Mathematical Modelling of the COVID-19 Epidemic in Northern Ireland in 2020

In this study, we investigate the dynamics of the COVID-19 epidemic in Northern Ireland from 1 st March 2020 up to 25 th December 2020, using several copies of a Susceptible-Exposed-Infectious-Recovered (SEIR) compartmental model, and compare it to a detailed publicly available dataset. We split the data into 10 time intervals and fit the models on the consecutive intervals to the cumulative number of confirmed positive cases on each interval. Using the fitted parameter estimates, we also provide estimates of the reproduction number. We also discuss the limitations and possible extensions of the employed model.


Introduction
The coronavirus disease 2019 , caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was early reported in China mainland. In January 2020, the World Health Organization declared the outbreak a Public Health Emergency of International Concern [1], and a pandemic in March 2020 [2]. At the time of this study, the number of confirmed cases was over 80 million, and the disease caused more than 1.8 million deaths worldwide [3]. Since the early signs of the pandemic, despite the challenges posed by this unknown disease, see [4], several mathematical and statistical models of varying complex-taking into account the social, cultural, demographic, economic and other granular issues that differ between populations yet may dramatically influence the dynamics of any epidemic [16] [17]. Our goal is to take advantage of the detailed publicly available datasets and to capture the history of the pandemic in Northern Ireland by fitting a compartmental model to empirical data. More specifically, we estimate the parameters of a system of ordinary differential equations so that its numerical solution for the cumulative positive cases will reflect the daily reported cumulative number of positive tests. The corresponding numerical solutions enable us not only to estimate the historical number of susceptible, exposed, infectious and recovered individuals but also to make some predictions of these numbers. Hence, the fitted model can be used to monitor and assess the state of the epidemic, its future evolution as well as the impact of specified levels of intervention measures. To the best of our knowledge, this is the first research work assessing the course of the pandemic in Northern Ireland using tools from mathematical epidemiology.

Data
In Northern Ireland (NI), the Department of Health (DoH) publishes daily updates of COVID-19 related data [18]. Although the first reported case of COVID-19 in NI dated from 11 th January 2020, there are no other confirmed cases until 5 th March 2020. Hence, this study focuses on data from 1 st March 2020 up to 25 th December 2020. The rationale behind the choice of this time frame is that 1) either the case reported on 11 th January 2020 had no effect on the course of the epidemic; 2) or there were some individuals, whose number will be estimated by the method describe in Section 3, already carrying the disease without symptoms hence being undetected before 1 st March 2020; 3) new lockdown measures had been introduced on 26 th December 2020; 4) data collection was less consistent between 26 th December 2020 and 31 st December 2020 5) no information is currently available on the effect of the new strains first detected in November 2020.
This study is based on data sets from the aforementioned period, and in particular, the time series of the daily number of confirmed cases i.e., positive tests or the daily incidence of infection, Figure 1. However, the basic model, described below, captures the number of infected at any given time 0 t ≥ , i.e., the prevalence of infection. To overcome this issue, we generated the number of cumulative cases from our data set, and extended the basic model so that it provides the estimation of the cumulative cases at any given time 0 t ≥ . Figure 2 depicts the cumulative number of confirmed cases in    As such, the first interval includes the period before the first nationwide lockdown was introduce in the UK on 27th March 2020. We let i D denote the data from the i th interval i T . For the purpose of this study, the population size of Northern Ireland, N, was kept at the estimated value of 1,893,700 [19] with no incoming or outgoing travel.

The Transmission Model
Compartmental models have been used widely to model the dynamics of various infectious diseases [20] [21]. The most commonly used mathematical descriptions in modelling the early phase of the outbreak of COVID-19 are variants of the SIR model derived in [22]. The SIR model assumes infectiousness immediately after the exposure to the causative agent. However, early studies on epidemiological and clinical features of COVID-19 identified a significant incubation period of 6 days on average, see e.g. [23] [24]. Therefore, in this study, we use different SEIR models on the consecutive time intervals to approximate the course of the pandemic in Northern Ireland in 2020. This approach enables us to model the number of exposed, infected but not yet infectious, individuals in the population. It also enables to capture the continuous growth in the number of new cases when strict intervention measures reduce drastically the contact rate (or the transmission probability). This model does not include several other possible features such as asymptomatic infections, the presence of variants of the virus or vaccination.
An SEIR model has four non-negative state variables, which are described in  Table 1, and the corresponding flow of the disease transmission is depicted in Figure 3.
The disease is transmitted, at a rate 0 β > , from an infectious individual to a susceptible one, who then becomes exposed. Individuals from compartment E move to compartment I at a rate 0 a > , where 1 a is the average incubation period. Subjects from compartment I progress to compartment R at a rate we consider the following model: The number of susceptible individuals at time t in the population.
The number of exposed individuals at time t in the population.
The number of infective individuals at time t in the population.

R(t)
The number of recovered individuals at time t in the population.
with the following initial conditions , , , , .
Moreover, we assume that is independent of i, and the parameters of the model reflect introduced or relaxed measures instantaneously i.e., in contrast with the formulations in [25] [26], we do not assume the exponential decay from one parameter value to the other. The latter assumption is motivated by the implementation of COVID-19 related interventions in NI without any transition period; although we recognise that this is an approximation of the real world behaviour. The transitions between the time intervals, i T , are arbitrary and do not necessarily match the introduction and release of control measures.

Basic Reproduction Number
The basic reproduction number, 0  , is the most commonly used metric to characterize the early phase of a disease outbreak. It quantifies the transmissibility of the disease, and it can be defined as the mean number of secondary infections induced by an infectious individual in a completely susceptible population. In general, if 0 1 >  an epidemic occurs, and the larger the values of 0  the more challenging to bring the outbreak under control. On the other hand, if , no epidemic occurs and the disease dies out. For the model (1), it can be shown that is a threshold number 1 , see [28]. Furthermore, when 0 1 >  , i.e. the occurrence of an outbreak, the system tends to a unique endemic equilibrium. For the model (2), we define 0, However, since we assume complete immunity, then in the subsequent time intervals the population is not fully susceptible anymore. Therefore, in Section 12, we provide the so-called effective reproduction number defined as 1 Although the formulae for Note that, by using the data,

Parameter Fitting
To fit the model (2) to the data, we fit the model variables , to estimate the number of initially exposed yet not infectious to generate the epidemic in NI. All the computations were made using Matlab [32] leveraging the functions ode23 and lsqcurvefit.

Result
Using our model, we present numerical solutions for the model (2) as well as the corresponding fitted parameters. Furthermore, we compare our estimates of 0  with the estimates provided by the DoH [33].

Early Spread of the Disease Pre-Lockdown Period
The DoH reported one positive test on 11 th January 2020. However, there were no additional cases reported until 5 th March 2020 when two positive tests were reported, and from this date onwards, the number of new cases were at least two.
We started our data-fitting with parameters 0 0 , a β and 0 γ as of 1 st March 2020, and we also fit 0 0 E , the number of exposed at 0 t = ; the initial guess for  Table 3, is presented in Figure 4 along with the number of positive cases, 0 D . Figure 5 shows components E, I and R for the numerical solutions of the model (2). In Figure 6, we present the course of the epidemic without lockdown (or any intervention), which results from computing the solution of (2) on 0 T using the parameters and the initial condition described above.    Table 3, 0 i = .

Dynamics of First Wave of the Pandemic
The first wave of the epidemic in NI is captured in the first 6 interval (~175 days) of the outbreak. Components i E , i I and i R of the numerical solutions of (2) for 0, ,5 i =  Table 3

Dynamics of the Epidemic during the Rest 2020
We now focus on the later phase of the epidemic. In Table 3, we provide the fitted parameter values and the estimates of 0,i  for the period of study, whereas for completeness, Table 4 presents the estimates of  Figure 11, whereas the estimates of t  are shown in Figure 12.

Discussion and Future Work
The mathematical modelling of infectious disease spread can have significant utility in providing information and insights that can, if used carefully, facilitate decision making, the implementation of public health strategies as well as allow the use of scarce health care resources, [38] [39]. Criticism of modelling has been levelled, [40], but usually by those who have misunderstood the limitations of modelling: their value is not in long-term-prediction but rather for outlining potential near-future scenarios and how they may be altered by interventions.
Clearly to be effective models need to be robust and tested against real word data.
Moreover, it has become clear that the multidimensional differences between populations (economic, social, cultural, demographic etc) profoundly influence epidemic dynamics [16] [17]. Hence it is crucial, as we have done, to model in P. A. Hall et al.    Table 3, for 0, ,9 i = .  Table 3.
local settings. Northern Ireland is lucky in having a well-developed and robust system for data capture that we have taken advantage of. Using this we successfully fitted SEIR models without vital dynamics to COVID 19 data spanning the Our study clearly highlights the power of mathematical modelling of epidemiological processes. For instance, by fitting only one variable, , c i I , of a sequence of a relatively simple SEIR models to real data, we obtained information about specific features of the disease. For example, by using Table 3, we can estimate that the average incubation time is between 3.4459 and 20 days and that the average recovery time is between 2 and 20 days. However, it is important to note that the mentioned extrema of these estimates, in particular for the infectious period, are the results of the enforced parameter bounds during the fitting process. Nevertheless, the values from these intervals are in good agreement with the widely accepted clinical properties of COVID-19 [34] [35] [36]. In addition, with our modelling approach, assuming only around 7 exposed but no symptomatic individuals at the beginning of the outbreak and clearly importing the disease, we were able to successfully reproduce numerically the evolution of the cumulative number of confirmed positive cases. This fact highlights the necessity of considering strict border control strategies in the case of an emerging pandemic [41], and the crucial role of effective contract tracing methods. Furthermore, the modelling process gives information about variables, the level of exposed and infectious subpopulations at any given time in our case, which are not or partially accessible by current testing methods and strategies. For instance, despite the decreasing number of exposed and infectious individuals, our solutions estimates that, when the easing of the lockdown restrictions was announced, around day 80, the number of exposed and infectious individuals were about 260 and 101 respectively. Although, our computed 0,i  for that and the following period, in agreement with D t  , is below 1 and shows decreasing tendency, shortly after the number of infectious individuals took an upward turn. This might suggest that the reproduction numbers on their own are not sufficient to assess the state of an ongoing pandemic, but in addition, one may wish to consider at least estimates of the number of active infectious individuals in the population.
The vaccination campaign has already been started in NI on 8 th December 2020. When data for level of vaccination become available, we aim to update our model to investigate the effects of the campaign on the dynamics of the outbreak in NI. The minimum level of vaccination, with vaccine giving 100% immunity, to achieve herd immunity is provided 0 1 R > , see [42]. Using which is significantly lower than the estimate (70% -80%) recently reported in the news, [43] As we mentioned in Section 2, there are many aspects of the disease one may wish to address, such as the level asymptomatic infective subpopulation that we currently do not capture. However, the employed model is not designed to capture this variable. In addition, the SEIR model does not reflect on the finer de-

Conclusion
We successfully fitted several copies of a Susceptible-Exposed-Infectious-Recovered (SEIR) compartmental model on consecutive time intervals between 1 st March 2020 and 25 th December 2020 to the cumulative number of confirmed positive cases of COVID-19 in Northern Ireland. We provided estimates of the basic reproduction number and the effective reproduction number. In addition, based on our parameter estimates, we discussed the required level of herd immunity to disrupt the chain of COVID-19 infection in Northern Ireland.