Estimation of susceptibility and infectiousness through proportionality factors

The proportionality factor is assumed to be age-specific and denoted by q_ij where i and j belong to some age classes. q_ij could be further split into heterogeneous components: where:

• The vector (a_i) represents age-specific differences in factors influencing transmission which are specifically related to the susceptibility of individuals, including, but not limited to, direct (immunological) susceptibility to infection upon exposure (e.g., due to age-specific heterogeneous risk behavior and/or compliance to non-pharmaceutical interventions not already captured by contact frequency, natural susceptibility from previous infection, differences in vaccination status, etc.). In order to distinguish it from direct susceptibility to infection, this vector will be referred to as q-susceptibility.
• The vector (h_j) describes age-specific differences in factors influencing transmission which are specifically related to the infectiousness of individuals, including, but not limited to, infectiousness after acquiring infection (e.g., due to differences in viral load upon exposure, proportion of asymptomatic individuals, differences in vaccination status, mask wearing, etc.). In order to distinguish it from infectiousness upon infection, this vector will be referred to as q-infectiousness.
• The remaining global proportionality factor captures any remaining residual effect and is of no relevance when considering relative q-susceptibility or q-infectivity.

In the remainder of this paper, we will talk about susceptibility or infectiousness when considering immunological aspects of disease transmission, while we will add the prefix q whenever quantities can carry additional effects related to susceptible and infectious individuals in order to avoid any ambiguity.

If we denote by w = (w_j) a vector representing the relative incidence within age class j (usually normalized such that ∑_j w_j = 1) and by M^T a matrix containing the social contact data (whose components m_ji represent the average daily number of individuals of age j who have a contact with a single individual of age i), then we have the following system: where R_t represents the reproduction number. The core matrix of this system, is called the next generation matrix and gives the number of new infections in a successive generation. Details concerning the construction of this matrix and the next generation principle can be found in Section Materials and methods.

Using our method, we compare the social contact matrix M^T extracted from the CoMix social contact survey in Belgium [12] and its derived relative incidence w to the incidence obtained from real-life data in Belgium coming from PCR positive tests [24]. This method allows for an estimation of either the relative q-susceptibility (a_i) or relative q-infectiousness (h_j) by age class, while assuming that the other set of parameters is known from the literature (i.e., holding one of the two vectors fixed). The chosen age groups are [0, 6) years, [6, 12) years, [12, 18) years, [18, 30) years and subsequent 10-year age classes up to 80+ years in order to account for the Belgian educational system. Due to the method, obtained results only have a relative interpretation, hence we present them under the assumption of a mean susceptibility of one for the first adult class [18, 30). The period of observation goes from 22 December 2020 to 26 May 2021 (and to 15 June 2021 for confirmed cases data). A detailed description of considered data, literature assumptions, fitting procedure and normalization method is presented in Section Materials and methods.

The estimated relative q-susceptibility for the whole period is presented in Fig 1 and implies that very young children in age group [0, 6) are about 0.182 (95% percentile bootstrap-based CI: 0.146–0.230) times as susceptible compared to the first adult age class [18, 30) with relative susceptibility equal to 1 (95% CI: 0.829–1.252). Primary school students aged [6, 12) have a relative susceptibility of 0.550 (95% CI: 0.427–0.629) and secondary school students aged [12, 18) a susceptibility of 0.603 (95% CI: 0.536–0.700). This shows an increasing q-susceptibility by increasing age up to [18, 30) after which the relative q-susceptibility tends to decrease slightly. Note, however, that this q-susceptibility captures not only differences in immunological susceptibility to infection and the rather low relative q-susceptibility in the [12, 18) age class could therefore be influenced by (compliance to) non-pharmaceutical interventions, over and above the age-specific contact frequencies.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Estimated relative q-susceptibility.

The estimation of q-susceptibility is performed by age class using the next generation principle under an assumption on age-specific infectiousness (0.54, 0.55, 0.56, 0.59, 0.7, 0.76, 0.9, 0.99, 0.99, 0.99). The calibration is performed on CoMix waves 12 to 23 (observation period: 22 December 2020 to 15 June 2021). Dots represent means and bars represent 95% percentile (nonparametric) bootstrap-based confidence intervals.

https://doi.org/10.1371/journal.pcbi.1009965.g001

The comparison of the relative incidence as estimated based on the positive PCR test data and the CoMix social contact data is presented in Fig 2. The social contact data are presented by waves starting with wave 12 on 22 December 2020 and an inter-survey wave interval of two weeks for subsequent waves (cf. details in Table A of S1 Appendix). The nationally collected data are represented in blue and the estimates coming from social contact data in two colors: in green, the initial estimate with a homogeneous proportionality factor (i.e., with a_i = 1 and h_j = 1 for all i, j) and in red, the estimate using heterogeneous q-susceptibility and infectiousness as presented in Fig 1. We clearly observe that estimates of the relative incidence under the homogeneous proportionality factor assumption (green) are very different from the empirical estimates (blue), especially for the young age groups. The relative incidence among adult age classes is estimated relatively well up to a constant, but the relative incidence for children coming from the homogeneous social contacts approach is clearly overestimated, except perhaps during times of school closure (see, e.g., wave 19). This finding provides a clear indication that SARS-CoV-2 transmission is different in children as compared to adults and that homogeneity assumptions should be avoided given that such assumptions could lead to erroneous projections.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Estimated relative incidence.

In blue: relative incidence based on Belgian PCR data. In green: estimated relative incidence based on the next generation principle under the assumption of a homogeneous proportionality factor. In red: estimated relative incidence based on the next generation principle with an estimated age-specific q-susceptibility under the assumption on age-specific infectiousness as in Fig 1. Dots represent means and bars represent 95% percentile (nonparametric) bootstrap-based confidence intervals.

https://doi.org/10.1371/journal.pcbi.1009965.g002

The result of estimating the q-infectiousness for the whole period, is depicted in Fig 3. The estimates also show a potential important heterogeneity concerning the proportionality factor on the infectiousness side. However, this reverse exercise provides less accurate results, with very large confidence intervals and some bootstrap estimates reaching zero, both being problematic when dealing with relative values. Those effects are the result of a lack of constraints for q-infectiousness. Indeed, while it is impossible to reach zero susceptibility for a specific age class when having at the same time non-zero incidence, it is technically allowed for age-specific infectiousness to be zero as the observed incidence could result from transmission from other age classes.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Estimated relative q-infectiousness.

The estimation of q-infectiousness is performed by age class using the next generation principle under an assumption on age-specific susceptibility (0.4, 0.39, 0.38, 0.79, 0.86, 0.8, 0.82, 0.88, 0.74, 0.74). The calibration is performed on CoMix waves 12 to 23. Dots represent means and bars represent 95% percentile (nonparametric) bootstrap-based confidence intervals.

https://doi.org/10.1371/journal.pcbi.1009965.g003

Exact estimates of the components of the q-susceptibility (a_i) and q-infectiousness (h_i) are provided in Tables E and I of S1 Appendix. Additional estimates under the assumption of homogeneity regarding infectiousness or susceptibility (i.e., estimating (a_i) under h_j = 1, ∀j or estimating (h_j) under a_i = 1, ∀i) are also presented in Figs D and L of S1 Appendix together with estimated values and the effect on the relative incidence. These additional estimates provide qualitatively similar results. Additional sensitivity analyses with regard to (h_j) and (a_j) showed that the variation of q-susceptibility estimates under different assumptions is clearly limited while sensitivity is greater concerning q-infectiousness estimates (see Figs B and C of S1 Appendix).

Time evolution of proportionality factors

Since proportionality factors capture several effects, they also capture time-dependent effects such as the reduction in susceptibility and infectiousness as a result of the vaccination campaign. In order to account for such a time evolution, we also performed the previous analysis using groups of two consecutive CoMix waves instead of the full period. The decision to consider two CoMix waves (28 days) together is motivated by the fact that a sufficiently long non-holidays period is required as social contacts in children are of importance and the heterogeneity of the transmission concerning adult classes is partially constrained by infection reported by children. Note that the gradual introduction of the alpha variant of concern might also interfere.

Estimates of the time-dependent q-susceptibility relying on the same (time-invariant) assumption with regard to the infectiousness vector are presented in Fig 4. A normalization of the relative values was performed over the different waves such that the average of the estimated factors for the age classes [0, 6), [6, 12) and [12, 18), i.e. , is assumed constant. This choice is motivated by the fact that the vaccination campaign was not including children during the entire study period, hence proportionality factors regarding susceptibility can be expected to be more stable for these age classes (however still influenced by the evolution of the proportion of susceptible individuals due to the ongoing epidemic). Thus, the results provide an estimate of the evolution of adults’ proportionality factors under an on average constant assumption for children [0, 18). A second normalization (global scaling) is performed under the assumption of a mean susceptibility of one for the first adult class [18, 30) for wave 12.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Estimated relative q-susceptibility over time.

The estimation of q-susceptibility is performed by age class and over time on groups of two consecutive CoMix waves, corresponding to a period of 4 weeks, under an assumption on age-specific infectiousness (0.54, 0.55, 0.56, 0.59, 0.7, 0.76, 0.9, 0.99, 0.99, 0.99). Dots represent means and bars represent 95% percentile (nonparametric) bootstrap-based confidence intervals.

https://doi.org/10.1371/journal.pcbi.1009965.g004

As indicated previously concerning the estimation during the complete time-period, a decreasing q-susceptibility is observed through adult age classes (see Fig 1), with the oldest age class (80+) being the least susceptible among all adults aged 18 years or older. This is a priori in contrast with usual assumptions regarding age-specific susceptibility to SARS-CoV-2 infection. However, in Fig 4, we clearly observe the highest relative q-susceptibility in the 80+ age class for the earliest waves as compared to all other age groups, or at least an equal q-susceptibility by considering the lower side of the confidence interval. Moreover, the q-susceptibility in the oldest age class decreases rapidly over time towards the lowest relative q-susceptibility equal to 0.446 (95% CI: 0.266–0.660) among the adult age groups. In general, the estimated q-susceptibility is almost similar across the different adult classes during the first period with the exception of the oldest class 80+ with an estimated relative q-susceptibility of 1.844 (95% CI: 0.920–3.127). Overall, q-susceptibility estimates of other age classes tend to decrease over time, albeit at a slower pace and to a lesser extent. This is in line with the implementation of the vaccination policy in Belgium, giving vaccination priority to residents of nursing homes (CoMix waves 13–16, see schematic timeline in Fig A of S1 Appendix) and the elderly in the general population (CoMix waves 18–21), while going gradually down from old to young throughout the study period.

Exact values of the estimates in Fig 4 are provided in Table F of S1 Appendix as well as time-dependent q-susceptibility and q-infectiousness under the various constraints mentioned above.

Time evolution of R_t using contact patterns

In order to check the utility and validity of the use of a heterogeneous proportionality factor, we illustrate its application by determining the reproduction number R_t, or more specifically the variation of the reproduction number over time, and comparing this evolution with R_t directly estimated from confirmed cases/hospitalizations data.

In Fig 5, the variation of the reproduction number computed from the CoMix data is compared to the reproduction number computed either from the number of cases [25] (panel a) or from hospitalizations [24] (panel b). Clearly, specific choices of q-susceptibility and q-infectiousness affect the computation and in Fig 5 we report results for the homogeneity scenario, heterogeneity scenario (corresponding to Fig 1) and temporal heterogeneity scenario (corresponding to Fig 4). A homogeneity assumption for q-infectiousness and q-susceptibility leads to a poor agreement with the reproduction number estimated from both confirmed cases and hospitalizations and is also characterized by a larger uncertainty. The use of the estimated heterogeneous reproduction factor agrees more with reality, with the use of temporal values is not leading to a substantial improvement.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Temporal reproduction number.

Reproduction number estimated from the CoMix data using the next generation approach in comparison to the reproduction number estimated from the number of confirmed cases (a) and from the number of hospitalizations (b). In green: estimated R_t under the assumption of a homogeneous proportionality factor. In red: estimated R_t with the estimated age-specific q-susceptibility under the assumption on age-specific infectiousness as in Figs 1 and 2. In blue: estimated R_t with the estimated temporal and age-specific q-susceptibility under the assumption on age-specific infectiousness as in Fig 4. Dots represent means and bars represent 95% percentile (nonparametric) bootstrap-based confidence intervals.

https://doi.org/10.1371/journal.pcbi.1009965.g005

Social contact data

Our study is based on Belgian social contact data collected within the CoMix survey [12, 13] during the COVID-pandemic between December 2020 and May 2021. These data are stored, processed and stratified by age by means of the online Socrates tool [13, 26, 27]. Participants were asked to fill in a contact dairy including all contacts made during a specific day, reporting the type of contact, location, and age of the contacted person, with a contact defined as an in-person conversation of at least a few words, or a skin-to-skin contact. The CoMix survey was repeatedly performed in different waves and different survey periods. More specifically, an initial survey period containing 8 waves was carried out between 4 March 2020 and 27 July 2020 targeting adults only. A second survey period, still ongoing in 2021, began on 11 November 2020 targeting participants of all ages. The waves are conducted with an interval of two weeks (14 days). For more detailed information on the CoMix survey and the stratification process, the reader is referred to [12, 13]. A detailed timetable of the CoMix waves and survey periods is presented in Table A of S1 Appendix. A schematic timeline of CoMix waves according to the evolution of the alpha variant of concern and vaccination campaign in Belgium is presented in Fig A of S1 Appendix.

We use the following notation. N_i denotes the number of individuals in the Belgian population of age i according to Belgian demographic data [28] and integrated into the Socrates tool [27]. In general, we use subscripts i as an index for the participant’s age, and j as an index for the contacted person’s age. The following observable quantities (dependent on the wave chosen) can be extracted from the survey:

• m_ij represents the average daily number of individuals of age j who are contacted by a participant of age i. The elements m_ij constitute a matrix M called social contact matrix.
• c_ij is the per capita contact rate per day for participants of age i with persons of age j in the population. The elements c_ij constitute a matrix C called the contact rate matrix. This matrix is related to the social contact matrix by the relation c_ij = m_ij/N_j.

In theory, due to the reciprocal nature of contacts, the total number of contacts between members of two age classes, as reported by participants in each of the age groups, must be equal, hence N_im_ij = N_ic_ijN_j = N_jc_jiN_i = N_jm_ji, which is equivalent to the condition that the contact rate matrix should be symmetric, i.e., c_ij = c_ji, ∀i, j. The social contact matrix M respects the relation N_im_ij = N_jm_ji, but is in general not symmetric due to differences in N_i and N_j. In practice, the observed total number of contacts and are not necessarily equal due to sampling bias, hence, we calculate the reciprocal social contact matrix by:

All these notations and definitions are similar to those described in detail in [29], except that the subscripts i and j and order of indices are inverted here such that the definition of the social contact matrix M corresponds to the default output of the Socrates tool [27].

Next generation principle

The social contact hypothesis [16] implies that the age-specific number of infectious contacts is proportional to the self-reported age-specific number of social contacts. There are two ways to interpret empirical social contact survey data in light of this hypothesis: either survey participants can be infected by their infectious contacts or participants can infect their susceptible contacts. Here, we consider the first interpretation as initial definition—since the CoMix survey did not specifically target infected persons, and symptomatic participants may have been less likely to participate in the survey. However, we will show that the two interpretations lead to the same mathematical result under the assumption of reciprocity of social contacts.

If we denote by w_j the incidence within age class j over a short observation interval (e.g. corresponding to a wave period), then v_j = w_j/N_j is the risk of being infected during the observation interval for that age class (incidence rate or force of infection). The new generation of infected people is given by: where q is a general proportionality factor completely defining the relationship between infection and contact events. The q-factor accommodates several effects such as susceptibility to infection, infectiousness upon infection, duration of the infectious period, type and effectiveness of contacts, seasonality, pre-existing natural and vaccine-induced immunity, etc.

The elements define a matrix K called the next generation matrix (or reproduction matrix) since k_ij represents the mean number of individuals of age i that are infected through a single individual of age j during their entire infectious period (for which the time between consecutive generations of infected individuals is chosen to be equal to the average duration of infectiousness).

Note that under the reciprocity assumption leading to a symmetric matrix C, the relation N_im_ij = N_jm_ji provides: corresponding to the second interpretation that survey participants (on the right side of the transpose contact matrix M^T) can directly infect their contacts (now on the left side) modulo the proportionality factor. This expression relying on the transpose of the social contact matrix obtained as a direct output of the Socrates tool, M, is chosen because of its better numerical stability.

The recurrence relation of the next generation matrix K: tends to a stable distribution due to the Perron–Frobenius theorem [30], i.e., with R_t corresponding to the reproduction number of SARS-CoV-2 [31] which is defined as the leading eigenvalue of the next generation matrix K. More specifically, estimation of the reproduction number R_t and the relative incidence w can be done by computing the leading eigenvalue and corresponding right-eigenvector of K. However, R_t depends on the proportionality factor q, which might be unknown, but the relative incidence w is independent of q and can therefore be directly extracted from the social contact data M^T. The reproduction number is initially the basic reproduction number R₀, but switches to the effective reproduction number as long as social contact data evolve and the proportionality factor q captures the depletion of susceptible. We emphasize here that the eigenvector w is only recovered up to a global constant and therefore individual components w_j have no meaning. What can be interpreted are relative ratios such as w_i/w_j, providing an estimate of the relative incidence in age class i as compared to the incidence in age class j. This vector is usually normalized such that ∑_i w_i = 1. In the same way, the incidence rate v_i can be recovered, in relative sense, as the leading left-eigenvector of M^T.

The switch from a homogeneous proportionality factor q to a heterogeneous q_ij is performed by assuming: where the vector (a_i) acts on the susceptible side, the vector (h_j) acts on the infectiousness side, and is a remaining global proportionality factor captures any residual effect. This remaining factor has no influence on the computation of the relative incidence w. However, due to the presence of , the vectors (a_i) and (h_j) only have a relative interpretation.

The heterogeneous next generation matrix is defined as:

We note that we are working here with a next generation matrix with small domain. There also exists a next generation matrix with large domain taking explicitly into account the different states of the disease and their duration for each age class [8]. However, the small domain approach is appropriate here since we do not work with a dynamical system and heterogeneity in disease duration is part of the effects captured by the proportionality factors.

Estimating relative q-susceptibility (a_i) and q-infectiousness (h_j) from COVID-19 age-structured indicators

The vectors (a_i) and (h_j) have an important impact on the determination of the leading right eigenvector in the system: The obtained relative incidence w* can be compared with the normalized relative incidence estimated from the observed incidence data in Belgium. Using this approach, we are able to determine q-susceptibility and q-infectiousness corresponding to SARS-CoV-2 transmission in Belgium. However, (a_i) and (h_j) vectors cannot be estimated simultaneously in a unique way from this process since there remains an indeterminacy [10, 17]. Nevertheless, the identifiability problem can be solved by imposing a constraint on one of the two vectors.

For this study, we choose each time a heterogeneous constraint coming from the literature as well as a homogeneous constraint (whose results are only presented in S1 Appendix). The heterogeneous constraints are defined from the following assumptions:

• For the assumption on infectiousness h_j (estimation of q-susceptibility parameters a_i): We consider the probability of an asymptomatic COVID-19 infection in case of SARS-CoV-2 exposure in the Belgian population to be as assumed in [19] using data from [2]. Assuming that the relative infectiousness of asymptomatic versus symptomatic individuals is 0.51 [19], we obtain the following constraint:
• For the assumption on susceptibility a_i (estimation of q-infectiousness parameters h_j): The assumption is taken from [1]:

Data and fitting procedure

We use Belgian data on daily incidence of COVID-19 confirmed by means of a positive PCR test, as provided by the Belgian Institute for Public Health, Sciensano [24]. In order to reduce testing biases, the period of study is restricted to a period with almost constant testing policy (mandatory testing for both symptomatic cases and asymptomatic close contacts or red zone travelers) and before biases are induced by the introduction of the EU Digital COVID Certificate, see [32] or Table A of S1 Appendix) for a summary). Since there is a delay between a change in social contact behavior and its effect on the relative incidence, we consider PCR test results for the period starting 7 days after the onset of a specific CoMix wave and lasting for 14 days thereafter.

Concerning social contact data, the initial CoMix survey waves (1 to 8) are discarded due to a variable testing policy and lack of information regarding child-child contacts. The three subsequent waves (9 to 11) are also discarded since, despite the introduction of measuring child-child contacts, the information was collected using a different survey formulation. CoMix waves 12 to 23 correspond to a period with constant testing policy, an identical survey design as well as without vaccination in children, which implies that the results with regard to age classes [0–6), [6–12) and [12–18) years are expected to be more stable. The start of wave 12 corresponds to 22 December 2020 when the vaccination campaign in adults has not been started and the last wave considered corresponds to 26 May 2021, with PCR tests considered up to 15 June 2021 (thus when vaccination of the oldest individuals in the Belgian population was nearly completed).

The estimation of the parameters a_i or h_i is performed using the statistical software package R. A minimum-distance estimation is performed using the Hellinger distance [33] (which is suitable for distributions) between relative incidences w* and . The optimization is done by means of a random search numerical method [34], starting from an initial homogeneous prior (a_i = 1 or h_i = 1, ∀i = 1…10). The process uses a Gaussian random walk with steps of length which are performed until no change in the distance is observed during 100 consecutive iterations. The sensitivity analysis is provided by repeating the process over 200 nonparametric bootstrap runs using the previous posterior as new prior. Uncertainty is quantified using means and 95% percentile confidence intervals (i.e., 2.5% and 97.5% quantiles of all bootstrap-based estimates).

Since q-susceptibility and q-infectiousness vectors represent relative values, a normalization process should be chosen for the representation of the results. We choose here the following (two-step) normalization process:

• The average or of the estimated factors for the age classes [0,6), [6,12) and [12,18) (children, no subject to vaccination during the complete period) is assumed to be constant across all bootstrap runs and wave groups if applicable (i.e. all combinations of two CoMix waves).
• The mean or (across the bootstrap runs) for the first adult age class [18, 30) is set to one. This constraint is chosen because we mainly want to compare susceptibility and infectiousness of children versus adults while the age class [18, 30) is one of the most stable ones in the bootstrapping process. Note that this second normalization step is only a global scaling, thus conserving the confidence interval around one for the age class [18, 30).

R_t evolution from contact patterns

Via the next generation approach, the ratio of the eigenvalues of two next generation matrices can be used to evaluate the relative reduction in the reproduction number. This can be done to compare the temporal reproduction number derived from the CoMix survey with independent evaluations of the reproduction number. We use as comparison the R_t computed from the daily number of cases [25] and the daily number of hospitalizations [24]. In order to account for the time delays associated with infections and hospitalizations (e.g. time to develop symptoms, time to hospitalizations, etc.), the reproduction number computed from the CoMix social contact data was shifted forward in time. A time shift of 7 (14) days is considered when comparing R_t estimates with the reproduction number computed from the number of confirmed cases (respectively hospitalizations). These time shifts are chosen in order to take account of a mean delay between infection and (symptomatic) testing as well as an additional delay between symptom onset and hospitalization [35]. As the reproduction number is known up to the overall constant , we fix the reproduction number for CoMix wave 12 to be equal to the reproduction number computed from infections or hospitalizations. Uncertainty due to sampling variability is estimated via 10000 nonparametric bootstraps.