Prioritization strategies, rollout speed, and vaccine efficacy.

The proposed model allows to investigate the impact of the behavioural response under different conditions. We consider six populations matching the characteristics of the countries under consideration and exploring the model with the same epidemic initial conditions. The experimental setup considers that each population has already experienced a previous wave of infections and that restrictive measures are in place to mitigate the spreading. Therefore, we set the basic reproductive number R₀ = 1.15, 0.5% of initially infected individuals, and 10% of immune individuals. In line with estimates of vaccine efficacy against COVID-19 we set VE_S = 70% and we choose VE_Symp such that VE = 90% [57], while we let vary the vaccine rollout speed r_V between 0.1% and 1% to cover the spectrum of real vaccination rollout speeds of the vaccination campaigns across the globe. Peru, for example, administered on average 0.05 daily doses per 100 people in the week commencing on the 8^th of March, Italy administered 0.30, and Serbia 0.70 [58]. In Fig 1B we show the relative deaths difference for the three vaccine prioritization strategies and the three vaccine rollout speed r_V = 0.1%, 0.5%, 1.0%. The relative deaths difference (RDD) is defined as the fraction of deaths averted thanks to the vaccine with respect to a baseline simulation without vaccine and, therefore, no behavioural response. For example, RDD = 0.2 indicates that 20% of deaths are averted. Note how this quantity may become negative if the behavioural response causes more deaths than those averted thanks to the vaccine. We refer the reader to the Materials and Methods section for further details. Moreover, we explore a range of behavioural responses running the simulations for a range of values of the parameter α. Starting from α = 0 (which implies no behavioural response), we perform simulations with increasing α values (implying stronger reactions), while keeping constant the other behavioural parameter (γ = 0.5).

As a first observation, across the different countries and rollout speeds considered, the strategy aiming to curb the severity of the pandemic (i.e., vaccine strategy 1) is indeed the most effective in reducing the number of deaths.

As illustrative and concrete example let us consider the case of Canada. When α = 0 (i.e., no behaviour response) and r_V = 1% with this strategy the fraction of averted deaths is 0.77 with respect to a baseline without vaccine. This fraction reduces to 0.72 with the homogeneous vaccination strategy, and to 0.70 with the strategy prioritizing younger individuals. When r_V = 0.5%, these fractions reduce to, respectively, 0.67, 0.64, 0.62. The ordering of the strategies slightly changes when r_V = 0.1%, in this case we obtain 0.28. 0.35, 0.36. Overall, the picture that emerges is consistent with previous studies on the allocation of COVID-19 vaccines among age groups showing that the strategy minimizing deaths is generally the one targeting the elderly, even though for some combination of the parameters the strategy targeting the younger might be preferable [43, 44].

When the behavioural response to the vaccine rollout is considered (i.e., α > 0), we note a consistent decreasing trend of the relative deaths difference for increasing values of α. This shows that the behavioural response impacts the unfolding of the epidemic, and that a relaxation of NPIs leads to a smaller fraction of averted deaths. Interestingly, a concerning effect also emerges: in some conditions, as non-compliance becomes larger, the benefit brought by the vaccine is nullified and the number of observed deaths increases with respect to the no-vaccine and no-behaviour change scenario (i.e., the relative deaths difference become negative). This is solely attributable to the behavioural reaction to the vaccination campaign which in turn is not efficient enough to balance behaviour relaxation. Indeed, this phenomenon is observed in particular when strategies that do not target a reduction in severity are employed, and when the vaccine rollout speed is low. In the case of Serbia, with r_V = 1% and a strategy aimed at reducing severity, the fraction of averted deaths goes from 0.84 when α = 0 to 0.73 when α = 10, with a potential loss of 0.11 attributable to the NPIs relaxation. This effect is more pronounced in the case of the vaccination strategy 2 (or 3): the fraction of averted deaths goes in this case from 0.76 (0.73) when α = 0 to 0.59 (0.50) when α = 10, with a potential loss of 0.17 (0.23) attributable to the NPIs relaxation. Similarly, lower vaccine rollout speed are impacted more significantly by behavioural responses. In the previous example, an immunization campaign with a r_V of 1%, 0.5%, and 0.1% would induce a relative deaths difference, in the case of the vaccination strategy 1, of 0.84, 0.75, 0.40 when α = 0. By setting instead α = 10, these figures would drop to 0.73, 0.44, −0.67 with a loss of 0.11 in the first, of 0.31 in the second, and of 1.07 in the third case.

In the results presented in Fig 1B, we observe that the most disadvantaged countries are the developing economies, Egypt and Peru. At first, this may seem counter-intuitive. Indeed, these countries have a lower fraction of individuals aged over 65. A possible explanation is given by the high activity of young individuals combined with the high inter-generational mixing. Moreover, it is worth stressing that these results are to be intended in relative terms: a relative worst performance in averting deaths does not necessarily imply a worst absolute performance. In other words, the relative impact of behavioral responses might be stronger in those countries, but their age pyramid might induce still a smaller number of deaths in absolute terms. On this point, it is important to remember that we are not considering possible modulations of the IFRs due to the heterogeneity in health care infrastructure, prevalence of co-morbidities and access to health care. In the S1 Text we provide an example highlighting this point by comparing Italy and Egypt. We observe also that the difference between countries is more profound for the vaccination strategy aimed at reducing transmission (i.e., strategy 3). Furthermore, behavioural relaxation widens the distance between the countries. With r_V = 0.1% and vaccination strategy 3, the gap between the relative deaths difference for Italy and Egypt is 0.24 when α = 0. This figure increases to 0.87 when α = 10.

In the S1 Text we repeat the analysis considering the fraction of averted infections instead of deaths. We find that the ranking of allocation strategies is inverted. Indeed, in line with previous findings [43], when considering averted infections the most efficient strategy is the one targeting the younger population first, followed by the homogeneous, and the strategy aimed at curbing severity. In the S1 Text, we also repeat the results presented in Fig 1B exploring different vaccine efficacy VE (50%, 70%, 90%) rather than rollout speed. Similarly to the case presented in the main text, also in these additional analyses we find that early behavioural relaxation reduces the fraction of averted infections, influences heterogeneously the prioritization strategies, and impacts more significantly lower vaccine efficacy or slower rollout.

To further explore how to mitigate the impact of behavioural relaxation once that the vaccination campaign started, in Fig 2 we systematically explored the interplay between vaccine efficacy VE and vaccination rollout speed r_V. The black dashed lines highlight the combinations of VE and r_V that achieve a 30% drop in observed deaths (taken as reference value), in the absence of a behavioural response (i.e., α = 0). In most countries, it can be achieved with r_V smaller than 0.2%. On the contrary, when even a mild behavioural response is active (red dash-dotted lines, α = 1), the rollout speed has to increase greatly when vaccine efficacy diminishes: for the case of Italy, a 30% drop in deaths would be achieved with a r_V of 0.4% when the vaccine efficacy is 90%, but in the case of VE = 60% the rollout speed has to grow up to 0.8% to achieve the same result.

Download:

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

Fig 2. Interplay between vaccine efficacy and rollout speed.

Contourplot of the relative deaths difference in the scenario with a mild behavioural response (α = 1) for different combinations of VE and r_V. We let VE_S vary between 30% and 70% and we choose different VE_Symp such that the overall efficacy VE vary between 50% and 90%. A 30% reduction of deaths is highlighted with a red dashdotted line. The black dashed line highlight the 30% death drop achieved with a vaccination campaign without the behavioural component (α = 0). Vaccination Strategy 1 is considered, parameters used are γ = 0.5, R₀ = 1.15, r = 1.3, 0.5% of initially infected, 10% of initially immune individuals, and simulations length is set to 1 year.

https://doi.org/10.1371/journal.pcbi.1009346.g002

Epidemic trajectories and impact of behavioural relaxation.

As a way to ground the model on more realistic epidemic scenarios, we calibrate it to the real epidemic trajectories of the six countries considered. The calibration is performed via an Approximate Bayesian Computation technique (ABC) [59] on weekly deaths for the period 2020/09/01–2020/12/31. In doing so, we estimate some key epidemiological parameters matching the context of each country, thus defining realistic initial conditions for the model. To capture the effects of non-pharmaceutical interventions on the contacts between people we consider data from the Google Mobility Report [60] and the Oxford Coronavirus Government Response Tracker [61] until week 11 of 2021. After the calibration step, the model evolves from 2021/01/01 up to 2021/06/01 with the vaccination rollout for each of the countries. Note that in some of these countries, the vaccination campaigns officially started in the second half of December, though mostly symbolically.

In Fig 3A we show key indicators resulting from the calibration. In particular, for the six countries we report the boxplot for the calibrated infection parameter β, the projected number of symptomatic infections per 100, 000, and the fraction of recovered individuals on the 2021/01/01, which is the start of the vaccination campaign in our simulations. We acknowledge a significantly high acquired immunity in the case of Peru, where the estimated attack rate as of 1^st January, 2021, is around 38%. Nonetheless, this figure is line with other available estimates [58]. We observe how the calibration allows for an heterogeneous representation of the epidemiological conditions of the different countries. In Fig 3A we also show the effects of restrictive measures on contacts. As a proxy, we consider the ratio between the leading eigenvalue of the contacts matrix (considering the restrictions) and the leading eigenvalue of the baseline contacts matrix (without restrictions). The leading eigenvalue of the contacts matrix influences the reproductive number of the disease (see Materials and methods), and more broadly it measures the intensity of contacts among people. By normalizing with respect to the baseline contacts matrix (with no restrictions) we can grasp the strictness of the measures in place. For example, a ratio of 0.3 would imply, in our simulations, a 70% reduction of the reproductive number with respect to the baseline without restrictions. The effect of restrictive measures on contacts varies over time up to week 11, 2021. In the case of Italy, for example, we observe the partial ease of the measures during January and February (i.e., ρ(C_t)/ρ(C_b) increases) followed by the tightening of measures in March aimed at curbing the third wave of infections. Afterwards, we keep the mixing levels as those observed for week 11 (more details in the Materials and methods section). This is a conservative assumption as we can imagine that the seasonality and impact of the vaccination campaign might induce a relaxation of NPIs as the number of cases and deaths, hopefully, will go down. In this scenario, the countries with the strictest measures in place after week 11, are Peru and Italy, and Canada, while Egypt and Ukraine show the most permissive measures. In the S1 Text, we repeat the analysis using real data to compute contacts reduction parameters for the whole period. The results are in line with the observations presented below.

Download:

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

Fig 3. Giving up NPIs during rollout may nullify the benefits brought by the vaccine.

A) We display for the different countries the boxplot of the calibrated infection parameter β, the projected number of symptomatic infectious cases per 100′000 and the fraction of recovered at the 2021/01/01, start of the vaccination campaign in our simulation. We also report the ratio between the leading eigenvalue of the contacts matrix considering restrictions and of the baseline contacts matrix with no restrictions. B) We display the median relative deaths difference for the calibrated model in the different countries. We consider the three vaccination strategies and two possible rollout speed: r_V = 1% (faster rollout), and r_V = 0.25% (slower rollout). We run the model over the period 2021/01/01–2021/06/01. Other parameters are γ = 0.5, α = 10², r = 1.3, VE_S = 70% (VE = 90%).

https://doi.org/10.1371/journal.pcbi.1009346.g003

In Fig 3B we study, for the calibrated model in the six countries, the difference between the fraction of averted deaths with respect to a baseline without vaccine in the case of a strong behavioural reaction to vaccine deployment (i.e., α = 10²) and in the case of no relaxation of NPIs (i.e., α = 0). This quantity, which we indicate with Δ_RDD, represents the additional fraction of deaths that occur due to lower compliance to COVID-safe behaviours in our simulations (see Materials and methods for more details). We study two vaccination rates. A fast deployment that manages to vaccinate r_V = 1% of the population daily (aligned with the preparedness plan for Influenza pandemic [62] and vaccination rates achieved by countries like Israel or Chile), and a slower rollout r_V = 0.25%. Worryingly, in the different countries, even in a scenario with several restrictions and a successful vaccination campaign in place, it is possible to witness a significant increase in deaths exclusively due to a relaxation of COVID-safe behaviours. As a result of the calibration step, the countries under investigation face different epidemiological conditions (e.g., effective reproductive number, immunity from previous waves, initial number of infected, estimated effects of restrictions). These translate into phenomenological differences with respect to the results of the previous analyses. In particular, Ukraine and Egypt are significantly impacted by behavioural relaxation in the case of the slower rollout r_V = 0.25%, while Peru, differently from the previous analysis, is less impacted. From Fig 3A we see that in the first two countries the estimated effect of restrictions on contacts (especially in the last weeks) is smaller with respect to the others. On the other hand, Peru features tougher NPIs, together with Italy and Canada, which are also less impacted by behaviour in Fig 3B. This underlines how, besides the characteristics of the vaccination campaign, also the epidemiological conditions and the measures in place are important factors influencing the behaviour-vaccine interplay. In the case of the faster rollout r_V = 1%, the obtained values of Δ_RDD are smaller and the differences between countries are less pronounced. A possible explanation is that when a fast rollout is employed the influence of epidemiological conditions and restrictions become less important because of the efficiency of vaccine administration. Furthermore, we observe that, for both r_V, the vaccination strategy targeting a reduction of severity (strategy 1) performs generally better in case of NPIs relaxation with respect to the other strategies. In the S1 Text we investigate more in detail the comparison between vaccination strategies in terms of robustness to behaviour relaxation considering both averted deaths and infections. We find that, when considering deaths, a vaccine allocation aimed at reducing disease severity (i.e., strategy 1) is always preferable. Instead, vaccination strategy 2 and 3 perform better when the number of avoided infections is used as evaluation metric. Information about vaccination strategy robustness against behavioural changes might be used to tune and design resilient rollout campaigns. Policy-makers might also consider to optimize vaccination strategies with respect to a combination of multiple metrics. In this direction, our analysis provide additional insights, and suggests that behavioural changes might play an important role possibly modifying the impact of vaccine prioritization strategies.

Model definition.

We propose an age-stratified compartmental model that incorporates both the vaccination process and the behaviour dynamics possibly linked to it. See Fig 4 for a schematic representation. Individuals are divided into 16 five-year age groups (last group is 75+). The virus transmission is modeled using the following approach. Susceptible individuals (S compartment) in contact with infectious compartments become latent (L compartment). After the latent period (ϵ⁻¹), L individuals enter the pre-symptomatic stage of the infection (P). Then, they can transition either in the asymptomatic (A) or the symptomatic stage (I) at rate ω (the length of time spent in the L and P compartments is the incubation period). The probability of being asymptomatic is f. After the infectious period (μ⁻¹), both I and A individuals enter the Recovered compartment (R_I, R_A). Alternatively, I individuals can also die and transit to the D compartment. Note how we include a delay of Δ days from the time individuals enter the compartment D and die (D^o compartment, the superscript stands for “observed”). The infectious compartments are P, A, I. The transmission rate is β and the force of infection is dependent on the age-stratified contact matrix , whose element C_ij represents the average number of contacts that an individual in age group i make with individuals in j per day. The matrix C has four location-specific contributions: contacts at home, workplace, school, and other locations. We adopt country-specific contacts matrices provided in Ref. [78]. We assume that P and A have lower infectiousness with respect to symptomatic I (βχ, with χ < 1). Similar approaches have been used in several modeling studies in the context of the current pandemic [70, 79].

Download:

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

Fig 4. Compartmental model.

We consider an extension of the classic SLIR model adding compartments for presymptomatic (P) and asymptomatic (A). We also design compartments for vaccinated (V), dead (D describes individuals that will die with a delay of Δ⁻¹ entering the compartment D^o), susceptible (S^NC) and vaccinated (V^NC) individuals that do not comply with COVID-safe behaviours. The vaccine offers a protection VE_S against infection and VE_Symp against symptoms that can lead to severe outcomes such as death. The transmission rate for susceptible is β and for susceptible non-compliant rβ (r > 1). The parameter α regulates the transition from compliant to non-compliant behaviours, while γ regulates the opposite flow. Arrows describe the transitions between compartments. For simplicity of visualization, we do not display the compartments mediating the different transitions. For example, the transition S → L is mediated by the infectious compartments (P, A, I, P^V, I^V, A^V). The compartmentalization is then extended to account for empirical age-structure and contact matrices.

https://doi.org/10.1371/journal.pcbi.1009346.g004

On top of the disease dynamics, we model both the vaccination process and the behavioural change that is possibly coupled to the vaccination. More in detail, after the start of the vaccination campaign at t_V, at each time step, a fraction of the susceptible population receive a vaccine and transit to compartment V. We introduce the rollout speed r_V as the number of daily available doses expressed as a percentage of the total population. We adjust the number of doses available per day considering the fraction of susceptibles among all those that can receive the vaccine (S, L, P, A, and R_A) and we assume that the remaining doses are wasted. We consider a “leaky” vaccine that reduces susceptibility with a certain efficacy VE_S and the probability of developing symptoms by VE_Symp. In other words, the infection rate for V individuals is β(1 − VE_S) and the probability of entering the infectious symptomatic compartment I^V from P^V is (1 − f)(1 − VE_Symp). Similar approaches have been used in previous works in the context of mathematical modeling of COVID-19 immunization campaigns [44, 80]. As mentioned above, we consider three vaccination strategies: one in which the vaccine is given in decreasing order of age (vaccination strategy 1 aimed at reducing the severity), one in which it is given homogeneously to the population (vaccination strategy 2), and one in which it is first given to individuals in age brackets 20–49 and then to the rest of the population (vaccination strategy 3 aimed at reducing the transmission of the virus) [10]. In parallel, we imagine a behavioural dynamics triggered by the presence of the vaccine. Indeed, susceptible individuals (both vaccinated and not) may start adopting less safe behaviours because reassured by the presence of an effective vaccine. This is encoded in the model with a transition from the compartment S (V) to a new compartment S^NC (V^NC)—NC stands for non-compliant—of individuals that protect less themselves and as a result get infected at a higher rate. The parameter r > 1 captures the increased risk of contagion for NC individuals. It modulates the contribution to the force of infection from the S^NC and V^NC compartments. In the simulations presented in the main text we used r = 1.3 imagining a scenario where the relaxation of COVID safe behaviors increases the transmission rate of 30%. This choice is informed by the estimated effects of NPIs such as face masks and social distancing on COVID-19 spreading, as reported in the literature [4, 20, 81, 82]. Nonetheless, in the S1 Text we perform a sensitivity analysis considering also r = 1.1, 1.5. We propose two mechanisms to model the behavioural transition. In the first mechanism, the transition towards non-compliance happens at a rate α and it is catalyzed by the cumulative fraction of individuals that received a vaccine (v_t, including both compliant and non-compliant). The opposite transition from S^NC (V^NC) to S (V) happens at rate γ and is catalyzed by the number of deaths per 100, 000 observed in the previous time step (, including both compliant and non-compliant). Indeed, an increase in deaths is frequently used—especially by media—as an indicator of the severity of the current epidemiological situation. Existing literature suggests that risk perception (in the form of number of infected individuals or deaths) and communication of such risk significantly affect adherence to personal mitigation strategies such as social distancing and wearing face masks [5, 83]. Expressing the number of deaths in proportion to the population allows us to compare countries of different size. Overall, this approach aims to depict the adaptive nature of human behaviour where individual choices are influenced by vaccination and pandemic progression [84]. In the second mechanism, S (V) individuals transit to the non-compliant compartment S^NC (V^NC) at a constant rate α. We also account for the possibility of non-compliant going back to safer behaviours again at a constant rate γ. To simplify the narrative, we present the results considering this second mechanism only in the S1 Text. The overall picture discussed above does not change significantly. Note that in order to avoid issues with transition probabilities large than one we model the rates as λ_X→X^NC = 1 − exp^−g(α), λ_X^NC→X = 1 − exp^−h(γ) with X = [S, V] and where the specific expression of the exponent depend on the two mechanisms described above: g(α) = αv_t, in the first one, and g(α) = α, h(γ) = γ in the second mechanism. Note how for small values of g(α), h(γ) the rate converges to the usual mass-action law. The model just described can be written down as the following system of differential equations for individuals in age group k: (1) Where is the force of infection for age group k.

The basic reproduction number is , where is the contacts matrix weighted by the relative population in different age groups. In the S1 Text we provide details on its derivation. We adopt the model integrating numerically the equations, thus it is deterministic. However, it is worth stressing that when we calibrate the model to the real epidemic trajectory in the six countries, we use a probabilistic framework through an Approximate Bayesian Computation technique. Said differently, the calibrated parameters are characterized by posterior probability distributions rather than exact values. For this reason, the results presented in Fig 3 (i.e., the median of model’s projections) are to be intended as an ensemble of multiple trajectories generated sampling from the posterior distribution of the fitted parameters. The model is implemented in the programming language python with the use of the libraries scipy [85], numpy [86], and numba [87]. The visualization of the results is realized with the library matplotlib [88].

In the Results section, we kept for simplicity γ = 0.5 and we let vary α. The choice of γ is informed by the maximum number of deaths observed in the countries of focus. In Italy, for example, the maximum number of deaths reported on a single day is around 1, 000. Therefore, this value of γ is such that, in a similar situation, non-compliant individuals would likely return to COVID-safe behaviours. As described in the previous paragraph, in the dynamic-rate behavioural mechanism we model the transition rate as (X = [S, V]), where is the number of deaths per 100, 000 individuals observed at time t − 1. Hence at the peak of deaths the transition rate towards compliance is . The sensitivity to this choice is discussed in the S1 Text, where we also report additional analyses to better understand the effects of the behavioural parameters. More in detail, we plot the rates and together with the actual number of NC individuals in time for several values of α, γ, v_t, and different epidemiological conditions.

In Table 1 we report a list of the model’s parameters together with their values used in the simulations and the related sources. We also indicate which values are optimized during the calibration step (described in the next sections).

Download:

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

Table 1. Model parameters.

https://doi.org/10.1371/journal.pcbi.1009346.t001

Relative deaths difference.

In the previous analyses, we compared the vaccination strategies in the six countries using different evaluation metrics. The relative deaths difference (RDD) represents the fraction of deaths that are avoided in a simulation with vaccines (and behavioural response) with respect to a baseline simulation without vaccines (and no behavioural response). We compute it as: Where we made explicit the dependence from α. By definition, a value of RDD = 0.2 indicates that 20% less deaths are observed in the simulation with vaccines administered. Negative values of RDD capture instead scenarios where the behavioral response induces more deaths that in the case without vaccines. An analogous quantity can be easily computed for infections. In Fig 3B we have also considered the difference between the fraction of averted deaths with respect to a baseline without vaccine in the case of behavioural reaction (i.e., α > 0) and in the case of no relaxation of NPIs (i.e., α = 0). We indicated this quantity with Δ_RDD(α), and we compute it as follows:

With respect to the simple RDD, the Δ_RDD provides a slightly different piece of information. Indeed, by subtracting the RDD obtained with α = 0 we isolate only the effects of behaviour. As a clarifying example, Δ_RDD(α) ∼ 0 refers to a negligible effect of behaviour on the fraction of averted deaths. Instead, Δ_RDD(α) < 0, indicates that due to behaviour relaxation more deaths are occurring. Also in the case of Δ_RDD(α), an analogous for infections can be easily derived.

Approximate Bayesian computation.

We calibrate the model for each country using the Approximate Bayesian Computation (ABC) rejection method [59, 100]. At each step of the rejection algorithm, a set of parameters θ is sampled from a prior distribution and an instance of the model is generated using θ. Then, an output quantity E′ of the model is compared to the corresponding real quantity E using a distance measure s(E′, E). If this distance is greater (smaller) than a predefined tolerance ψ, then the sampled set of parameters is discarded (retained). After accepting N sets, the iteration stops and the distribution of accepted parameters is an approximation of the real posterior distribution P(θ, E). The free parameters of our model and the related prior distributions are:

• The transmission rate β. We explore uniformly values of β such that the related R₀ is between 0.8 and 2.2. The basic reproduction number of SARS-CoV-2 is higher [101], but we consider lower values since our calibration starts on 2020/09/01 when restrictions were in place to mitigate the spreading.
• The delay in deaths Δ ∼ [14, 25]. Indeed, for COVID-19 the average time between symptoms onset and death is about 2 weeks [97] and we also account for possible additional delays in death reporting.
• The initial number of infected individuals. We explore uniformly values between 0.5 and 15 times the number of reported cases in the week before the start of the simulation. We then assign these individuals to the infected compartments (L, P, A, I) proportionally to the time spent there by individuals (ϵ⁻¹ for L, ω⁻¹ for P, and μ⁻¹ for I, A), and we split between I and A individuals considering the fraction of asymptomatic f.

The model is calibrated on the period 2020/09/01–2020/12/31 using the weighted mean absolute percentage error on weekly deaths as an error metric with a tolerance ψ = 0.3 (ψ = 0.4 for Egypt for convergence issues) and 5, 000 accepted parameters set.

Model initialization.

The number of individuals in different age groups is initialized considering the 2019 United Nation World Population Prospects [102]. We consider 16 age brackets of five-years, except for the last one that includes 75+ individuals. As specified above, the initial number of infected individuals is calibrated considering the total number of confirmed cases in the week before the start of the simulation according to the data issued from the European Centre for Disease Prevention and Control [103]. To initialize the number of non-susceptible individuals (placed in the R compartment) we compute the average of several publicly available projections of total COVID-19 infections up to 2020/09/01 (i.e., start of the simulation) from different modeling approaches [58]. Both the initial number of infected and of non-susceptibles is assigned homogeneously across age groups. Other parameters used are ϵ⁻¹ = 3.7 days, μ⁻¹ = 2.5 days, ω⁻¹ = 1.5 days χ = 0.55, f = 0.35 in line with current estimates of COVID-19 infection dynamics parameters [90–92, 95, 96]. We use the age-stratified Infection Fatality Rate (IFR) from Ref. [56].

Modeling the effects of NPIs on contact matrices.

In our model, we incorporate the implementation of top-down NPIs by changing the contacts patterns defined by the contacts matrix C. As mentioned, we use the country-specific contacts matrices provided in Ref. [78]. These are made by four contributions: contacts that happen at school, workplace, home, and other locations. In a baseline scenario, the overall contacts matrix C is simply the sum of these four contributions. Following Ref. [104], we implement the reductions in contacts, due to the restrictions, multiplying the single contribution by a reduction factor η_i(t). Thus, in general the overall contacts matrix at time t become: (2)

For simplicity, we assume no changes to contacts at home, though lockdowns tend to increase them [64]. For the contacts locations work and other locations we use data from the Google Mobility Report [60]. In detail, we consider the fields workplaces percent change from baseline to model reduction in the contacts location work and the average of the fields retail and recreation percent change from baseline and transit stations percent change from baseline for other locations. A general entry of the report p_l(t) represents the percentage change on day t of total visitors to a specific location l with respect to a pre-pandemic baseline. From p_l(t) we derive a contacts reduction coefficient as follows: η_l(t) = (1 + p_l(t)/100)². Indeed, the number of potential contacts in a specific location is proportional to the square of the the number of visitors. We model contacts reduction in school considering instead the Oxford Coronavirus Government Response Tracker [61]. More in detail, we use the ordinal index C1 School closing. The index ranges from a minimum of 0 (no measures) to a maximum of 3 (require closing all levels). We turn this quantity into the contacts reduction coefficient in school as follows: η_s(t) = (3 − C1 School closing)/3. We use these datasets to inform contacts reductions up to week 11, 2021. After, we assume that contacts remain at the same level of week 11. Note how we let vary both the transmission rate β and the contact matrix. The former describes the risk of infection given contacts with infectious individuals. This is function of the disease (which is assumed to be the same, we don’t consider multiple or emergent new strains possibly more transmissible) and of the protective behaviours such as social distancing and use of face masks. The latter describes variations to the number and types of contacts induced by top-down NPIs as for example remote working, schools closure and lockdowns. By splitting the contributions to the force of infection of transmission rate and contact matrix we are able to take into consideration different behavioural attitudes which, given the same number of contacts, might lead to higher or lower risks of infection. This allows us to consider explicitly both top-down and bottom-up NPIs.

Calibration results.

In Fig 5, we report the results of the calibration. It is important to stress how our goal is not to develop a predictive model aimed at forecasting the pandemic trajectory. The fit is used to ground the model and to define the epidemic conditions at the start of the vaccination campaign in the six countries. In fact, our aim is to understand the possible interplay between behaviours and vaccine rollout which is also function of the epidemic progression. In the Figure we report the official and simulated weekly number of deaths (median, 50% and 95% confidence interval). Despite its simplicity and approximations, the model is able to reproduce the evolution of the pandemic in the six countries capturing well its progression after the summer. In the S1 Text we also report the posterior distributions for the parameters.

Download:

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

Fig 5. Calibration results.

For each country we represent the observed and simulated weekly deaths (median, 50% and 95% confidence intervals).

https://doi.org/10.1371/journal.pcbi.1009346.g005