5 Data and Methodology
5.3 Model development
5.3.1 Baseline Model: SARS-CoV-2 transmission model
To model the dynamics of SARS-CoV-2 in Portugal, we developed an extension of the classic SEIR model - an age-structured deterministic compartmental model for the Portuguese population. The model distinguishes (i) contact rates between age groups of 10-years age bands, (ii) different categories of infectiousness (subclinical and clinical infections), (iii) disease’s severity with individuals requiring hospitalisations or intensive care, and lastly (iv) the possibility of individuals recovering or dying by COVID-19 disease. Individuals are classified based on their infection and disease status as susceptibles (S), exposed (E), infectious (I), hospitalised (H), critical cases in intensive care units (ICU), recovered (R), and deceased individuals due to COVID-19 disease (D). The infectious state (I) encompasses distinctly subclinical and clinical infections, with subclinical infections representing the individuals who never develop clinical symptoms throughout the infection (i.e., the infectious asymptomatics (IA)), whereas clinical infections include all individuals who develop illness during the infection. Clinical infections are composed of two periods: a presymptomatic period (IP) in which individuals are already infectious but have not developed symptoms yet, and a symptomatic infectious period (IS) in which individuals present clinical symptoms. We assume that all asymptomatic individuals recover from infection, and recovered individuals (R) cannot be re-infected during the time of our study. Furthermore, we assumed that hospitalisations only occur among symptomatic cases, critical cases in ICUs occur from hospitalised patients, and deaths by COVID-19 disease occur among patients in H or ICUs states. Additionally, we assume realistic distributions for the time individuals remain in the latent (E), infectious (IA, IP, IS) and hospitalised (H, ICU) states. For this, we employed the method of stages detailed in section 3.3. Specifically, based on Davies et al. [58], we assumed a latent period with a mean duration of three days and four sequential stages (𝑛𝐸), and both subclinical and clinical infectious periods with a mean duration of five days and four sequential stages each (𝑛𝐴, 𝑛𝑃, and 𝑛𝑆) (with a distribution similar to the one illustrated in Figure 3.3B).
The hospital discharge period, in hospitals and ICUs, was also modelled with a mean duration of 9 days
29 and 19 days, respectively, and 17 sequential stages each (𝑛𝐻 and 𝑛𝐼𝐶𝑈) [61]. Figure 5.4 shows the flux diagram of the compartmental model developed together with the transition rates between the different compartments.
Figure 5.4. SARS-CoV-2 transmission model and the transition rates between compartments. Transition rates are provided above the connecting arrows between compartments. Here, ε-1 represent the duration of time in the latent state; γ-1 parameters represent the duration of time in the infectious period (IA, IP, IS); 𝑛 parameters represent the number of stages in each state (E, 𝐼𝐴, 𝐼𝑃, 𝐼𝑆, H, ICU); and 𝜆𝑖 represents the force of infection in age group-i (see eq.(5.8)). Individuals in age group-i are initially susceptible (Si) and become exposed (Ei) after an effective contact with an infectious person. After a latent period, a fraction 𝑓𝑎,𝑖 becomes infectious asymptomatic, whereas (1- 𝑓𝑎,𝑖) will develop a clinical infection (moving firstly to the infectious presymptomatic state (𝐼𝑃), and later to the infectious symptomatic state (𝐼𝑆)). Among the symptomatics, a fraction 𝑓𝐻 will require medical care, moving into the H compartment, whereas (1-𝑓𝐻) will recover from infection. Among the hospitalised group: a fraction 𝑓𝐼𝐶𝑈 will require intensive care treatment, moving into the ICU compartment and a fraction 𝑓𝐻𝑑 will die. Among critical cases in ICU group, a fraction 𝑓𝐼𝐶𝑈𝑑,𝑖 will die and a fraction (1-𝑓𝐼𝐶𝑈𝑑) will recover. Further details about model notations and parameters values are given in Table A.1, Appendix A.
Our model allows the transmission rate of an infectious individual to vary depending on their category of infectiousness (IA, IP, IS) and on their age group. Since multiple systematic reviews suggest asymptomatic carriers may transmit the infection, and their attack rate is lower than clinical infections, we assume that asymptomatic infectiousness is 42% lower than the infectiousness of clinical cases as estimated in Byambasuren et al. [31]. In particular, we assumed that such reduction is compared to the infectiousness of preclinical cases. This is because, in reality, most symptomatic individuals are already diagnosed and isolated (or self-isolated) due to their evident clinical COVID-19 symptoms, which reduce their overall number of contacts and, consequently, their attack rate. Therefore, based on previous estimation [59], we
30 assume a reduction of 66% in the number of contacts made by a symptomatic case. In practice, this assumption was incorporated in the model by multiplying by 1/3 the term 𝛽𝑆𝐼𝑗𝑆 in the force of infection (see eq.(5.8)). Note that we assumed that hospitalised individuals (in H and ICU states) cannot transmit the infection since they are isolated from the remaining community. We denote the transmission probability for individuals in IA, IP, IS states as 𝛽𝐴, 𝛽𝑃, 𝛽𝑆, respectively. With these assumptions we are able to infer 𝛽𝐴 and 𝛽𝑆 by estimating only the value of 𝛽𝑃, as follows:
𝛽𝐴 = 0.58 𝛽𝑃 (5.2)
𝛽𝑆 = 𝛽𝑃 (5.3)
We also allowed that transmission probability varies with the patient’s age for each infectiousness category.
For infectiousness, we grouped the eight age classes into four groups: [0-19], [20-39], [40-59], ≥ 60, such that the transmission probability was the same across the aggregated groups as follow:
𝛽𝑃0−9 = 𝛽𝑃10−19 (5.4)
𝛽𝑃20−29= 𝛽𝑃30−39 (5.5)
𝛽𝑃40−49= 𝛽𝑃50−59 (5.6)
𝛽60−70𝑃 = 𝛽+70𝑃 (5.7)
where the infectiousness in equations (5.4)-(5.7) are denoted, hereafter, as 𝛽0−19𝑃 , 𝛽20−39𝑃 , 𝛽40−59𝑃 , and 𝛽+60𝑃 , respectively.
The probability that a susceptible individual in age group-i will acquire the infection from an infectious individual in age group-j per unit of time is given by:
𝜆𝑖(𝑡) =
∑ 𝐶𝑖𝑗(𝑡)(𝛽𝐴𝐼𝑗𝐴 + 𝛽𝑃𝐼𝑗𝑃 + 13𝛽𝑆𝐼𝑗𝑆)
8 𝑗
𝑁𝑗 , 1 ≤ 𝑖, 𝑗 ≤ 8 (5.8)
where 𝜆𝑖 denotes the force of infection in age group-i, 𝐶𝑖𝑗 represent the contact matrix formulated in section 5.2 (eq.(5.1)), and 𝑁𝑗 represents the total population size in age group-j.
Different age groups may also differ in their underlying parameters, particularly in the fraction of asymptomatic infections, hospitalisations, critical cases in ICU, and deaths, with children and young adults having a higher asymptomatic rate and older individuals having a higher hospitalisation and mortality rate.
We assume that the average time in E, IA, IP, IS, H, and ICU states are the same across all age groups. Each compartment of the mathematical model described in this subsection is represented by a differential equation, and the whole system is represented by an ODEs system shown in eq.(A.1) Appendix A. All notations and parameters used in the model are given in Table A.1-3, Appendix A.
31 The model is formulated assuming that all demographic changes, such as births and natural deaths rates, have a negligible impact on the epidemic dynamics. As a consequence, the population size remains approximately constant throughout the epidemic. In addition, since asymptomatic and mild infections have been constantly undercounted in Portugal, as shown by the serological studies, we assume that the total number of infections (cumulative cases) estimated by the model is given by the sum of all the individuals who develop clinical symptoms plus a fraction 𝑓𝑇 who have an asymptomatic infection, which we set as 𝑓𝑇
= 25%. The 𝑓𝑇 value was chosen based on the first serological study developed by the National Health Institute between May 2020 and July 2020, which estimated that approximately 44.5% of infections were asymptomatic [74], which in turn, we assumed that roughly half of these infections were detected by the epidemiological surveillance system. For the sake of simplicity, we assume that the rate of detection of asymptomatic infections 𝑓𝑇 is constant throughout the simulation; however, we are aware that this value might have changed over the course of the epidemic due to different testing policies.
5.3.2 Extension of the Baseline Model: Modelling Contact Tracing
In this subsection we extend the age-specific transmission model described above (in subsection 5.3.1) to capture the effect of tracing and isolating contacts in reducing the burden of infection. It is important to note that this modelling extension will aid to simulate and analyse the impact of extra contact tracing efforts beyond the level that was achieved in Portugal between March 2020 and February 2021 (our period of study) and which is inherently captured in the baseline model, as it will be described in the following chapters.
In practice, the contact tracing strategy aims to detect contacts that might develop infection and quarantine them for a specific period of time (usually for 14 days from March 2020 to February 2021). From a modelling perspective, tracing and quarantining/isolating involve removing potentially infectious individuals from the population, which can be addressed by adding further compartments into the model.
Considering this approach, we included five additional compartments for tracing and isolating infected contacts, which might occur in the latent period (QE), in the asymptomatic period (QIA), in the presymptomatic period (QIP), and in the symptomatic period (QIS)
–
although they are confirmed cases, we assumed the tracing program might also detect them. Figure 5.5 shows the flux diagram for the extended SARS-CoV-2 transmission model (with traced and isolated compartments represented within the red box).32 Although tracing and isolating contacts is also applied, in general, to susceptible individuals (i.e., people who might have had potential contact with an infectious case but have not acquired the infection), we have not included them in this work. The reason for this is that removing susceptible individuals in this well-mixed SEIR type-model will not significantly affect the dynamics of SARS-CoV-2 transmission, mainly because the removed fraction is very small compared to the overall population size. At the same time, this assumption enables us to maintain a simpler transmission model4. Moreover, it is expected that isolated people maintain similar infection progressions and disease stages to those not traced. For this reason, we set that the transition rates in the traced/quarantined compartments (specifically, the average time spent in isolated compartments QE, QIA, QIP, QIS)to the same values as the infectious not traced (i.e., E, IA, IP, IS).
4 The inclusion of traced and isolated susceptible individuals might be a relevant factor in studies that aim to analyse the impact of contact tracing at an economic level, such as in cost-benefit studies.
Figure 5.5. Extended SARS-CoV-2 transmission model and the transition rates between compartments for traced and isolated contacts. Transition rates are provided with the connecting arrows between the traced compartments represented inside the red box, whereas compartments outside represent the model diagram in Figure 5.4. Here, 𝑓𝑄𝐸 is the fraction of exposed traced contacts moving to QE state; 𝑓𝑄𝐼𝑎 is the fraction of asymptomatic traced contacts moving to QIA state; 𝑓𝑄𝐼𝑝 is the fraction of presymptomatic traced contacts moving to QIP state; and 𝑓𝑄𝐼𝑠 is the fraction of symptomatic traced contacts moving to QIS state; γQ-1 parameters represent the duration of time in each infectious traced state; 𝑛Q parameters represent the number of stages in each state (QE, Q𝐼𝐴, Q𝐼𝑃, Q𝐼𝑆) , 𝑓𝑄𝑎 is the fraction of exposed traced contacts that develop an asymptomatic infection and 𝑓𝑄𝐻 is the fraction of symptomatic traced individuals who require medical care treatment moving to H state. More details of the model and states transition are given in Table A.2, Appendix A.
33 We assume that the fraction of tracing and isolating contacts (𝑓𝑄𝐸, 𝑓𝑄𝐼𝑎, 𝑓𝑄𝐼𝑝 and 𝑓𝑄𝐼𝑠) are the same across different age groups since the contact tracing policy is applied equally to the population in general. Most traced and isolated contacts recover from infection, with the exception of a fraction 𝑓𝑄𝐻 of symptomatic isolated cases that may require hospitalisation. Note that by fixing the traced parameters as 𝑓𝑄𝐸 = 0, 𝑓𝑄𝐼𝑎 = 0, 𝑓𝑄𝐼𝑝= 0, and 𝑓𝑄𝐼𝑠 = 0, we obtain the baseline transmission model from subsection 5.3.1.