key: cord-0767937-n71fq8q4
authors: Ahmed, D. A.; Ansari, A. R.; Imran, M.; Dingle, K.; Ahmed, N.; Bonsall, M. A.
title: Mechanistic modelling of coronavirus infections and the impact of confined neighbourhoods on a short time scale
date: 2020-07-30
journal: nan
DOI: 10.1101/2020.07.28.20163634
sha: 6c8c1c78a2c1ce5f3d42bbcb7ed4afb03ded78d7
doc_id: 767937
cord_uid: n71fq8q4

Background: To mitigate the spread of the COVID-19 coronavirus, some countries have adopted more stringent non-pharmaceutical interventions in contrast to those widely used (for e.g. the state of Kuwait). In addition to standard practices such as enforcing curfews, social distancing, and closure of non-essential service industries, other non-conventional policies such as the total confinement of highly populated areas has also been implemented. Methods: In this paper, we model the movement of a host population using a mechanistic approach based on random walks, which are either diffusive or super-diffusive. Infections are realised through a contact process, whereby a susceptible host may be infected if in close spatial proximity of the infectious host. Our focus is only on the short-time scale prior to the infectious period, so that no further transmission is assumed. Results: We find that the level of infection depends heavily on the population dynamics, and increases in the case of slow population diffusion, but remains stable for a high or super-diffusive population. Also, we find that the confinement of homogeneous or overcrowded sub-populations has minimal impact in the short term. Conclusions: Our results indicate that on a short time scale, confinement restrictions or complete lock down of whole residential areas may not be effective. Finally, we discuss the possible implications of our findings for total confinement in the context of the current situation in Kuwait.

The novel coronavirus SARS-CoV2 referred to by the World Health Organization in a localized directional persistence [25, 26, 27, 28] . This means that individuals 50 in the short term are more likely to keep moving in the same direction than to 51 perform abrupt turns. In the absence of directional persistence, the CRW reduces 52 to the Simple Random Walk (SRW), which can be considered as a special case, so 53 that the movement is uncorrelated and completely random [29, 30] . In the case of 54 a population of non-interacting individuals, such movement processes are known to 55 be diffusive, particularly at large spatial scales [31, 32] . In movement ecology, the 56 . 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 preprint this version posted July 30, 2020. . https://doi.org/10.1101/2020.07.28.20163634 doi: medRxiv preprint CRW is supported by empirical evidence from animal movement data, and thus fre-57 quently used to model animal movement paths [33, 28, 34, 35] . However, in the case 58 of more complicated movement types, such as that observed for humans, the CRW 59 does not provide an adequate description, but can still serve as a null model. To the 60 best of our knowledge, no epidemiological studies have considered host movement 61 as a CRW -even in disease ecology. 62 Another conceptual tool for modelling movement is the Lévy Walk (LW), where 63 the individual performs short steps forming clusters, with the occasional longer step 64 in between them [36, 37, 38] . If the LW is oriented during the clustering phases, the 65 corresponding movement type is referred to as the Correlated Lévy Walk (CLW). 66 In contrast to the CRW, the movement pattern is much faster, and super-diffusive. 67 It is now generally accepted that some animal species perform LWs [39, 40, 41] , 68 particularly in context-specific scenarios such as foraging, and known to describe 69 an efficient searching strategy where resources are scarce and randomly distributed 70 [42, 43, 44] . Alongside this, there is growing empirical evidence that human move-71 ments may also exhibit Lévy type characteristics. Such inferences have been reached 72 from studies on the daily movement patterns of humans, traces of bank notes, mo- 73 bile phone users' locations and GPS trajectories [45, 46, 47, 48, 49] . Therefore, a 74 LW description could be useful to study a wide variety of challenging issues; such as 75 traffic prediction, urban planning, and in the context of our study, epidemic spread In this paper we use a mechanistic description based on RWs to model the move-81 ment of susceptible and infectious hosts in 2D space. We consider the early stage of 82 epidemic development, during the incubation phase prior to the infectious period, 83 where it is assumed that the virus cannot be further transmitted. We demonstrate 84 how different modes of host movements can lead to varying levels of infections. 85 In addition, we consider various confinement scenarios for both homogeneous and 86 overcrowded populations, where the movement is restricted to a certain area. Thus, 87 we reveal whether confinement is effective in mitigating disease spread, at least on 88 a short time scale.

Random walk framework 91 The movement of a walker in 2D space along a curvilinear path in continuous 92 space-time, x = x(t) = (x(t), y(t)) can be modelled using a discrete time random 

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The copyright holder for this preprint this version posted July 30, 2020. . https://doi.org/10.1101/2020.07.28.20163634 doi: medRxiv preprint where (∆x) i = (∆x i , ∆y i ) is a random step vector for the i th step along the walk.

Any 2D RW can also be described in polar co-ordinates, by expressing the step 101 vector in terms of step lengths l and turning angle θ (i.e. the angle between two 102 consecutive headings), using the transformation:

(2) with inverse transformation:

where atan 2 (∆y, ∆x) is equal to arctan ∆y ∆x for ∆x > 0 and to arctan ∆y ∆x ± π 105 for ∆x < 0. The 2D RW can then be characterized by the statistical properties of 106 the probability distributions of step length λ(l) and turning angle ψ(θ).

Simple random walk 108

The earliest models of movement based on RWs are uncorrelated and unbiased, 109 referred to as simple random walks (SRW). This means that the direction of move-110 ment is independent of previous directions moved and completely random [53, 23] .

For our modelling purposes, we consider each component of the step vector to be 112 distributed as a zero-centered Gaussian distribution with the same scale parameter 113 σ, so that: which quantifies the mobility of the walker [54] .

It can readily be shown that the corresponding step length and turning angle 117 distributions are given by:

where λ(l) is the Rayleigh distribution and ψ(θ) is the uniform distribution ranging 119 from −π to π, see [52] for a derivation. For this step length distribution, the mean 120 step length and second moment is:

Correlated random walk 122 A correlated random walk (CRW) allows for short term directional persistence, so 123 that the movement direction is the same as that of the previous step. For a bal-124 anced CRW, the probability of left and right turns are equal, and the turning angle 125 distribution is now considered as a zero centered symmetric circular distribution.

An example of such, is the von-Mises distribution:

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The copyright holder for this preprint this version posted July 30, 2020. . https://doi.org/10.1101/2020.07.28.20163634 doi: medRxiv preprint where κ is the concentration parameter and I m (κ) is the modified Bessel 1 2π π −π cos(mθ)e κ cos θ dθ. Note that, other types of circular distributions can also 130 be used, for e.g. the wrapped Cauchy or wrapped normal distributions [55] . The 131 mean cosine of the distribution of turning angles quantifies the strength of the short 132 term directional persistence, defined as:

and in the particular case of the von-Mises distribution, this reads:

Note that, the SRW corresponds to a special case of the CRW when c = 0 (or iden- . 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 preprint this version posted July 30, 2020. . https://doi.org/10.1101/2020.07.28.20163634 doi: medRxiv preprint lengths is finite, and therefore the movement process is scale-specific and diffusive prior, is that the end tail decays much more slowly (known as a fat or heavy tail), 154 according to the power law:

where µ is the Lévy exponent. As a result, the walker can execute rare but longer 

Without loss of generality, we choose to rely on the folded Cauchy distribution 164 for step lengths:

which has quadratic decay in the end tail λ(l) ∼ 1 l 2 corresponding to Lévy exponent 166 µ = 2. Alongside this, we consider the distribution of turn angles to be the von 167 Mises distribution, see equation (7) with mean cosine given by equation (9). The 168 case 0 < c ≤ 1 now corresponds to a CLW and c = 0 to a LW.

To compare between a CRW and CLW, with identical distributions of turn an-170 gles, it remains to simply relate λ(l). This can be done by considering the survival 171 probability P(l > L) = δ, i.e. the probability of occurrence of move lengths longer 172 than some characteristic scale length L, and considering δ and L to be the same 173 for both distributions. In addition, by imposing an optimization constraint such as 174 minimizing the L 2 norm, one can compute an optimal value for δ, and therefore a 175 relationship between scale parameters. As an example, to 'fairly' compare between 176 distinct movement types, such as a CRW with step length distribution given by 177 equation (5) and a CLW with distribution given by equation (11), one gets: [58] compute a relationship between distribution scale-parameters, but consider the probability that move lengths do not exceed L, i.e. P(l < L) = = 1 − δ. In either case, the result in equation (12) is the same.

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The copyright holder for this preprint this version posted July 30, 2020. 189

How the population disperses in space can be actualised by modelling the individual 190 movement paths using a n step RW, given by the equation:

and different movement behaviours can be simulated using the movement rules In addition, an infectious host is introduced into the susceptible population at the 201 centre of the domain at location ξ 0 = (0, 0), whose movement is modelled as a RW 202 in 2D space:

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The copyright holder for this preprint this version posted July 30, 2020. to those susceptible individual(s) that are within a close spatial proximity of less 208 than a distance r from the infectious host [21] , with condition:

Since the coronavirus is highly infectious, we assume that the probability of disease is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 30, 2020. Susceptible individuals that come into close contact of either of these infectious hosts are also infected, with interaction radius r = 1. (c)-(d) An alternative scenario is considered where confinement restrictions are imposed, by partitioning the residential area into four neighbourhoods of equal population density. Each sub-population of N = 1000 individuals, and also the corresponding infectious host, is thus confined to each neighbourhood of dimensions d = 250 by d = 250. A 'no-go' boundary condition is also prescribed to the inner boundaries, as described in §. As a result, potential contacts and hence the transmission of the virus is also confined. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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We found that infection levels increased more rapidly for a slowly diffusive sus-280 ceptible population with increase in short term persistence in host movement, but 281 . 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 preprint this version posted July 30, 2020. It is plausible that the exact timing may depend on the population characteristics 319 (movement, spatial structure etc.) and the rate of initial spread, and this would 320 constitute an important research line of enquiry in a future study. Such informa-321 tion is vital for stakeholders (government, health officials, policy makers etc.), as 322 some countries are past their (first) epidemic peak, and a second wave of the pan-323 demic is predicted [80, 81] . One important aspect of this study is that no further 324 disease transmission was assumed once individuals in the susceptible population 325 were infected, which is adequate on a short time scale. However, to investigate the 326 long-term effects, one would need to account for further transmission from newly 327 . 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 preprint this version posted July 30, 2020. i.e. the average number of secondary cases per infectious case in a population [82] .

In terms of contact specifics, we assumed that those individuals who were in close 333 spatial proximity of the infectious host to be instantly infected -which could be 334 justified in the case of highly contagious viruses. A more realistic scenario would 335 assign a transmission probability, so that the virus is transmitted to only a pro-336 portion of those individuals who come into close contact. This would allow disease 337 modellers to identify and quantify 'near misses' and to explore possible alternative 338 epidemic outcomes given shifts in epidemiological parameters [11] . Moreover, if con-339 tact rates and transmission probabilities can be estimated from epidemic/movement 340 data, mechanistic models could prove to provide a powerful modelling framework 341 for a broader category of diseases [11] .

Our findings indicate that infection levels can vary depending on the movement 344 rules that govern host-host movement. In the case of a slowly diffusive susceptible 345 population, the level of infection increases, but remains the same for a highly or 346 super diffusive population. We also found that in the short-term, prior to when 

where the subscript 'a' is included here, to distinguish between the asymptotic 361 MSD and the actual MSD in equation (A1). Note that, the asymptotic MSD grows 362 . 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)

The copyright holder for this preprint this version posted July 30, 2020. . https://doi.org/10.1101/2020.07.28.20163634 doi: medRxiv preprint linearly with n and therefore the RW becomes diffusive in the large step limit, and 363 can be related to the diffusion coefficient D [83, 84, 85, 23, 86] , through the relation:

Another useful metric is the sinuosity index S, which quantifies the amount of 365 turning in a walkers' movement path (tortuosity), defined as:

where u = E[l] ∆t is the mean speed [87]. On combining equations (A4)-(A5), an 367 equivalent expression for the sinuosity index can be written as:

In the particular case of a balanced CRW with Gaussian increments, this index is 369 given by:

where the moments are computed in equation (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)

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