key: cord-1005941-8zvvh8yg
authors: Kamiya, T.; Alvarez-Iglesias, A.; Ferguson, J.; Murphy, S.; Sofonea, M. T.; Fitz-Simon, N.
title: Estimating time-dependent infectious contact: a multi-strain epidemiological model of SARS-CoV-2 on the island of Ireland
date: 2022-03-25
journal: nan
DOI: 10.1101/2022.03.25.22272942
sha: a062ee5582d65b6245dba2608ada5cb999703787
doc_id: 1005941
cord_uid: 8zvvh8yg

Mathematical modelling plays a key role in understanding and predicting the epidemiological dynamics of infectious diseases. We construct a flexible discrete-time model that incorporates multiple viral strains with different transmissibilities to estimate the changing infectious contact that leads to new infections during the COVID-19 pandemic. Using a Bayesian approach, we fit the model to longitudinal data on hospitalisations with COVID-19 from the Republic of Ireland and Northern Ireland during the first year of the pandemic. We describe the estimated change in infectious contact in the context of government-mandated non-pharmaceutical interventions in the two jurisdictions on the island of Ireland. We also take advantage of the fitted model to conduct counterfactual analyses exploring the impact of lockdown timing and a more transmissible new variant. We found substantial differences in infectious contact between the two jurisdictions during periods of varied restriction easing and December holidays. Our counterfactual analyses reveal that implementing lockdowns earlier would have decreased subsequent hospitalisation substantially in most, but not all cases, and that an introduction of a more transmissible variant - without necessarily being more severe - can cause a large impact on the health care burden.

During an epidemic, behavioural changes are encouraged, and sometimes mandated, 19 to curtail infectious disease transmission. These changes aim to reduce the number of 20 contacts between people indiscriminately (e.g., closure of schools, workplaces, commer- 21 cial establishments, roads, and public transit; restriction of movement; cancellation of 22 public events; maintenance of physical distances in public) and reduce the chance of in- 23 fection upon contact (e.g., use of personal protective equipment). Furthermore, tracing 24 and isolating known infectious cases can limit the contact between infectious and sus-25 ceptible individuals. These actions are collectively referred to as non-pharmaceutical 26 interventions (NPIs) and are often mandated by governments. Slowing the surge of 27 infection (or "flattening the curve") affords an opportunity to reduce infection-induced 28 mortality and mobility, alleviate health care burden and wait out an epidemic until 29 pharmaceutical solutions (i.e. treatment and vaccines) become available. Implemen-30 tation of mandated NPI in historical outbreaks, including during the 1918 influenza 31 pandemic, was crucial for preventing excess death in the United States [1] . NPIs have 32 also been mandated globally during the COVID-19 pandemic. 33 Mathematical modelling and quantitative analyses of empirical data plays a pivotal 34 role in understanding and predicting epidemiological dynamics. Mechanistic epidemi- 35 ological models have been widely applied to study the dynamics of SARS-CoV-2, and 36 to make predictions of clinical outcomes under alternative scenarios (e.g. an assumed 37 decrease in physical contact [2] ). Despite their public health benefits, social distanc- 38 ing measures have been shown to incur high costs in several domains, including in 39 economy [3] , mental health [4] , and civil liberty [5] . Thus, it is crucial to quantify 40 infection contact, or its derivative quantities like the effective reproductive number, 41 R -to monitor changes in infection burden, achieve desired public health outcomes 42 and improve policy transparency and public engagement. While it is not possible to 43 measure infectious contact directly, fitting a mathematical model to longitudinal data 44 on observed processes such as reported cases and hospital admissions allows estimation 45 of inter-individual infectious contact and its derivatives [6, 7] . 46 A large contingency of epidemiological models follows a rich tradition of ordinary 47 differential equation (ODE) models [8] , which track the spread of infection and often 48 immunity in a population. Specifically, ODE models assume that waiting time pro-49 cesses (such as infectious period and time to hospitalisation) are memoryless, that is to 50 say, that the waiting time until an event (such as recovery and hospitalisation) does not 51 depend on the elapsed time. Seen at the population level, this assumption deduces that 52 times spent by individuals in each compartment are distributed exponentially, implying 53 large individual variability. While mathematically convenient, the lack of memory is 54 unsupported for certain epidemiological processes [9] , and empirical evidence indicates 55 other probability distributions with smaller individual variability and non-monotonic 56 densities (e.g., gamma-, Weibull and log-normal distributions) are better equipped to 57 describe those processes. Previous studies have also demonstrated that quantitative 58 predictions of epidemiological outcomes depend on the assumed probability distribu-59 tion in a variety of systems [10] [11] [12] [13] , including SARS-CoV-2 [7] . As such, it is pertinent 60 to incorporate realistic waiting time distributions, particularly when one aims to ob-61 tain quantitative and short-term, rather than qualitative and long-term insights from 62 2 demographic turnover. We ignored the dynamics of recovered hosts who were assumed 106 to have minimal influence on the transmission during the period investigated.

Our model is parameterised by θ, the probability of hospitalisation, ∆ s , the daily 108 probability of infection with strain s, and a discrete random variable H that charac-109 terises a set of probabilities governing daily transitions to hospital. ∆ s is informed by 110 a discrete random variable, Z, that characterises a set of probabilities governing daily 111 transitions into infected compartments, and is defined in Section 2.1.1. Z denotes the 112 time in days from exposure of the infector to exposure of the infectee for a randomly 113 chosen infectee-infector pair (i.e., generation interval), and can be viewed as the av-114 erage relative contribution of each day to the individual reproduction number. The 115 probability that infection occurs at infector age i,

where the cumulative distribution function F Z (i) = P (Z ≤ i).

The event transmission of infection does not affect the individual's stay in the 118 compartment; however, for transition out of the Y compartments, the event going to 119 hospital at infection age i is conditional on still being in the compartment at infection 120 age i − 1. Given H, the random variable representing time from infection to hospital 121 admission in hospitalised patients, we denote by η i the probability of hospitalisation at 122 infection age i given the individual was still not hospitalised at infection age i − 1, that initial conditions as in [7] we derive the following:

4 . 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 March 25, 2022. ; centring Figure 1 : Discrete-time model of SARS-CoV-2 community transmission consists of susceptible S and infectious (J and Y ) compartments of pathogen strain s. Each square represents a group of individuals with an identical contribution to the epidemiological dynamics. Infection with strain s occurs with probability ∆ s per day. Individuals in the components Y are infectious patients to be hospitalised. Once infected, individuals progress to the next square each day (J and Y ), capturing the memory effect of the infection age. After spending n j days, infectious hosts in J are no longer infectious. Alternatively, a fraction, θ of infectious hosts (in Y ), is admitted to the hospital with a delay specified by the probabilities η 1 , ..., η ny , where η i is the probability that the individual is admitted to hospital on the day i, conditional on their being infectious for i − 1 days. The grey arrows indicate the daily transition of individuals from one square to another that occurs with probability 1.

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Similarly, expressions can be derived for more than two strains. 153 It follows then that the number of susceptibles on the next day is expressed as:

Of those exposed to either strain of the infection, the proportion θ will develop 154 severe symptoms and eventually be admitted to the hospital (Fig. 1 ).

For less severe cases that do not result in hospitalisation, J, the infection progresses towards recovery until they are no longer infectious on the day n j :

Those that develop severe symptoms, Y , are admitted to hospital with the probability 156 η i on the i-th day following exposure.

It follows then that the number of hospital admissions on day d + 1 equals . 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 March 25, 2022. ; https://doi.org/10.1101/2022.03.25.22272942 doi: medRxiv preprint respectively by the Central Statistics Office COVID data hub for the ROI [24] , and the Successive invasions of new variants have so far characterised the COVID-19 pan-169 demic. Our study tracks two strains that circulated in the island of Ireland in the 170 first 12 months of the pandemic: i.e., the original strain (initially detected in Wuhan,

We used publicly accessible data on the frequency of the Alpha strain in the ROI [26] , 173 and NI [27] , respectively. 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 March 25, 2022. ; https://doi.org/10.1101/2022.03.25.22272942 doi: medRxiv preprint to the pre-pandemic, pre-intervention baseline (eq. 1 & 2) and estimated this quantity 209 using a piece-wise function consisting of weekly intervals. Specifically, we estimated 210 the ratio in each area a (NI and ROI), per week w (i.e., c a,w ) as a function of ϕ a,w , 211 the log proportional change in the contact rate from the previous week. We index w 212 from the date of the first public health intervention in either jurisdiction, which took 213 place in ROI on 2020-03-12 (Supporting Information S1: Table S1 & S2); hence the 214 preceding, pre-intervention contact ratios are defined as 1.0.

With this formulation, hierarchical Bayesian inference with priors on the ϕ a,w allows 216 us to estimate the time-varying weekly contact ratios with minimal prior information 217 specific to the modelled system. Specifically, we used a prior ∼ N (0, ϵ), where ϵ is 218 a hyperparameter specifying the standard deviation of ϕ, such that c a,w would equal 219 c a,(w−1) in the absence of signals from epidemiological data ( Table 1) . A priori, this for- (Table S2) . Two days later, the first official case in the ROI was also con-230 firmed from a traveller from Northern Italy (Table S1 ). Initially, most known cases 231 are travel-related, and contact tracing may successfully contain infections. As our 232 model solely tracks community transmission, we started our simulations on the first 233 day that community transmission was detected on the island of Ireland: 2020-03-05 234 (Table S1 ). Coincidentally, the exponential growth of confirmed cases appears to have 235 begun around 2020-03-05 in both ROI and NI [24, 31] . We account for the uncertainty 236 of the beginning of community transmission by estimating the initial infectious den-237 sity independently in the two jurisdictions ( 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 March 25, 2022. ; Table 1 : Description of model parameters and their fixed values, or prior distributions used in Bayesian statistical inference. We assigned an informed prior for R 0 , τ 2 and a generic, weakly informative prior forĪ s,a (0), ϵ and measurement error parameters. 

The counterfactual infectious density on day d follows from equation 1, and we denote thisĪ *

where the J * , Y * denote the counterfactual numbers in these compartments on day 

We estimated a rapid decline in infectious contact ratios during the first month of the 314 pandemic before a strict lockdown was implemented ( Fig. 3 ; Tables S1 & S2). The 315 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. 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 March 25, 2022. approximately 70-80% of the pre-pandemic baseline in July, August and September.

In NI, the contact ratio rose to a peak around the end of July. We detected higher 324 infectious contact in NI than the ROI in mid-June (Fig. 3; bottom panel) . Of potential 325 relevance, we note that all non-essential retail outlets were allowed to open earlier in NI 326 than the ROI during this period from 2020-06-12 and 2020-06-29, respectively (Tables 327 S1 and S2). Infectious contact in NI decreased through August but elevated again to 328 about 90% of baseline by the end of September with no parallel increase in the ROI. (Table S1 ). This period coincides with an (Fig 3; green) followed by the first lockdown in late March (Fig 3; yellow) and 350 second lockdown in November (Fig 3; green) . 

Of note, we found that a counterfactual simulation to bring forward the second 368 lockdown date by seven days showed a non-conclusive impact on the cumulative hos-369 pitalisation in the subsequent 30-day period in either jurisdiction (judged by the 95% 370 predictive interval of the difference containing zero; Fig. 4 ). The second lockdown was 371 preceded by a declining trend in contact ratios while the contact during the lockdown 372 remained relatively higher than the first or third lockdown (Fig. 3) . To assess the the impact of the Alpha strain, which arrived later and is more 382 transmissible than the original strain, we compared the fitted model ( Fig. 3; orange) 383 to a counterfactual simulation without the Alpha strain, in which we assumed the 384 same estimated contact ratio ( Fig. 3; blue) . We detected a statistically distinguishable Date lnRR (cumulative hospital admissions due to Alpha) Figure 5 : Counterfactual analysis shows the extent to which the Alpha strain elevated the burden of hospitalisation. To compute the impact of the Alpha strain, the counterfactual simulation (blue) assumes that the Alpha strain never invaded either jurisdiction. The crosses indicate data, and the coloured bands correspond to 95% predictive intervals of the fitted model (orange) and counterfactual scenario (blue), respectively, incorporating uncertainty in parameter estimation and sampling (left panels). The relative difference in cumulative hospital admissions between the two scenarios is estimated as log response ratio (lnRR). The grey band indicates the 95% predictive interval (right panels).

. 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. We presented a generic, epidemic model parameterised for SARS-CoV-2 to fit lon-414 gitudinal hospitalisation data, one of the most reliable and available data types [6] .

Of most relevance to COVID-19 in 2022, our model lacks human age structure and 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) 

. 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) 

. 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. 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 March 25, 2022. ; https://doi.org/10.1101/2022.03.25.22272942 doi: medRxiv preprint

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