key: cord-0313003-wx5mar2i
authors: Nielsen, B. F.; Eilersen, A.; Simonsen, L.; Sneppen, K.
title: Lockdowns exert selection pressure on overdispersion of SARS-CoV-2 variants
date: 2021-07-06
journal: nan
DOI: 10.1101/2021.06.30.21259771
sha: 7d09205c7b44be3422d0401a1eafbf6121a14b6d
doc_id: 313003
cord_uid: wx5mar2i

The SARS-CoV-2 ancestral strain has caused pronounced superspreading events, reflecting a disease characterized by overdispersion, where about 10% of infected people causes 80% of infections. New variants of the disease have different person-to-person variations in viral load, suggesting for example that the Alpha (B.1.1.7) variant is more infectious but relatively less prone to superspreading. Meanwhile, mitigation of the pandemic has focused on limiting social contacts (lockdowns, regulations on gatherings) and decreasing transmission risk through mask wearing and social distancing. Using a mathematical model, we show that the competitive advantage of disease variants may heavily depend on the restrictions imposed. In particular, we find that lockdowns exert an evolutionary pressure which favours variants with lower levels of overdispersion. We find that overdispersion is an evolutionarily unstable trait, with a tendency for more homogeneously spreading variants to eventually dominate.

to be seen how this reduced variability affects the transmission 23 patterns of the virus. 24 The altered viral load distributions seen in persons in- or not they also have tyrosine at the 501 position. However, 36 the difference in variance was most pronounced for those sam-37 ples which had the deletion as well as the 501Y mutation. 38 Similarly, an analysis of samples with the N501Y mutation 39 show that they have a higher median viral load as well as a In this paper, we use a mathematical model to study the 51 competition between idealized variants which differ in their 52 level of overdispersion (k) and their mean infectiousness. Our 53 focus is on exploring whether overdispersion confers any evo-54 lutionary (dis)advantages, and whether non-pharmaceutical 55 interventions which restrict social network size and transmis-56 sibility change the fitness landscape for variants with varying 57 degrees of overdispersion. While it is evident that a higher 58 mean infectiousness confers an evolutionary advantage to an 59 emerging pathogen, it is not a priori obvious if a competitive 60 Significance One of the most important and complex properties of viral pathogens is their ability to mutate. The SARS-CoV-2 pandemic has been characterized by overdispersion -a propensity for superspreading, which means that around 10% of those who become infected cause 80% of infections. However, evidence is mounting that this is not a stable property of the virus and that the Alpha variant spreads more homogeneously. We use a mathematical model to show that lockdowns exert a selection pressure, driving the pathogen towards more homogeneous transmission. In general, we highlight the importance of understanding how non-pharmaceutical interventions exert evolutionary pressure on pathogens. Our results imply that overdispersion should be taken into account when assessing the transmissibility of emerging variants. Survival chance (%)

The epidemic spreads in an unrestricted setting (homogeneous mixing contact structure) B) The epidemic spreads in a situation with limited social connectivity (modeled as an Erdos-Renyi network of average connectivity 10). The survival chance is computed by simulating several outbreaks, each starting from a single infected individual in a susceptible population. This initial individual is infected with a variant of a given overdispersion. For each outbreak, the variant is recorded as having survived if it does not go extinct within 10 generations. The dashed white line indicated parameters for which the variant has a 5% chance of surviving. The biological mean infectiousness (horizontal axis) has been scaled such that it equals the basic reproductive number (R0) in the homogeneous mixing scenario of panel A. For details on these calculations, see the Materials and Methods section.

when all contacts are susceptible. This is in contrast to the 121 effective reproductive number (known variously as R, Rt and 122 Re), which is affected by population immunity. Note that R0 123 as well as Re are context dependent, since behaviour (and 124 mitigation strategies) will affect e.g. the number of contacts 125 that a person has and thus the reproductive number. Another 126 parameter entirely is the (biological) mean infectiousness, by 127 which we mean the rate at which transmission occurs when an 128 infected person is in contact with a susceptible person. This is 129 a property of the disease and not of the social environment. In 130 Fig. 1 , the independent variables are thus the mean infectious-131 ness and the dispersion parameter, both of which are assumed 132 to be properties of the disease. The details of the calculation 133 can be found in the Materials and Methods section.

In the unmitigated scenario (Fig. 1A) , the procedure is rel-135 atively straightforward. A single infected individual is initially 136 introduced, with a personal reproductive number z drawn from 137 a negative binomial distribution P NB [Z; R0, k] with mean value 138 R0 and dispersion parameter k. Thus, this individual gives 139 rise to z new cases, and the algorithm is reiterated for each of 140 these subsequent infections.

In the case of a lockdown scenario, in terms of restrictions 142 of the number of social contacts (Fig. 1B) , the algorithm is 143 slightly more involved. In this case, a degree c (the number of 144 contacts) is first drawn from a degree distribution (in this case 145 2 | Nielsen et al.

. CC-BY 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 6, 2021. the probability that a single infected individual will infect i 160 others). Then the extinction risk d is the sum:

where the first term on the right hand side is the extinction until one below the threshold is obtained. Since the disease is 203 highly heterogeneous, this process is analogous to "removing" 204 a potential superspreading event and replacing it with a much 205 lower personal reproductive number (typically z = 0). This is 206 exactly why the intervention is rightly called targeted. Their 207 approach is thus based on viewing superspreading entirely as 208 an event-based phenomenon, where one can directly remove 209 superspreading events above some threshold size, and instead 210 let the individuals take part in other less risky events. Our 211 approach, on the other hand, assumes superspreading to be 212 due to a combination of high individual biological infectious-213 ness and opportunity, e.g. a large number of social contacts. 214 These two viewpoints are complementary in obtaining a com-215 prehensive description of superspreading phenomena, rather 216 than mutually exclusive (17). . CC-BY 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 6, 2021. ; https://doi.org/10.1101/2021.06.30.21259771 doi: medRxiv preprint have already managed to gain a foothold, and so have moved 220 past the initial risk of stochastic extinction. This is a separate 221 aspect of "fitness", distinct from the initial survival ability 222 described in the last section. Fig. 2 Interventions exert selection pressure 283 As the observed differences in the viral load distributions of 284 the Alpha (B.1.1.7.) variant and the ancestral strain suggest, 285 overdispersion is not a fixed property, but rather one that may 286 evolve over time. Furthermore, the SARS-CoV-2 pathogen 287 has been estimated to mutate at a rate of approximately 2 288 substitutions per genome per month (23), translating to about 289 one mutation per three transmissions. In Fig. 4 , we explore 290 the consequences of overdispersion as an evolving feature of 291 the pathogen. In these simulations, the virus has a mutation 292 probability of 1/3 at each transmission. When it mutates, the 293 overdispersion factor is either increased (by a factor of 3/2) or 294 decreased (by a factor of 2/3). Thus, we assume no drift on 295 the microscopic scale, but one may arise macroscopically due 296 to selection pressure from the environment. It should of course 297 be noted that while the assumed mutation rate is realistic for 298 SARS-CoV-2, many mutations will be neutral and only very 299 few mutations will affect transmission dynamics. As such, the 300 present model will likely overestimate the magnitude of the 301 drift in overdispersion. It is however conceptually robust -302 decreasing the mutation rate merely slows down the drift, but 303 the tendency remains.

In our simulations, we find that there is always a tendency 305 4 | Nielsen et al.

. CC-BY 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 6, 2021. ; https://doi.org/10.1101/2021.06.30.21259771 doi: medRxiv preprint for overdispersion to decrease (i.e. for the k value to increase), leading to more homogeneous disease transmission. This makes 307 sense, since we have already established that heterogeneous 308 disease variants are more likely to undergo stochastic extinc-309 tion (Fig. 1) and that they have a competitive disadvantage 310 as soon as contact structures are anything but well-mixed 311 (Fig. 3) . In the absence of any interventions, the tendency 312 to evolve towards homogeneity is quite weak (Fig. 4A) , but 313 when a partial lockdown is instituted, the picture changes In these simulations, random mutations occur which alter the level of transmission overdispersion in a non-directed fashion. However, external evolutionary pressures are seen to drive the disease towards developing more homogeneous spreading patterns. The filled red curve shows the combined incidence of all strains. The purple curve shows the average dispersion factor k in the infected population (with higher k corresponding to a more homogeneous infectiousness). The shaded purple area shows the 25% and 75% percentiles of the distribution of dispersion factors in the infected population. A) The pathogen evolves in an open society with no restrictions imposed (homogeneous mixing contact structure). B) Partial lockdown, with an average social network connectivity restricted to 15 persons. C) No restrictions on social network, but infectiousness lowered by other means (e.g. face masks).

We use an individual-based (or agent-based) network model of . CC-BY 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 6, 2021. factor of 3/2 (i.e. k → 3k/2) or decreasing it by a factor of 2/3 452 (i.e. k → 2k/3). On a microscopic level, the dispersion level thus 453 performs an unbiased (multiplicative) random walk. The value of 454 this step-size parameter is arbitrarily chosen, and as such the simula-455 tions can only be regarded as qualitative and conceptual. However, 456 although no intrinsic bias is built into the mutation mechanism, 457 external selection pressures may drive the level of overdispersion in 458 the population up or down, as is explored in Fig. 4 . 459 In Fig. 4C , the average infectiousness of the strain is lowered so 460 as to produce an initial growth rate that is identical to that of 4A, 461 namely 11% per day in the first 14 days of the epidemic. 

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