The mathematical modelling of infectious disease has a long history and has in recent years been a key element of the investigation of infections in human and animal populations [1] - [6]. Moreover, mathematical modelling is now firmly established as a key tool in public health and in health care planning, and in guiding the responses to infections [7] [8]. Soon after the report of the presumptive respiratory tract zoonosis caused by the novel coronavirus in 2019 [9] [10] [11] that had the potential to cause multi-system illness and death and with the clear person-to-person transmission it became clear that there was a potential for a worldwide pandemic of a novel virus in a virgin population [12] [13] [14] [15]. Considerable scientific and medical interest quickly led to rapid progress in the development of insights into the pathogenesis of COVID-19 disease with the definition of its cellular receptor [16] and basic characteristics [17]. Since then, despite the challenges posed by this unknown disease, see [18], mathematical modelling of the epidemic has been reported [19] - [31] and used to guide and inform public health, economic and political decisions [32] [33] [34]. Epidemiological data soon indicated that the epidemic had very different characteristics in different populations and the importance of local factors (social, cultural, demographic, economic, transport infrastructure, housing, etc.) in determining epidemic development became clear [33] [34]. Such factors argue for the crucial need for the use of modelling in the context of local environments and we took advantage of a detailed data set from the Northern Ireland Department of Health as a platform for our modelling efforts. We recently described our basic SEIR model and demonstrated it provides a robust description of the Northern Ireland COVID-19 epidemic in 2020 [35].

As with other respiratory infections, such as influenza, the spread is seen across spatial scales. The majority of transmission events occur in household and household-like settings such as nursing homes, prisons, communal housing for workers, etc. [36]. Early in the pandemic, it became clear that there was a variable but significant pre-symptomatic period for many (if not all) infected individuals. Furthermore, studies from a range of locations have provided support for this idea and the duration of pre-symptomatic infection appears to range from 5 to 11 days [37]. Over and above this, there is clear evidence for asymptomatic infection without any symptoms reviewed in [38]. Most of the reports of this asymptomatic phenomenon were retrospective in nature and cross-sectional with limited longitudinal data. Since they were usually serendipitous in nature, the studies have significant methodological failings including poor symptom definition, inadequate follow-up and concerns about testing protocols [39]. Nevertheless, the available evidence points to asymptomatic infection being a highly prevalent phenomenon. While we remain ignorant of much of the biology underpinning this phenomenon viral loads seem similar [40] and asymptomatic transmission of infection is well documented [37] [38] [39] [40] [41] [42]. It follows that both pre-symptomatic and asymptomatic cases are potential of great significance in driving infection [36] [43].

An additional area of uncertainty in defining the dynamics of the COVID-19 pandemic has been the consideration of uncertainty in testing. Broadly COVID-19 testing incorporates three techniques: 1) PCR detection of viral nucleic acid; 2) antigen detection, and 3) antibody detection. For all of those techniques, there is a range of commercial kits available. However, Axell-House et al. [44] argued that many current studies evaluating test performance characteristics were not methodologically robust with sub-optimal statistical methods for the estimation of test performance characteristics. Similar concerns have been voiced elsewhere [45] - [54]. Moreover, there has also been a concern about the use of differing C t threshold values in PCR tests [53] [55]. These methodological issues remain unresolved and so the comparison of reports from different jurisdictions and health care systems is problematic. This is significant since that error (or at least uncertainty) in tests will be significantly impacting prevalence data, i.e., how many positives there are in a population [54].

Here, we extend our recently described SEIR model of COVID-19 in the setting of Northern Ireland to investigate the impact of differing levels of asymptomatic transmission and the impact of test uncertainty on the model. More specifically, by using our model, introduced in Section 2.2, we are able to model the course of the COVID-19 epidemic in Northern Ireland in 2020, including the level of asymptomatic cases.

We successfully fitted (2) to the number of individuals tested positive for COVID-19. Our model, incorporating a compartment for asymptomatic individuals, provides estimates on the size of this subpopulation. However, it is very difficult to obtain realistic picture about the level of asymptomatic infections without mass-testing. Nevertheless, our incidence-based model indicates that the probability of being infectious without symptoms ranged between 0.0754 and 0.25 following the pattern of Non-Pharmaceutical Interventions (NPIs) in NI in 2020. Detailed presentation of our estimates with confidence interval can be found in Table 7, which are in good agreement with the recent study [77].

In Tables 3-8, we present the parameters provided by the fitting algorithm described in Section 2.4. For those parameters, confidence intervals, computed by the method described in Section 2.5, are also provided in Tables 3-8. Based on these parameters, we also provide estimates of the basic and the effective reproduction numbers using (4) and the procedure described in Section 2.7.

In Figure 4 and Figure 5, we plotted the solution of (2) when i = 0 , that is for the period between 1^st to 29^th March 2020. Our estimate of R 0 for this period is 3.3089; also R 0,0 L ≈ 2.6190 and R 0,0 U ≈ 3.6277 . Furthermore, in Table 9, we present the weekly estimates of R t D for the period of study, provided by the Department of Health of Northern Ireland. We can use these estimates to approximate required size of the immunised population (via infection or vaccination) to prevent large subsequent waves of infection. Namely, since the minimum level of vaccination, with vaccine giving 100% immunity, to achieve herd immunity is:

h c = 1 − 1 R 0 , (6)

provided R 0 > 1 , see [78]. Using R 0,0 from Table 10, we obtain the following estimate of the herd immunity.

h c = 1 − 1 R 0,0 ≈ 0.6978. (7)

Furthermore, by using R 0,0 L and R 0,0 U , we get a lower and an upper estimate for herd immunity:

h c L : = 1 − 1 R 0,0 L = 0.6181     and     h c U : = 1 − 1 R 0,0 U = 0.7243.

Notice that:

h c D : = max i = 0, ⋯ ,9 ( 1 − 1 R t D ) = 0.4 (8)

which is significantly lower than the estimate (70% - 80%) recently reported in the news [79]. Furthermore, in Table 10 and Table 11, we collected the parameter estimates, and initial values of E and A, respectively. The latter indicates that approximately 5 exposed and 6 asymptomatic individuals on 1^st March 2020 in NI could have triggered the epidemic in the country. Using the values from Table 10 and Table 11, we plotted the result of a simulation to explore the course of a hypothetical epidemic without any NPIs in NI in 2020 (Figure 6).

Similarly to Figure 4, Figure 7 depicts the fitted solution together with data points for i = 1, …, 10. Also, Figure 8 and Figure 9 shows simulation results when i = 0, …, 5 and i = 0, …, 10, respectively.

In Tables 3-8, we present the fitted parameter values for i = 0 , ⋯ , 10 together with the 95% confidence intervals. Using these parameter values, we computed numerical solutions of (2) using (3) for i = 0 , ⋯ , 10 . The results of the solutions are plotted in Figures 10-12. More specifically, Figure 10 shows the solution components for the first 29 days of the epidemic. The solution components for the first wave of the epidemic, the first 169 days, are plotted in Figure 11. Furthermore, in Figure 12, we plotted solutions of (1) by using (3) for i = 0 , ⋯ , 10 . In addition to the solution profiles, in Figures 10-12, we also plotted results of the procedure, described in Section 2.6, to illustrate the effects of testing uncertainty. Finally, in Table 12, we provide the number of solutions satisfying (5).

Table 13 summarizes the findings on R 0, i ,   i = 0, ⋯ ,10 , calculated by using (4). In addition, the table contains the values of R 0, i L and R 0, i U , obtained by the method described in Section 2.7 where i = 0 , ⋯ , 10 . The findings on R 0, i ,   i = 0, ⋯ ,10 are visualized in Figure 13.

We fitted our model (2) to the daily incidence of COVID-19 in Northern Ireland

in 2020; the data are provided by the Department of Health, and they are publicly available. The main finding of this study is that the proportion of asymptomatically infectious subpopulations ranged between 5% and 25% in Northern Ireland in 2020. Our estimate of the basic reproduction number, R 0 , is 3.3089. This implies that around 70% of the population of NI should acquire immunity via

infection or vaccination. This estimate is in the range of estimates reported in other studies, see e.g., [80]. Also, by using R 0, i L and R 0, i U from Table 13, we obtained a lower and an upper estimate for herd immunity ( 0.6181,0.7243 ) .

A justifiable criticism of our modelling is the lack of precision in the definition of asymptomatic COVID-19 infection. Indeed this has been a source of

concern and confusion since the beginning of the pandemic [37] [38]. The initial case definition focused entirely on fever and cough. In particular, through the Zoe self-reporting phone app, see [81] [82], anosmia rapidly became recognized as a very common feature of COVID-19 infection. Subsequently, a wide range of other symptoms has been associated in some (but by no means all) COVID-19 patients [77] [83]. These include fatigue, muscle and chest pain, nausea, headache, breathlessness, abdominal pain, diarrhoea, hoarse voice, skin lesions and swelling (particularly on lips and face), finger and toe lesions, eye pain and a host of other features [77] [83]. Indeed the range of features is so broad yet also so variable in any given patient that none of the clinical features have significant diagnostic specificity and merely raise the possibility of COVID-19 infection and thus require laboratory test confirmation [83] [84]. Thus from an initially restrictive case definition, there is now evidence of extraordinary pleiotropism of effects of the virus. So what should be the definition of asymptomatic COVID-19 infection? Is it lacking the cardinal features of the infection, or is it lacking any of the possible clinical features? This complex issue remains unresolved. However, what is clear is that there is a substantial subset of patients who have either no symptoms or symptoms that are so mild and non-specific that only the most suspicious might consider COVID-19 infection [37] [38] [39] [40] [43]. It thus remains the case that consideration of an asymptomatic compartment is an important aspect of disease modelling. Moreover, it should be emphasized in studies as the asymptomatic (or very mildly symptomatic) are likely to be an important conduit for the infection of others [36] [37] [43] [85]. It is also the case that since COVID-19 has a long (and often variable) incubation period and that this may include a pre-symptomatic phase, where the patient is possibly shedding virus, the potential role of this phase in onwards transmission needs to be considered [36].

Methodological issues relating to false positive and false negative results in COVID-19 testing have been of concern throughout the pandemic [44] - [55]. Of particular note are the lack of precise information about false positive and false negative rates and the concern about the over-sensitivity of PCR when high numbers of cycles are used [53] [55]. To investigate the possible significance of this we sought to model the potential impact of test uncertainty and found that diagnostic variability (due to any cause, as we cannot model specific methodological issues) can dramatically impact the modelling predictions. This is particularly evident in Figure 11 where considerable variation in case rates emerges depending on test performance. Clearly, in our retrospective analysis, this is probably not a major concern but it does highlight its potential impact on modelling. Indeed where mathematical models are used to prospectively forecast the dynamics of the epidemic, and critical outcomes such as the number of hospitalizations and the need for ITU (Intensive Therapy Unit) beds, then much greater attention to the operating characteristics of COVID-19 testing is essential. Indeed, from the mathematical epidemiological perspective, it appears to be a much-neglected issue and warrants much greater attention.

There are many aspects of the ongoing epidemic of COVID-19 we did not consider in this study. For instance, as reported in [86] [87] [88], environmental differences, seasonal weather patterns might have significant effects on the spread and the severity of the disease. Also, our model (2) captures only the symptomatic and the asymptomatic spread, however, as we mentioned in the introduction, the presymptomatic spread is yet another potentially important source of infection. Last but not least, the vaccination programme in NI started in December 2020, and a good understanding of its effects on the epidemic requires a careful assessment, in particular, the subsequent NPI relaxation strategies. A recent UK-wide modelling of vaccination and control strategies [89], suggests that only a carefully designed combination of immunization and NPI relaxation has the potential of driving R 0 below 1.

Since the completion of this study and it had the original submission of a series of variants of COVID-19 which has emerged and some have become highly prevalent. Indeed, variants such as Delta and, more recently, Omicron have novel properties, in terms of infectivity and pathogenicity, which give them markedly different epidemiological characteristics. Delta and then Omicron have in turn become dominant in many populations. These variants are more likely to be transmitted by vaccinated persons compared to earlier variants, despite very good individual protection against severe diseases courses after vaccination [90] [91] [92]. Strikingly Omicron appears to be more infectious with a shorter incubation period and milder disease. Preliminary reports suggest that it may have a higher frequency of asymptomatic illness. Such data suggest that our analysis of the early phases of the pandemic, where Alpha and Beta variants predominate, may not be directly analogous to the current situation, underscoring the rapid evolution of COVID-19. Notwithstanding, to these important new variants and their properties, our general strategy remains valid and the information reported is highly relevant. Further studies of novel variants in the Northern Irish and other contexts are essential, building on the work reported here.

Moutari, S. would like to acknowledge the support from Innovate UK through the Knowledge Transport Partnership scheme (Projects S2377MPH and S2417MPH).

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. (2022) Estimating the Level of Asymptomatic COVID-19 Infections in Northern Ireland in 2020. Open Journal of Modelling and Simulation, 10, 190-218. https://doi.org/10.4236/ojmsi.2022.102011