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In this study, we investigate the dynamics of the COVID-19 epidemic in Northern Ireland from 1<sup>st</sup> March 2020 up to 25<sup>th</sup> December 2020, using sever al copies of a Susceptible-Exposed-Infectious-Recovered (<i>SEIR</i>) compart mental 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.

The coronavirus disease 2019 (COVID-19), 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 [

The COVID-19 pandemic reached England around late January 2020, and about one month later, it was present in all the four regions of the United Kingdom (UK). The epidemic reached its first peak in the UK in April 2020, and since then various lockdown measures have been introduced and eased. The rationale behind these lockdown measures is to reduce the chances of introducing new infections from outside, to control the spread of the disease domestically and to reduce the pressure on the National Health Service. In the UK, although there is a nation-wide effort to mitigate the challenges posed by COVID-19, major public services affected by the pandemic, including public health services and education, are the responsibility of the devolved administrations, namely England, Scotland, Wales and Northern Ireland. This results in some variations in terms of public health policies, for example, the timing of the launch of test-and- trace strategies and the details of lockdown. This heterogeneity is confirmed not only by the distribution of the levels of the infection across the four regions of the UK but also by their own different roadmap for easing lockdown restrictions after the first wave of the pandemic. For example, England and Northern Ireland moved faster to open some businesses including non-essential retails and restaurants before Wales and Scotland.

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.

In Northern Ireland (NI), the Department of Health (DoH) publishes daily updates of COVID-19 related data [^{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, ^{st} March 2020 to 25^{th} December 2020.

Furthermore, we divide the period from 1^{st} March 2020 to 25^{th} December 2020 into 10 overlapping intervals T i , i = 0, ⋯ ,9 , such that

• the first 9 intervals, T i = [ t i , l , t i , r ] , i = 0 , ⋯ , 8 with t i + 1 , l = t i , r , have an equal length of 30 days; and

• the last interval, T 9 = [ t 9, l , t 9, r ] with t 9 , l = t 8 , r , has a length of 39 days.

As such, the first interval includes the period before the first nationwide lockdown was introduce in the UK on 27th March 2020. We let D i denote the data from the i^{th} interval T i . For the purpose of this study, the population size of Northern Ireland, N, was kept at the estimated value of 1,893,700 [

Compartmental models have been used widely to model the dynamics of various infectious diseases [

An SEIR model has four non-negative state variables, which are described in

The disease is transmitted, at a rate β > 0 , from an infectious individual to a susceptible one, who then becomes exposed. Individuals from compartment E

move to compartment I at a rate a > 0 , where 1 a is the average incubation period. Subjects from compartment I progress to compartment R at a rate γ > 0 , where 1 γ is the average duration of infection. Furthermore, we assume that the

population size N is independent of time, and the exposure to the pathogen offers immunity for the time period of the study. Notice that the model is without vital dynamics, that is we do not consider the effect of birth and death processes, hence N = S + E + I + R . Thus, the described flow of the disease transmission results in

S ˙ = − β S I N E ˙ = β S I N − a E I ˙ = a E − γ I R ˙ = γ I (1)

To approximate the dynamics of COVID-19 in NI in 2020, we use several copies of the model (1) with an additional non-negative variable I c ( t ) . As mentioned in Section 0, the DoH provides the daily number of new COVID-19 cases; however, the model (1) does not include a variable for such data. Therefore, we introduce the non-negative variable I c , i ( t ) , which represents the cumulative number of infected individuals in the population at any time t in the interval T i , i = 0 , ⋯ , 9 . The variable I c , i ( t ) does not affect the transmission dynamics; its role is to capture the cumulative number of COVID-19 cases.

To describe the dynamics of the outbreak, in each time interval T i = [ t i , l , t i , r ] , i = 0 , ⋯ , 9 , we consider the following model:

Variable | Meaning |
---|---|

S(t) | The number of susceptible individuals at time t in the population. |

E(t) | The number of exposed individuals at time t in the population. |

I(t) | The number of infective individuals at time t in the population. |

R(t) | The number of recovered individuals at time t in the population. |

S ˙ i = − β i S i I i N E ˙ i = β i S i I i N − a i E i I ˙ i = a i E i − γ i I i R ˙ i = γ i I i I ˙ c , i = a i E i (2)

with the following initial conditions

( S i 0 , E i 0 , I i 0 , R i 0 , I c , i 0 ) = ( ( N − 2 , 2 , 0 , 0 , 0 ) , i = 0 , ( S i − 1 ( t i − 1 , r ) , E i − 1 ( t i − 1 , r ) , I i − 1 ( t i − 1 , r ) , R i − 1 ( t i − 1 , r ) , I c , i − 1 ( t i − 1 , r ) ) , 1 ≤ i ≤ 9. (3)

Moreover, we assume that N = S i + E i + I i + R i is independent of i, and the parameters of the model reflect introduced or relaxed measures instantaneously i.e., in contrast with the formulations in [

The basic reproduction number, R 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 R 0 > 1 an epidemic occurs, and the larger the values of R 0 the more challenging to bring the outbreak under control. On the other hand, if R 0 ≤ 1 , no epidemic occurs and the disease dies out. For the model (1), it can be shown that

R 0 = β γ (4)

is a threshold number^{1}, see [

For the model (2), we define

R 0 , i = β i γ i . (5)

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

R t , i = S ( t ) N R 0 , i . (6)

Note that, by using the data, S ( t ) N ≈ 0.9734 for t = t 9 , r .

To fit the model (2) to the data, we fit the model variables I c , i to the data D i , from the i^{th} interval T i , as follows. For each time interval T i , i = 0 , ⋯ , 9 , numerical solutions of the model (2) with initial condition (3) are generated while the corresponding nonlinear curve-fitting problem was solved in the least-squares sense. The ranges of these non-negative parameters β i , a i and γ i , used in the fitting process, are given in

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

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 β 0 , a 0 and γ 0 as of 1^{st} March 2020, and we also fit E 0 0 , the number of exposed at t = 0 ; the initial guess for E 0 0 was 2.

The first interval, T 0 , ends on 30^{th} March 2020, i.e., three days after the UK nationwide lockdown measures were introduced. After fitting the model, the estimated number of exposed individuals in this time interval, denoted E ˜ 0 0 , is about

Parameter | Meaning | Range |
---|---|---|

β i | Transmission rate | 0 - 1 |

a i − 1 | Average incubation period (days) | 2 - 20 |

γ i − 1 | Average infectious period (days) | 2 - 20 |

7.238, and the values of the corresponding fitted parameters are presented in the first row in

i | Days | β ˜ i | a ˜ i | γ ˜ i | R 0, i |
---|---|---|---|---|---|

0 | 1 - 30 | 0.5607 | 0.2902 | 0.1898 | 2.9539 |

1 | 30 - 59 | 0.4842 | 0.2058 | 0.5000 | 0.9684 |

2 | 59 - 88 | 0.4344 | 0.1880 | 0.5000 | 0.8688 |

3 | 88 - 117 | 0.1999 | 0.0500 | 0.3803 | 0.5255 |

4 | 117 - 146 | 0.0215 | 0.0500 | 0.0500 | 0.4297 |

5 | 146 - 175 | 0.1487 | 0.2543 | 0.0500 | 2.9742 |

6 | 175 - 204 | 0.1015 | 0.2020 | 0.0500 | 2.0298 |

7 | 204 - 233 | 0.4950 | 0.1456 | 0.2973 | 1.6649 |

8 | 233 - 262 | 0.3377 | 0.0759 | 0.5000 | 0.6753 |

9 | 262 - 300 | 0.1441 | 0.0735 | 0.0782 | 1.8419 |

Notice that the value of R 0,0 in

The first wave of the epidemic in NI is captured in the first 6 interval (~175 days) of the outbreak. Components E i , I i and R i of the numerical solutions of (2) for i = 0 , ⋯ , 5 , with initial conditions ( S i 0 , E i 0 , I i 0 , R i 0 , I c , i 0 ) = ( N − E ˜ 0 0 , E ˜ 0 0 , 0 , 0 , 0 ) , i = 0 and (3) when i > 0 , are presented in ^{th} March 2020) and gradually relaxed over the summer of 2020. Our numerical solutions suggest that these measures were very effective in slowing down the spread of the disease. Furthermore, our solutions indicate that the number of exposed and infectious individuals started to increase sharply around day 145 (27^{th} July 2020), i.e., about two months after the lockdown measures were eased in NI. During this period of 175 days, R 0 , i < 1 , as shown in

Day | Date | R t D |
---|---|---|

87 | 26-05-2020 | 0.8 - 1.0 |

94 | 02-06-2020 | 0.7 - 0.9 |

101 | 09-06-2020 | 0.5 - 0.9 |

108 | 16-06-2020 | 0.6 - 0.9 |

115 | 23-06-2020 | 0.6 - 0.9 |

122 | 30-06-2020 | 0.3 - 0.8 |

129 | 07-07-2020 | 0.5 - 1.0 |

136 | 14-07-2020 | 0.5 - 1.0 |

143 | 21-07-2020 | 0.9 - 1.5 |

150 | 28-07-2020 | 0.5 - 1.0 |

157 | 04-08-2020 | 1.2 - 1.4 |

164 | 11-08-2020 | 1.5 - 1.7 |

171 | 18-08-2020 | 1.2 - 1.6 |

178 | 25-08-2020 | 1.2 - 1.6 |

185 | 01-09-2020 | 1.1 - 1.4 |

192 | 08-09-2020 | 1.1 - 1.4 |

201 | 17-09-2020 | 1.0 - 1.3 |

206 | 22-09-2020 | 1.4 - 1.8 |
---|---|---|

213 | 29-09-2020 | 1.2 - 1.6 |

220 | 06-10-2020 | 1.3 - 1.8 |

227 | 13-10-2020 | 1.4 - 1.8 |

234 | 20-10-2020 | 1.0 - 1.3 |

241 | 27-10-2020 | 0.8 - 1.1 |

248 | 03-11-2020 | 0.6 - 0.9 |

255 | 10-11-2020 | 0.7 - 0.95 |

262 | 17-11-2020 | 0.9 - 1.1 |

269 | 24-11-2020 | 0.75 - 0.95 |

276 | 01-12-2020 | 0.9 - 1.1 |

283 | 08-12-2020 | 0.9 - 1.1 |

290 | 15-12-2020 | 1.0 - 1.2 |

297 | 22-12-2020 | 1.05 - 1.25 |

We now focus on the later phase of the epidemic. In

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, [

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 first two waves of the epidemic. More specifically, we split the data into 10 time intervals, T i , i = 0 , ⋯ , 9 , and fitted the models to the cumulative number of confirmed positive cases on each interval, and the estimated parameters reflect changes in the level of lockdown measures.

Our study clearly highlights the power of mathematical modelling of epidemiological processes. For instance, by fitting only one variable, I c , 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

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

h c = 1 − 1 R 0 , (7)

provided R 0 > 1 , see [

h c = 1 − 1 R 0 , 0 = 0.6615. (8)

Notice that

h c D : = max i = 0 , ⋯ , 9 ( 1 − 1 R t D ) = 0.4 (9)

which is significantly lower than the estimate (70% - 80%) recently reported in the news, [

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 details of society and interactions therein. For instance, one can obtain a better picture of the disease spread by incorporating the age structure of the considered population and the various levels of interaction between the age groups. Also, during lockdown, various limitations on social interactions were introduced, and possible hubs of disease spread were retail units of essential items. To assess the effectiveness of interventions in this and similar sectors, one can build a model reflecting the features of relatively short, indoor mixings of various subpopulations. In addition, in the case of a developing and ongoing epidemic, the economic impact of imposed control measures should be considered. Metapopulation models considering the spatial distribution and movement patterns within the population can be powerful tools in designing, implementing and updating movement restricting interventions affecting commuter networks vital to maintaining a healthy economy even in the case of an epidemic. Furthermore, during the last week of December, the level of the rates of infection is less likely to be captured by the data due to changes in behaviour that can affect testing. In the week up to 4^{th} January, there was a steep increase in the reported positive tests, which could result in increased hospital admissions. There was a steady increase in the reported positive cases despite the restriction measures introduced by the local authorities on 26^{th} December 2020 to control the spread of the disease. This surge in the daily incidence of infection may be attributed to the new strain of COVID-19 also known as Variant of Concern (SARS-CoV-2 lineage B.1.1.7), detected in November 2020, or socializing or both. SARS-CoV-2 lineage B.1.1.7 is rapidly spreading across England as well as the regions of the UK. In a recent study [

Despite its limitation, our model robustly captures the essential elements of the first 12 months of the NI COVID-19 epidemic and is a platform for further development. Developing further our model by considering additional features of the COVID-19 pandemic and the refined details of the population in NI, we aim at improving the quality of the parameter estimates and the details of information offered by compartmental modelling of the spread of communicable diseases.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Hall, P.A., Kiss, G., Kuhn, T., Moutari, S., Patterson, E. and Smith, E. (2021) Mathematical Modelling of the COVID-19 Epidemic in Northern Ireland in 2020. Open Journal of Modelling and Simulation, 9, 91-110. https://doi.org/10.4236/ojmsi.2021.92006