Introduction

At the early stage of a pandemic, it is crucial to comprehend the global transmission potential of the virus in the population, quantifiable by, e.g., the basic reproduction number, denoted as \(R_0\). This number delineates the per-generation count of infections caused by a typical infected individual in a fully susceptible population. The reproduction number is influenced by epidemiological characteristics, such as susceptibility and infectivity to the virus in the population. Susceptibility pertains to the set of individual characteristics that allow for infection, such as lower immunity, and may be tied to age. Infectivity, on the other hand, refers to both the virus and individual traits that facilitate transmission, including a higher viral load. Notably, in the case of SARS-CoV-2, both susceptibility and infectivity at the onset of the pandemic were positively correlated with age1,2. Consequently, understanding the impact of age-dependent susceptibility and infectivity is paramount for developing tailored public health interventions.

The extensive use of mathematical modeling and statistical analysis in the study of COVID-19 data has played a significant role in guiding public health policymakers during critical phases of the pandemic. These methods have been employed in an effort to tackle the challenge of quantifying and understanding the impact of non-pharmaceutical interventions, such as lockdowns, school closures, and guiding the strategic allocation of COVID-19 vaccines3,4. Among these methods, the Next Generation Approach (NGA) stands out as a tool for inferring relevant epidemiological parameters in a simple and straightforward manner, especially when compared to traditional compartmental models. Importantly, the NGA maintains an intuitive epidemiological interpretation, making it advantageous in the pandemic context, where the estimation of these parameters can rapidly and effectively inform public policies5,6,7. This approach has proven valuable in previous applications, such as obtaining optimal strategies for vaccine allocation in the Netherlands8,9, studying the evolution of age-specific parameters and an analysis of the reproduction during the early phases of COVID-19 epidemic in Belgium5,6.

The next-generation approach employs the use of the so-called next generation matrix (NGM or \(\textbf{K}\)), which directly derives from traditional compartmental models (e.g., SIR model). In a population divided into epidemiologically relevant groups, such as age groups, each matrix entry (represented by \(k_{ij}\)) corresponds to the average number of new infections in group i potentially generated by one individual in group j after one generation. The largest eigenvalue of \(\textbf{K}\) corresponds to the basic reproduction number \(R_0\). The sum of all the elements in a column represents the average number of infections that one infected individual in group j could potentially generate per-generation in the population, i.e., the group-specific transmission potential (\(\mathbf {k_{\cdot j}}\)). In the context of the NGA, the mean generation time includes the period during which an individual can infect, thus creating ’offspring’ that will subsequently infect during their generation. This concept creates a parallelism with demographics. In this study we formulate the NGM as the product of a social contact matrix10,11, with diagonal matrices related to the susceptibility and infectivity profile of the population.

This approach also facilitates the estimation of sensitivity indicators5, which quantify how changes in epidemiological parameters would affect \(R_0\) and relative incidence of the disease-i.e., the distribution of infections across heterogeneous classes. It enables us to estimate the effects of changes in the characteristics of a single group, on the reproduction number. These changes can occur by means of a public health intervention, which can encompass alterations in their infectivity and susceptibility profiles. The resulting changes in the \(R_0\) and relative incidence are expressed in terms of elasticity values5,12, quantifying the proportional impact of the targeted intervention.

In Portugal, the first COVID-19 cases were reported on March 2, 2020. The number of cases continued to rise until an eventual decline associated with the closure of schools followed by a nationwide lockdown. This decline occurred in early April 2020, marked by a sharp reduction in COVID-19 cases. While several modeling studies have investigated the impact of the first lockdown in March 20203,4,13 and the rollout of the vaccine3,13 in Portugal, to our knowledge, no study has systematically evaluated the importance of each age group and their associated susceptibility and infectivity profiles in disease transmission and age-specific relative incidence. In this study, we apply the next-generation approach to analyze the early outbreak of COVID-19 in March 2020 in Portugal. Our focus is on understanding the contribution of each age group and their associated susceptibility and infectivity profiles to disease transmission. Additionally, we propose a novel indicator within this methodology termed Relative Impact (RI), which gives a relative measure of the impact of an intervention on the number of infected individuals, compared with no intervention in the reference scenario. This method, along with the new indicator, could be particularly valuable in providing prompt responses during a pandemic. Specifically, we calculate sensitivity indicators for the basic reproduction number to measure the relative contribution of each age group towards \(R_0\). Moreover, we employ the RI indicator to simulate possible interventions, evaluating the impact of a reduction in susceptibility and infectivity for different age groups. In the methods section, we describe the main sensitivity indicators associated with \(R_0\) and RI, along with their epidemiological interpretation. The mathematical formulation is presented in the supplementary material.

This study, focusing on the early outbreak of COVID-19 in Portugal (01-03-2020 to 30-03-2020), aims to demonstrate how the Next Generation Approach can efficiently inform the dynamics of disease spread by summarizing data into interpretable epidemiological indicators in a fast and straightforward manner. This is particularly crucial in a pandemic context. The results obtained have the potential to inform public health policy during outbreaks of diseases with similar characteristics or can be adapted for different epidemic settings.

Methods

The transmission process is described through the next generation operator14 which, by means of linearization around the disease-free equilibrium, allows us to approximate the process of new infections in a matrix form. An explicit formulation of the next generation operator can be derived for any compartmental model7. In the setting of a susceptible-infectious-removed model with a discrete age structure, this process is given by a \(n\times n\) matrix \(\textbf{K}\), where n corresponds to the number of age classes. \(\textbf{K}\) is referred to as the next generation matrix, in which \(k_{ij}\) corresponds to the average number of infections created by an individual in age group j, in age group i throughout the course of its infection, \(i,j=1,...,n\)7. Matrix \(\textbf{K}\) can be expressed as

$$\begin{aligned} \textbf{K}=qAM^{\intercal }H, \end{aligned}$$
(1)

where A and H are diagonal matrices with entries related to the susceptibility and infectivity for each age group, respectively. Entry \([A]_{ii}\) corresponds to the per contact probability of infection, of an individual in group i, given an infectious contact and \([H]_{ii}\) gives the per contact probability of an individual in group i infecting a susceptible individual. These quantities arise from the social contact hypothesis15 and can be obtained up to a proportionality constant q, usually referred to as q-susceptibility and q-infectivity, respectively6. The parameter q accounts for the infectious period and other aspects that might affect the resulting transmission, such as environmental (e.g. seasonality) or social/behavioural (e.g. risk perception) and is calibrated without loss of generality. Matrix \(M^{\intercal }\) corresponds to the contact structure, see the supplementary material for more details.

Matrix \(\textbf{K}\) is used to relate the total number of infections in two consecutive generations11:

$$I_t=\textbf{K}I_{t-1}, \; t \in \mathbb {N},$$

where \(I_t\) denotes the vector of infected by age group in generation t. The entries of \(\textbf{K}\) are positive, yielding a dominant eigenvalue \(\lambda _1>0\) , which can be interpreted as the basic reproduction number \(R_0\)14,16. Moreover, \(\textbf{K}\) has positive (all entries \(>0\)) right (\(\textbf{w}_1\)) and left (\(\textbf{v}_1\)) eigenvectors associated with \(\lambda _1\). Eigenvectors can be interpreted up to a non-zero constant. We scale \(\textbf{w}_1\) and \(\textbf{v}_1\) such that \(||\textbf{w}_1||_1=1\) and \(\langle \textbf{w}_1,\textbf{v}_1\rangle =1\). Thus \(\textbf{w}_1\) can be interpreted as the vector of the relative incidence by age group (see supplementary material).

Changes in the epidemiological setting, e.g., vaccination programs, non-pharmaceutical interventions and depletion of susceptibles, will produce a perturbation in the elements of \(\textbf{K}\) and thus in the basic reproduction number and relative incidence. Quantifying these changes is vital to understand the future potential of epidemic growth and the age-specific incidence.

Sensitivity measures assess the response of \(R_0\) to a perturbation in a given epidemiological parameter. Cumulative elasticities5 are sensitivity measures that can be interpreted as the relative contribution of a specific age group towards the \(R_0\)12,17,18, i.e. it can ascertain which age groups are the main drivers of transmission. For age group i, \(i=1,...,n\), it is computed as

$$E_{i\rightarrow R_0}=w_1^{(i)}v_{1}^{(i)},$$

where \(w_1^{(i)}\) and \(v_{1}^{(i)}\) denote the ith entry of the right and left eigenvectors, respectively.

Elasticity measures can also be obtained for the entries of the relative incidence vector \(\textbf{w}_1\). In this case, sensitivity measures display how the vector \(\textbf{w}_1\) shifts given a perturbation in \(\textbf{K}\).

We can also arrive at a combined measure by considering changes to both \(R_0\) and \(\textbf{w}_1\). Given a proportional perturbation \(p_x \in \mathbb {R}\) on a epidemiological parameter x,

$$RI_{j}\approx 1+p_x\Bigg [ E_{x\rightarrow w_1^{(j)}}+m \;E_{x\rightarrow R_0}\Bigg ].$$

The ratio \(RI_j\) can be interpreted as an approximation to the relative number of infections, in age group j, in the mth generation after the implementation of an intervention that affected x by \(p_x\), compared to no intervention, given that no other perturbation in \(\textbf{K}\) occurred up to the mth generation. We note that RI is a projection of the relative measure of the effect of an assumed intervention, where we take the NGM without any alteration as the counterfactual reference.

In Section 6 of the supplementary material, we demonstrated that the error in our approximation of the index RI is influenced by two factors: the magnitude of the considered perturbation, \(p_x\), and the number of generations, m. By presenting a numerical example, we illustrate that there is an optimal value for the pair \((p_x, m)\) that minimises this error, which resulted in considering \(m=3\) generations. These results are depicted in Figs. S1, S2 and S3. The supplementary material details the mathematical formulation of RI.

Data

We partitioned the Portuguese population in 8 age groups: 0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70+ years old. Data on the number of SARS-CoV-2 positive cases by age group in Portugal was obtained between the period of 2020-03-01 and 2020-03-30 and was used to estimate the relative incidence during the early exponential phase of the pandemic. These data was shared by the group at the Portuguese Directorate General of Health.

Table 1 Relative incidence vector (\(\textbf{w}_1)\) and infectivity data (diagonal entries of H) for each age group related to the population residing in Portugal. Values for the q-susceptibility parameter (diagonal entries of \(\alpha =qA)\) were estimated via a method presented in the supplementary material.
Full size table

The basic reproduction number was set at 2.53,19,20,21, which refers to the transmission potential observed during the early outbreak period in Portugal. Table 1 summarises the relative incidence and infectivity data as well as the estimated q-susceptibility profile of the population. The estimation method is detailed in the supplementary material. Infectivity values were informed by previous modelling works5,6. Pre-pandemic social contact data for the population residing in Portugal was not available. However, a research article22, leveraging contact patterns from the POLYMOD project23 and health survey data, estimated a synthetic social contact matrix for Portugal using Bayesian techniques. This study resulted in a matrix featuring 16 age groups. This was then condensed to include only the 8 age groups considered above, using dimension reduction methods24.

Results

We commence by introducing the next generation matrix, illustrated in Fig. 1a. This matrix was computed using formula (1), detailed in the methods section. We considered that the Portuguese population was partitioned in 8 age groups: 0–9, 10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70+ years old. Social contact data, infectivity and q-susceptibility data for the Portuguese population was used to inform the NGM. Notably, the age group pair exhibiting the highest transmission is observed among those aged 70 years and above, primarily infecting individuals within the same age group. Following closely are individuals aged 30–39 years, predominantly infecting others within the same age bracket. The pattern observed in the next generation matrix mirrors the outcomes derived from social contact matrices, indicating that assortative contacts22 lead to assortative transmission.

Fig. 1
Fig. 1
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Next generation matrix \(\textbf{K}\) (a). Transmission parameters \(R_0\), \(\mathbf {k_{\cdot j}}\) and \(\mathbf {k_{i \cdot }}\) represent the basic reproduction number, average number of infections caused by an individual in age group j in a generation and sum of infections per generation in age group i caused by one individual of each age group, respectively (\(j=1,...,8\)) (b). Proportion of infections belonging to each age group, (relative incidence vector \(\textbf{w}_1\)) (c).

The relative incidence is illustrated in Fig. 1c. It reveals that, at the onset of the epidemic, the incidence of infections was concentrated in older age groups, with individuals above 50 years old accounting for approximately 50% of the confirmed infections. The column sums of the next generation matrix (\(\textbf{K}\)), denoted as \(\mathbf {k_{\cdot j}}\) for \(j=1, \ldots , 8\), correspond to the average number of infections produced by an infected individual within a given age group in each generational step. These values are depicted in Fig. 1b, alongside the basic reproduction number. The results indicate that, given infection, individuals aged 30–39 years exhibit the highest transmission potential, followed by those aged 40–49 and 50–59 years. In contrast, the youngest individuals (0–9) demonstrate the lowest transmission potential.

Figure 1b also presents age group estimates for the cumulative elasticity index \(E_{i\rightarrow R_0}\). These measure the proportional contribution of each age group towards overall transmission (\(R_0\)). The results indicate that \(R_0\) is least elastic to changes in age groups 0–9 and 10–19 years old, and most elastic to changes in age groups 30–39 and 40–49 years old, followed by those 50–59 years old. This suggests that alterations in the behavior or epidemiological characteristics of individuals aged 0–19 years result in the smallest variation in \(R_0\), while changes in the 30–59 age groups lead to the greatest variations. Specifically, individuals aged 30–39 and 40–49 years contribute equally (22%) to variations in \(R_0\). However, individuals aged 30–39 years exhibit the highest transmission potential, with an average of 3.3 infections when compared to 40–49 years old, with 2.85. The similar contributions, despite distinct transmission indexes (\(\mathbf {k_{\cdot j}}\)), result from individuals aged 30–39 being less infected compared to those aged 40–49. This observation is evident when examining the row sums of the matrix \(\mathbf {k_{i\cdot }}\), \(i=1, \ldots , 8\), which indicate the sum of infections per generation in group i originating from a single infected individual in each age group.

Fig. 2
Fig. 2
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Relative impact (RI) given a 10\(\%\) proportional reduction in q-susceptibility (a) and infectivity (b) for each age group. RI can be interpreted as the projected relative impact of a given intervention (simulated as a reduction to q-susceptibility or infectivity), on the relative number of infected after \(m=3\) generations, where the counterfactual scenario assumes no intervention.

The relative impact (RI) enables the projection and comparison of the impact of changes in epidemiological parameters on the relative number of infected individuals per generation, where the counterfactual reference is given by the NGM presented in Fig. 1a. We perform these projections to simulate potential interventions related to alterations in the susceptibility and infectivity of each age group. These interventions can include the use of personal protective equipment (PPE), e.g. face masks, which, for COVID-19, have been estimated to reduce transmission by at least 12–18%25. This reduction is achieved through a combination of reduced susceptibility due to mask usage by those who are susceptible and reduced infectivity of those who are infected. Figure 2a displays the RI assuming a 10% reduction in q-susceptibility, for each age group. The results reveal that the greatest decrease in RI is observed when PPE is used by either age groups 30–39, 40–49 or 50–59, aligning with the age groups to which \(R_0\) is most elastic, see Fig. 1b. As a direct effect of the use of PPE, it is anticipated that, in the absence of other changes, the number of infected individuals after three generations will decrease by approximately 17% and 16% in the age groups 30–39 and 40–49, respectively, when considering no use of PPE as the reference scenario. As an indirect effect, a decrease of approximately 3%-6% is expected in the remaining age groups. Generally, the direct effects of reducing q-susceptibility outweigh the indirect effects for all age groups.

Similar to the results obtained for q-susceptibility, RI is more pronounced when a 10% reduction in infectivity occurs in age groups between 30 and 59 years old, as displayed in Fig. 2b. However, the impact of reducing infectivity is generally smaller than the impact of reducing q-susceptibility, especially as a direct effect on the group where the reduction occurs.

Discussion

Computing sensitivity and elasticity indicators tied to the next generation matrix provides insights into the underlying dynamics of disease transmission. In a structured population, one can measure the contributions of specific groups to both transmission changes and age-specific relative incidence. Furthermore, it can be used to evaluate the theoretical effectiveness of a given public health intervention.

In this case study, we examined the age group profile of the Portuguese population concerning characteristics related to susceptibility and infectivity to the SARS-CoV-2 virus. Using cumulative elasticities, we demonstrated that the population aged 30–39 and 40–49 years contributes the most to new infections, followed by those aged 50–59 years, with a combined cumulative elasticity value of \(64\%\). These results are in line with similar studies. In particular, in a paper studying the resurgence of COVID-19 cases in mid 2020 in the United States26, the authors estimate that individuals 20–49 years were responsible for 65 out of 100 infections, and those 35–49 years old presented the highest transmission potential. They also concluded a limited role of children in disease spread. Furthermore, a modelling study for the first 2 years of pandemic in the Netherlands27 indicates that seroprevalence among children and adolescents was low, when compared to adults.

Looking at those 40–49 and 70+ years, we observe that although they have similar values for relative incidence, with those 70+ years having a marginally higher value, these occur due to distinct transmission dynamics. Those 40–49 years have a higher amount of contacts and lower susceptibility, while those 70+ have much lower contact rates but higher susceptibility. Resulting in a higher contribution to transmission from those 40–49 when compared to those 70+ years old, as displayed by their elasticity values. An intervention, such as the use of PPE, which reduces q-susceptibility by 10% in those aged 30–39, 40–49  or 50–59 results in a 16%-17% decrease in infections within the same group and a 3%-6% reduction in the remaining groups after three generations. Resulting in the greatest decrease in infections when compared to applying the same intervention in the remaining age groups. The inverse also holds: a proportional increase in q-susceptibility in these groups would lead to a greater increase in new cases compared to the remaining groups. Similar conclusions can be drawn when examining changes in infectivity. These results highlight that these three groups are the main drivers of new infections. Additionally, proportional variations in q-susceptibility and infectivity in the age group 30–39 would result in more significant changes to both \(R_0\) and RI, compared to the remaining age groups. This result emphasizes how individuals in this age range are the primary drivers of transmission. The previous results also highlight that, while individuals aged 70+ are the most infected, they are not be the most effective target for interventions aimed at reducing transmission.

We highlight that this analysis depends on having data regarding contact patterns of individuals, as well as their q-susceptibility and infectivity profiles. However, such data may not be readily available during the early stages of an outbreak. As an alternative, some authors have proposed using pre-pandemic contact data28,29, mobility data30, or synthetic contact matrices22 to inform contact patterns. Moreover, a limitation of using reported incidence data to inform q-susceptibility, is the fact that individuals with a clinical presentation are more likely to be tested. Consequently, infections in younger age groups, which are more likely to be asymptomatic but still infectious31, may be underrepresented in the testing data32,33. This may result in an underestimation of q-susceptibility and thus may result in an underestimation of their contribution to disease spread. However, our q-susceptibility values are in accordance with literature, where in the early phases of the COVID-19 epidemic, children have been shown to be less susceptible and those above 60 years old are more susceptible to infection2. Moreover, the under ascertainment impact was shown to be limited when looking at the sensitivity analysis5. When available, sero prevalence data can also be used to inform relative incidence, providing an alternative to inform on infection rates.

The approach presented in this study offers a notable advantage by requiring minimal epidemiological data associated with an outbreak to inform transmission patterns. Additionally, it is easily extendable to other diseases, provided that the associated next generation matrix derived has similar properties. Importantly, this approach can be applied cross-sectionally as new data becomes available during an outbreak, allowing for the continuous assessment of the main contributors to disease spread and the projection of the effects of future interventions. In this context, the dominant eigenvalue of the next generation matrix would indicate the effective reproduction number.

Future work entails conducting a similar analysis on the evolution of the reproduction number throughout the COVID-19 pandemic in Portugal and several other European countries. This analysis will assess how different transmission profiles lead to changes in the reproduction number across these countries, facilitating comparisons of epidemics and aiding in the development of country-specific strategies to combat future outbreaks with characteristics similar to COVID-19. Implementing this proposed analysis requires updating the structure of \(\textbf{K}\) to accommodate changes in the epidemiology of COVID-19, including the introduction of new variants and associated characteristics, the spectrum of susceptibility in the population due to previous infection, vaccination and booster doses, waning immunity, as well as changes affecting social contacts.