policies in a city with different numbers of identical neighbourhoods. In the right panel we simulate a city with neighbourhoods of identical density, but with different family composition, placing families with largest size in the southwest corner, and reducing family size as the location gets closer to the northeast corner. The horizontal lines indicate the proportion of infected triggering a lockdown or a reopening. Steady state outcomes and infections by location are available in Bisin and Moro (2022).

Moreover, we find that when neighbourhoods are heterogeneous in their demographic structure, the location of the initial cluster is a crucial determinant of the infection dynamics and steady-state outcome. When the infection starts in neighbourhoods with large families (proxying for a lower socio-economic status), it affects locally a large fraction of people, achieving quickly a local herd immunity that spares, to some extent, the rest of the city from a higher level of infections (Figure 2, Panel B).

The model’s network and demographic structure also allow us to study various selective lockdown policies, where lockdowns are imposed by the location of social interactions and/or by the agents’ demographic characteristics. We concentrate on two such policies. In the first one (not reported here), we limit the lockdown to the city, and we let firms/schools decide whether to operate remotely. In the second one, we limit the lockdown to the old, a policy aiming at reducing total fatalities while bearing limited economic costs (the old are not economically active but suffer a higher fatality rate than other demographic groups). While these selective lockdowns have positive direct effects in terms of economic costs, indirect effects across locations and/or demographic types could, in principle, substantially limit their advantage over general lockdowns. This outcome does not appear to occur in our simulations. The city-only lockdown does not induce a much larger fraction of infected at steady-state than the general lockdown, while the old-only lockdown is very successful in limiting old agents’ deaths. It is also interesting that the lockdown of the old at home is equivalent in terms of infection and fatality rates to one where the old are locked down in nursing homes with about ten patients each (Figure 3, Panel B). This is a consequence of the indirect interactions across locations hurting the more susceptible demographic group.

Figure 3 Old-only lockdown

Pandemic modelling with network structure and behaviour 4             

Note: The figure on the left illustrates the dynamics of active cases (Asymptomatics (A)+Symptomatics(Y)) as a fraction of the total population, under a cautious lockdown policy (5-2% rule) when old are isolated at their home. The figure on the right illustrates the fractions of infected among all and old (as a fraction of the population) when the old are isolated in nursing homes of different size (the dots represent outcomes when old are isolated at their homes). Steady state outcomes and infections by location and agent type are available in Bisin and Moro (2022).

Last, but not least, we illustrate the implications of what economists refer to as the Lucas Critique in the context of epidemiological models. Policy evaluations and interventions disregarding that agents’ and firms’ “decision rules vary systematically with changes in the structure of series relevant to the decision-maker” (Lucas 1976) might lead to policy decisions that are very costly in terms of their effects on the epidemic dynamics. The costs of disregarding behavioural responses are clearer if we consider that the lockdown and opening thresholds are calibrated to flatten the infection curve to avoid hitting a constraint on the available health care resources. In this case, our simulations imply under-utilisation of health care resources after lockdown (that is, the lockdown will turn out to be stricter than necessary) but possibly an over-run of these resources after reopening as in the examples from Figure 4.

Figure 4 Re-calibrated re-openings

Pandemic modelling with network structure and behaviour 5

Note: The figure illustrates the dynamics of active cases (Asymptomatics (A)+Symptomatics(Y) ) as a fraction of the total population with naive policymaking. Panel (a) illustrates a city-wide lockdown policy at 3.5% city-wide infections, with a target peak at 12.9 reached without assuming behavioural responses (solid green line). The dashed line illustrates results from adjusting the reopening policy at 1.1% infections after reaching the first peak. Panel (b): as in (a), but local lockdowns based on neighbourhood-level infections. Steady state outcomes and infections by location are available in Bisin and Moro (2022).

References

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Aguirregabiria, V, J Gu, Y Luo and P Mira (2020), “A Dynamic Structural Model of Virus Diffusion and Network Production: A First Report”, CEPR Discussion Paper No. 14750.

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Azzimonti, M, A Fogli, F Perri and M Ponder (2020), “Pandemic Control in ECON-EPI Networks”, Federal Reserve Bank of Minneapolis Staff report 609, August.

Baldwin, R (2020), “It’s not exponential: An economist’s view of the epidemiological curve”, VoxEU.org, 12 March.

Baqaee, D, E Farhi, M J Mina and J H Stock (2020a), “Reopening Scenarios”, NBER Working Paper 27244.

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Bisin, A and A Moro (2021), “JUE Insight: Learning Epidemiology by Doing: The Empirical Implications of a Spatial-SIR Model with Behavioral Responses”, Journal of Urban Economics 103368.

Bisin, A and A Moro (2022), “Spatial-SIR with Network Structure and Behavior: Lockdown Rules and the Lucas Critique”, arXiv working paper 2103.13789.

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Ferguson, N, D Laydon, G N Gilani, N Imai, K Ainslie, M Baguelin, S Bhatia, A Boonyasiri, Z C Perez, G Cuomo-Dannenburg and others (2020), “Imperial College COVID-19 Response Team: Impact of non-pharmaceutical interventions (NPIs) to reduceCOVID-19 mortality and healthcare demand”, Imperial College London 2020.

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Lucas, R E (1976), “Econometric policy evaluation: A critique”, Carnegie-Rochester conference series on public policy 1: 19–46.

Mossong, J, N Hens, M Jit, P Beutels, K Auranen, R Mikolajczyk, M Massari, S Salmaso, G S Tomba, J Wallinga and others (2008), “Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases”, PLoS Medicine 5(3), e74.

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Endnotes

1 Along these lines, Acemoglu et al. (2020), Baqaee et al. (2020a), Baqaee et al. (2020b), Bognanni et al. (2020), add a demographic structure like we do; and Azzimonti et al. (2020) adds a network structure. Ellison (2020) allow for heterogeneity of the contact process between subpopulations.

2 We follow Engle et al. (2021); but see also, e.g. Fenichel (2013) and Weitz et al. (2020) in epidemiology and Geoffard and Philipson (1996), Goenka and Liu (2012), Acemoglu et al. (2020), Aguirregabiria et al. (2020), Argente et al. (2020), Bethune and Korinek (2020), Farboodi et al. (2020), Fernandez-Villaverde and Jones (2020), Greenwood et al. (2019), Toxvaerd (2020) in economics.

3 See Bisin and Moro (2022) for details. We acknowledge the uncertainty in the literature concerning many epidemiological parameters pertaining to this epidemic. Our approach is less damaging when aiming at understanding mechanisms and orders of magnitude rather than at precise forecasts.

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