key: cord-0288663-qzt9i6ad
authors: Wren, L.; Best, A.
title: How local interactions impact the dynamics of an epidemic
date: 2020-11-28
journal: nan
DOI: 10.1101/2020.11.24.20237651
sha: 7cbce7693122d60f53aaca54a4de94e76f514dd7
doc_id: 288663
cord_uid: qzt9i6ad

Susceptible-Infected-Recovered (SIR) models have long formed the basis for exploring epidemiological dynamics in a range of contexts, including infectious disease spread in human populations. Classic SIR models take a mean-field assumption, such that a susceptible individual has an equal chance of catching the disease from any infected individual in the population. In reality, spatial and social structure will drive most instances of disease transmission. Here we explore the impacts of including spatial structure in a simple SIR model. In particular we assume individuals live on a square lattice and that contacts can be 'local' (neighour-to-neighbour) or 'global' or a mix of the two. We combine an approximate mathematical model (using a pair approximation) and stochastic simulations to consider the impact of increasingly local interactions on the epidemic. We find that there is a strongly non-linear response, with small degrees of local interaction having little impact, but epidemics with susbtantially lower and later epidemics once interactions are predominantly local. We also show how intervention strategies to impose local interactions on a population must be introduced early if significant impacts are to be seen.

One common approach to incorporating a degree of regular spatial structure, 23 and particularly 'local' near-neighbour interactions, in to infectious disease 24 models is to use a lattice-based probabilistic cellular automata (Sato et al, 25 1994; Rand et al, 1995) . These stochastic individual-based models have also 26 been combined with an analytic pair-approximation method (Matsuda et al, 27 1992; Sato et al, 1994) , where the full spatial dynamics are approximated by a 28 set of ordinary differential equations based on the classic SIR model. Such 29 models have been applied to infectious disease systems both with (Keeling 30 et al, 1997; Webb et al, 2007a,b; Best et al, 2012) and without (Keeling, 1999; 31 Sharkey, 2008) demography. These have found that local interactions reduce 32 the value of R 0 , slowing or even preventing an epidemic that would occur or community (local interactions), but also travelling some distance for work, (Kamo et al, 2007) . 54 Most recently, this multiscale method has been applied to a human 55 epidemiology model with equal births and deaths (Maltz and Fabricius, 2016) , 56 showing that pronounced (but damped) oscillations in infection may result 57 after a sudden shift to local interactions. However, this simple mechanism to 58 investigate the impacts of varying the degree spatial structure has yet to be 59 applied to simple human epidemic models over short-term scales such that 60 demography does not impact the dynamics. 61 In this study we present a combination of a stochastic individual-based model 62 and a pair approximation of epidemics on a lattice. We explore how changing 63 the proportion of local-to-global interactions alters the course of an epidemic 64 and investigate whether increasing the degree of local interactions -which we and McKendrick, 1927) , with no demographic processes (births/deaths). We 71 first consider the model under a mean-field assumption with no local 72 interactions. All individuals in the population are either susceptible (S), 73 infected (I) or recovered (R). The total population size N = S + I + R is 74 constant (assume N = 1 for consistency with what follows), meaning we only 75 need to track the dynamcis of S and I densities, given by the following 76 ordinary differential equations,

Transmission is assumed to be density-dependent with coefficient β, while 78 recovery occurs at rate γ and immunity is assumed to be permanent. To account for spatial structure and local transmission, we use a 81 pair-approximation model (Matsuda et al, 1992) . Assume individuals live on a 82 square lattice, where each site is always occupied by one susceptible, infected 83 or recovered individual. We define the probability that a site is occupied by a 84 susceptible individual as P S , an infected individual as P I and a recovered 85 individual as P R . The dynamics of these 'singlet' densities mirror those of the 86 mean-field model above, with the following ordinary differential equations,

• local: Lβq S/I P I .

This system of equations is not closed, since to calculate the conditional 99 probabilities we need to know the 'pair' density, P SI , e.g. the probability that 100 a randomly chosen pair of neighbouring sites are a susceptible and an infected.

The dynamics of these pair densities are governed by an additional set of 102 ordinary differential equations,

and P RR = 1 − P SS − P II − 2P SI − 2P SR − 2P IR . Again, this system of 104 equations is not closed as we have further conditional probabilities that 105 depend on 'triplets' (e.g. q I/SI = P SII /P SI ). One can appreciate that this 106 pattern will continue and that the equations will never form a closed system.

We thus apply a 'pair approximation' (Matsuda et al, 1992) where we assume 108 that, for example, q I/SI = q I/S , allowing us to close the system. 

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The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20237651 doi: medRxiv preprint mean-field (global) case where L = 0, this is simply given by R 0 = β/γ. When 114 interactions are fully local with L = 1, we have R 0,l = βq S/I /γ. In the limit 115 where the population is indeed entirely disease-free, the environs density 116 q S/I = P S = 1, and the two basic reproductive ratios will be equal. However, 117 in the early stages of an epidemic the environs density q S/I rapidly shrinks as 118 the contact network is formed, meaning it quickly becomes that R 0,l < R 0 , 119 leading to a slower epidemic (Matsuda et al, 1992; Keeling, 1999) . Given the 120 total reproductive ratio will be,

it is clear that the initial growth rate of an epidemic will be slower the greater 122 the degree of local interactions. chosen randomly from the lattice for it to occur to (e.g. recovery requires an 132 infected host to be selected). After an event occurs, the lattice is updated and 133 a new tau-leap calculated for the next event. This approach is fully spatially 134 explicit, unlike the approximation present in the mathematical methods above.

Code for the models are provided as electronic supplementary material. 136 6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

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The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20237651 doi: medRxiv preprint models. As we might expect, when L is small the pair approximation appears 158 to present a reasonable 'average' of the stochastic model runs. As L becomes 159 larger we find that, while the pair approximation often sits within the most 160 central runs, for larger R 0 at least, it tends to predict that the epidemic peak 161 is rather earlier and higher than seen in most of the fully spatially-explicit is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

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% infected by day 300 % infected by day 300 % infected by day 300 % infected at peak Day of peak 198 Obviously, the larger R 0 is, the faster the disease will be able to spread 199 9 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20237651 doi: medRxiv preprint through the population and therefore the faster it will die out (with no 200 susceptible individuals left to infect).

In general, the pair approximation appears to be a good fit to the results from 202 the stochastic model and is almost always within a standard deviation of the 203 mean, but this fit appears to be least good as L approaches 1. The pair 210 We now explore how enforcing movement restrictions, resulting in more 211 localised interactions, might impact the spread of an epidemic. We assume 212 that initially a population has predominantly global interactions (L = 0.1). 213 We then assume that when a threshold of percentage infected (here, 5%) is 214 reached, interactions immediately switch to being predominantly local 215 (L = 0.9) and remain so until the infected percentage returns below the 216 threshold. Figure 3 shows that compared to the case where interactions 217 remain predominantly global throughout (red), if movement restrictions are 218 imposed (blue) the peak of the epidemic is reduced, but less substantially than 219 if interactions had always been predmoniantly local, particularly for the lower 220 R 0 (see figure 1 and table 1). Interestingly, we also see a second wave 221 emerging for lower R 0 once restrictions are lifted since the herd-immunity 222 threshold has not been reached, suggesting further and/or longer restrictions 223 may need to be imposed. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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R 0 = 2 R 0 = 5 L = 0.1 constant 15.8% 48.0% L = 0.9 constant 4.0% 35.3% L Varying 7.6% 36.6% implemented, but the mean of these results is close to the pair approximation.

When the higher L = 1, the PA results change dramatically, due to the 236 emergence of the second peak. As the higher L increases towards 0.9, it can be 237 seen that the impact of the epidemic is mitigated, with the biggest change 238 seen in the peak infected proportion of the population almost halving from 239 L = 0.1 to L = 0.9.

This relative lack of impact is because of the speed with which the lattice becomes correlated in the early stages of an epidemic. The correlation between 11 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20237651 doi: medRxiv preprint S and I sites on the lattice is given by,

At the start of an epidemic with predominantly global interactions then the lattice is uncorrelated and C SI = 1. During the early stages, the correlation rapidly approaches a quasi-equilibrium as the contact network forms (Keeling, 1999) , which we show in the appendix can be approximated as, is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101 https://doi.org/10. /2020 'lockdown' scenarios (for example due to social distancing, regular 282 hand-washing, wearing masks, etc). We found that restrictions that both make interactions led to pronounced (damped) oscillations and significant periodic 300 outbreaks as the system was effectively moved such that it was no longer at its 301 steady state. Further investigation in to the use of movement restrictions as a 302 control mechanism is needed to explore the best strategies. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10.1101/2020.11.24.20237651 doi: medRxiv preprint simulations (Juul et al, 2020), such analytic approximations may provide a 312 useful guide. 313 We have deliberately focussed on the simplest possible epidemic model in this 314 study, with the only two mechanisms being transmission and recovery. This considered to make the model appropriate for specific infections or systems. A 324 standard extension for many disease models is to add an exposed 325 compartment, separating out those that are infected from those that are also 326 infectious (see Keeling and Rohani, 2008) . It may also be instructive to 327 consider the dynamics if immunity to infection wanes over time, since the 328 non-spatial model would then yield an endemic equilibrium, unlike our model.

If we wish to consider a disease persisting over the long-term, we should not 330 only add demographics but also consider seasonal-forcing (Aron and Schartz, 331 1984; Schwartz, 1985; Altizer et al, 2006) . Finally, more realistic spatial and 332 social networks would be needed for any conclusions around CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 28, 2020. ; https://doi.org/10. 1101 /2020 

control of covid-19: a mathematical modelling study