# data[].attributes.attachment_link.readable_text
A new take on optimal vaccine prioritization: when age-based strategy is efficient?

Abstract

The rapid spread, high mortality rate, and hospital overcrowding made the SARS-CoV-2 virus a significant challenge for humanity. The emergent situation urgently called for investments in vaccine development as a critical measure to mitigate this issue. However, the financial costs and time required for vaccine development posed considerable obstacles to achieving comprehensive vaccination coverage. Therefore, understanding the most effective immunization strategy was paramount to contain the virus's spread and optimize resource allocation in public policies. Age-based prioritization was applied in many regions, and one of the objectives of this work is to study under which circumstances such a strategy is efficient. Our results show that age-based vaccination is a poor strategy for disease eradication, although it has a fair performance in diminishing the number of deaths. Furthermore, other schemes, such as PageRank-based immunization, are much better options in general. We believe these results are of relevance, not only for the past SARS-CoV-2 pandemic, but also for future epidemic outbreaks.

Similar content being viewed by others

Article Open access

Discover the latest articles, books and news in related subjects, suggested using machine learning.

1 Introduction

As COVID-19 transitions from a pandemic to an endemic stage [ 1, 2], the scientific community is reviewing the effect of vaccination [ 3, 4, 5], its socioeconomic aspects [ 6, 7, 8, 9, 10, 11, 12, 13], and consequences of mandates and policies [ 14, 15]. Moreover, researchers are still studying optimal vaccination strategies and non-pharmaceutical interventions for preventing further outbreaks [ 16, 17, 18, 19, 20]. This work falls between these two broad directions, that is, an examination of what was done and a study of what would have been convenient to do. Indeed, we review the appropriateness of a vaccination strategy that was used in many countries, that is, vaccine prioritization based on individuals' age [ 21, 22, 23]. For such an evaluation, we compare its results with other prioritization schemes, many of them already analyzed in the vast literature on vaccination, and a few new ones introduced in this paper. It might be argued that, since the pandemic is already gone, this study may only be of historical value and have no practical importance. However, many researchers (see, e.g., [ 4]) believe that factors like climate change and ecosystem destruction will contribute to a higher frequency of pandemic outbreaks in coming years (evidenced by such a tendency in the last few decades). Thus, the thorough study of the errors and successes during the COVID-19 pandemic is of fundamental importance for adequate preparedness for future epidemics.

For our study, we need to resort to the broad area of mathematical epidemiology, which has a long history [ 24, 25]. In the last two decades [ 26, 27, 28, 29] and especially during and after the last pandemic of COVID-19 [ 30, 31], the mathematical modeling of disease propagation has been of great relevance as a public policy decision tool. Thus, with such an aid, but also with the experience of previous pandemics and the insight of experts' opinions, governments throughout the world set up preventive measures such as restrictions to mobility, city- or region-wide lockdowns, mask use, etc. [ 32]. Some of these actions had serious side effects, such as the deterioration of the economy [ 33], education [ 34], and a variety of mental health issues [ 35].

In view of the global picture, the production of effective vaccines appeared as an ideal solution. In record times, leading health companies around the world developed vaccines for some of the prevalent variants of the SARS-CoV-2 virus. However, it was the entire world population that required those vaccines in a short time in order to come out of the crisis. Natural production limitations, worldwide distribution problems, and economical issues related to the purchase of vaccines led most governments to propose different prioritization strategies. In some countries, groups of essential workers (medical caretakers, law enforcement, teachers, etc.) were vaccinated first, followed by persons with comorbidities. In other countries, prioritization was based on individuals' age, which is directly related to the mortality due to the disease.

In this work, we focus on vaccination strategies for the optimization of certain performance metrics, such as the number of deaths, the total number of hospitalization days, and the number of vaccines needed to eradicate the disease. As it is explained in detail in Sect. 2, the study of vaccine prioritization on networks is not new, not even in the context of the SARS-CoV-2 disease. However, the contribution of our work can be summarized as follows:

1. 1.
We base our analysis on the simulation of the virus propagation and the vaccine intervention on artificially generated networks that faithfully mimic the age-stratified contact frequencies and degree distribution of real-world social networks. For that purpose, our approach uses a recently introduced graph generation algorithm [ 36] and reproduces contact data from the POLYMOD surveys [ 37], conducted in eight European countries.

2. 2.
We introduce a disease spread model, based on the SEIHARD contagion model [ 38] and especially devised to reproduce the behavior of COVID-19. The model includes factors, such as population age, contact time among individuals, and the influence of vaccines. Furthermore, simulations use parameters specific for SARS-CoV-2.

3. 3.
For vaccine prioritization, we test a large number of metrics (\(>60\)). Many of these metrics are common node centrality measures, such as degree, closeness, PageRank, and betweenness [ 39]. But we also use less common measures, like Laplacian centrality [ 40], gravity [ 41], and node utility [ 42]. We also introduce several new metrics specifically tailored for the problem at hand.

4. 4.
We study under which circumstances and for which goals age-based prioritization is an efficient strategy.

The remainder of the paper is organized as follows. Section 2 presents a review of the related literature. Section 3 introduces the methodology used. In particular, Sect. 3.1 presents the main concepts of contact networks and describes the graph generation algorithm. The new contagion model is explained in Sect. 3.2 and the vaccine prioritization strategies are described in Sect. 3.3. Simulation results are presented in Sect. 4. Finally, we close with some discussion and conclusions in Sect. 5.

2 Related work

There is a vast literature on vaccination strategies, including several review papers. For example, the well-known work of Pastor-Satorras et al. [ 28] reviews epidemic models in networks, including vaccination strategies. Wang et al. [ 29] focus exclusively on the dynamics of vaccination. Noh et al. [ 30] and Espinosa et al. [ 31] present more recent reviews, which include the COVID-19 pandemic.

Somewhat arbitrarily, we can classify network-based papers on vaccination strategies between those published before and those published after the COVID pandemic. Among the first, we find the work of Pastor-Satorras and Vespignani [ 43], where it is shown that random immunization does not lead to the eradication of a disease in scale-free networks under the Susceptible-Infected-Susceptible (SIS) model. Moreover, the authors showed that targeting nodes with higher degrees is an efficient immunization strategy. Prioritization of highly connected individuals is also studied by Dezső and Barabási [ 44]. One problem with this type of strategy is that it may not be simple to know the actual degree of each node. In order to alleviate this difficulty, Cohen, Havlin, and ben-Avraham [ 45] introduced the acquaintance immunization strategy, where randomly chosen acquaintances (neighbors) of a fraction of the population are vaccinated. Thus, highly connected nodes have a higher probability of being vaccinated. Holme [ 46] assumed that a node can have knowledge of the contacts of its neighbors and, thus, suggested two different strategies. The first one is an extension of that of Ref. [ 45], where the neighbors of randomly selected nodes with the highest number of connections outside the neighborhood are vaccinated. The second one is similar, but instead of choosing nodes at random, the neighborhood of the last vaccinated node is considered. Holme showed that, in many situations, the second strategy outperformed others. A related approach was proposed by Gómez-Gardeñes,Echenique, and Moreno [ 47], where the nodes with higher degree within a fixed distance from a randomly selected vertex were vaccinated. Other similar strategies that focus on the vaccination of neighbors of randomly selected nodes were presented in Ref. [ 48].

In practice, a widely used technique to stop the propagation of a disease is contact tracing, that is, the identification of individuals that might have been exposed by following the contacts of those that are known to be infectious. Although contact tracing is usually accompanied by isolation and treatment of the followed contacts, it is simple to think of a variation that vaccinates those contacts. It has been shown to be effective for, e.g., severe acute respiratory syndrome (SARS) [ 49] and sexually transmitted diseases [ 50, 51]. Kiss, Green, and Kao [ 52] studied the effectiveness of contact tracing in scale-free networks and found that it is insufficient if the disease spreads too rapidly. The interested reader can find more information on contact tracing in Refs. [ 53, 54, 55, 56].

Christley et al. [ 57] were concerned with finding the most vulnerable-to-infection node by computing several centrality measures: node degree, eccentricity, betweenness, and random walk betweenness. They found that a simple metric such as the degree is a good proxy for infection vulnerability. Variations of betweenness for prioritization were analyzed by Schneider et al. [ 58]. Clustering, closeness, and other metrics were studied in Ref. [ 59]. Results by Hébert-Dufresne et al. [ 60] suggest that coreness is a poor centrality measure for selecting nodes to be inoculated. Miller and Hyman [ 61] also studied prioritization by PageRank, finding that it outperforms the degree-based strategy in general. Miller and Hyman also proposed a new metric, combining the probability of a node being infected by a contact with the degree of each of the nodes in the neighborhood, that fared well in comparison with other centrality measures. In Sect. 3.3.2 we shall also present new metrics of similar characteristics, but adapted to the contagion model used in this paper.

Chen et al. [ 62] proposed a different approach. If vaccination is understood as a way of cutting off pathways of disease spread, a possibility is to look for the optimization of the number of nodes that need to be inoculated for 'disconnecting' the network. Thus, Chen and colleagues divided the network into clusters of approximately the same size and connected by sets with the smallest possible number of nodes; these connecting nodes were vaccinated. In a similar vein, Salathé and Jones [ 63] studied networks with community structure and found it efficient to inoculate nodes that act as bridges between communities. Since we may think of the problem of vaccination as that of fragmenting and, hence, attacking a network, the literature on vaccination can be extended by the large number of papers on the vulnerability of complex networks [ 64, 65, 66, 67, 68, 69].

Topological properties are not the only measures that may be used for vaccine prioritization. It is a reasonable choice to vaccinate the population with high mortality rates. Another alternative is to inoculate those individuals with higher morbidity, that is, who are more vulnerable to becoming infected (e.g., school children present a high influenza morbidity). Bansal, Pourbohloul, and Meyers [ 70] compared the efficacy of contact-based (i.e., targeting higher degree nodes), morbidity-based and mortality-based prioritization for the case of influenza. They found that for strains with low transmissibility, both contact-based and morbidity-based strategies were more effective in diminishing the mortality than mortality-based vaccination. However, this situation reverts when the disease transmissibility increases.

To the best of our knowledge, before the SARS-CoV-2 pandemic, most of the literature of network-based studies on vaccine prioritization assumed a homogeneous population. Moreover, they used theoretical analyses of large random or scale-free networks, performed simulations on Erdős-Renyi, Barabási-Albert, or Watts-Strogatz graphs, or simulations on a few real-world networks. Papers with non-homogeneous populations and networks that account for different age-dependent contact patterns were rare (the work of Bansal, Pourbohloul, and Meyers [ 70] is one exception).

After the onset of the COVID-19 pandemic, the interest in vaccination strategies experienced a new sudden increase [ 30, 31]. It was realized early on that an individual's age had to be explicitly considered in modeling the propagation. For example, Matrajt et al. [ 71] and Bubar et al. [ 72] found that vaccinating the elderly diminished the number of deaths, but targeting young adults was more efficient in mitigating the spread of the disease. Network-based modeling by Chen et al. [ 73] showed that, under certain conditions, inoculating young adults can be more efficient even in terms of mortality. Simulated networks in the paper of Chen et al. [ 73] were based on a generalization of the Chung-Lu generation algorithm [ 39], where the mean number of contacts between individuals in different age strata was known. However, for the exact age-dependent degree distribution, either Poisson or power-law with exponent 2.5 was used. In our work, on the contrary, we use degree distributions extracted from population contact surveys.

Based on an agent-based model of the state of Virginia, Chen et al. [ 74] found that individuals' degree and social proximity time were more effective than the age-based strategies in reducing the number of infections, hospitalizations, and deaths. The interesting work by Prieto Curiel and González Ramírez [ 75] also analyzed node betweenness as a prioritization strategy, but they used arbitrary age distributions and artificial scale-free and small-world networks. Petrizzelli, Guzzi, and Mazza [ 76] showed that eigenvalue centrality can also be useful for vaccine prioritization, but they based their conclusions on results from a single real contact network and a few artificially generated graphs. In this paper, we study vaccination strategies by simulating several networks, all of them with the same contact pattern extracted from real-world data.

3 Methodology

3.1 Contact network as a graph

A network or graph \(G(\mathcal {N},\mathcal {L})\) is a tuple where \(\mathcal {N} = \{\nu _1,\nu _2,...,\nu _N\}\) is a nonempty set of elements known as vertices or nodes and \(\mathcal {L}\) is a set of ordered pairs from \(\mathcal {N}\) called links or edges. Let us define the graph's order as \(N = |\mathcal {N}|\) as the number of nodes in G and its size as \(L= |\mathcal {L}|\). In our problem, nodes represent individuals, N is the size of the population under consideration, and each edge constitutes a contact between two persons.

The graph topology can be described by the adjacency matrix A of order \(N\times N\), defined as \(A_{i,j} = 1\) if \((\nu _i,\nu _j)\in \mathcal {L}\) and \(A_{i,j} = 0\) otherwise. Whenever the adjacency matrix is symmetric, the graph is undirected. As person-to-person contacts are bidirectional, networks in this paper are undirected.

An alternative representation of a graph is a collection of neighborhood sets, one for each node. Two vertices are neighbors of each other if they are linked, and we denote the set of neighbors of a node \(\nu \) as \(\eta (\nu )\).

It may be useful to add labels or weights to edges. In our case, a weight on an edge represents the daily contact time between the two linked individuals. Nodes can have properties themselves, representing non-topological characteristics, e.g., the age of the individual corresponding to the vertex. Formally, we write a weighted graph as a tuple \(G(\mathcal {N},\mathcal {L},\Theta ,\Xi \left( \nu _i,\nu _j\right) )\), where \(\mathcal {N}\) and \(\mathcal {L}\) are as above, and \(\Theta : \mathcal {N}\mapsto \mathcal {S}\) assigns a property in \(\mathcal {S}\) to each node and \(\Xi \left( \nu ,\mu \right) : \mathcal {L}\mapsto \mathcal {R}\) maps each link to an attribute in \(\mathcal {R}\). If \(\mathcal {R}\) contains scalar values, we can define a weighted adjacency matrix W by \(W_{i,j}=\Xi (\nu _i,\nu _j)\) if \((\nu _i,\nu _j)\in \mathcal {L}\) and \(W_{i,j}=0\) otherwise. For the sake of notational simplicity, let us define \(w_{\nu ,\mu } = \Xi (\nu ,\mu )\).

3.1.1 Generation of realistic contact networks

The use of graphs in the study of COVID-19 is very intuitive, as the spread of infectious diseases occurs through complex networks of human interactions. In order to apply this approach, it is most relevant to have access to data on social contacts, including information on age, contact time, risk factors, etc. In the first decade of the century, the European Commission created the POLYMOD project [ 37] to collect such information in order to be able to predict future patterns of disease dissemination in Europe [ 77]. This project conducted a survey between May 2005 and September 2006 in eight European countries. Survey participants were chosen to be representative of the population in terms of geographical distribution, age, and sex, but children and teenagers were oversampled. Each survey participant had to fill out a contact diary, which included the time of interaction and age of the other individuals (for more details, see Mossong et al. [ 77]).

Even though the POLYMOD project collected very important data, it does not contain all information about the network of contacts among all survey participants. On the contrary, the diaries filled out by the participants correspond to a set of star-shaped or ego networks. In order to bridge this difficulty, Manzo and van de Rijt [ 78] proposed a synthetic network formation algorithm based on the configuration model [ 39]. The mean number of daily contacts of each individual is taken as the corresponding unweighted degree of a node. As such, the configuration model allows the formation of a synthetic network where the nodes have the same degree as that inferred from the POLYMOD survey. However, it is known that networks generated in such a way have low clustering, lower than expected in usual social networks [ 39]. For that reason, Manzo and van der Rijt proposed to increase the network clustering by connecting each pair of neighbors of every node with probability p. Thus, as p increases, the graph clustering increases.

Gonçalves [ 36] extended the algorithm of Manzo and van de Rijt [ 78] to take into account age-dependent differences in the degree distribution and the probability of connection between individuals in different age groups. In this work, we make use of the Gonçalves algorithm, resorting to the contact information in the POLYMOD study and the age distribution of the Brazilian population [ 79]. Furthermore, edge weights were randomly generated by sampling normal distributions with mean and variance obtained from the daily contact time reported in POLYMOD for each type of inter-age connection.

3.2 Contagion model

There are many contagion models in the literature on COVID-19. Looking for a balance between model completeness and simplicity, in this work we base our modeling scheme on the approach by Eikenberry et al. [ 38]. Although Eikenberry and colleagues were concerned about the effectiveness of mask use as a barrier for the spread of the disease, we disregard the use of masks and focus our attention on vaccination. Moreover, we adapt the macroscopic model of Eikenberry et al. [ 38] for its use in a heterogeneous network of contacts where each node can be in any of the following states: Susceptible ( S), Exposed ( E), Symptomatic/Ill ( I), Hospitalized ( H), Asymptomatic/Ill ( A), Recovered ( R), and Dead ( D). We shall refer to this model as SEIHARDS, due to the initials of the node states. There is an incubation time before a person who has been exposed ( E) to the disease becomes ill. We assume that a symptomatic ill ( I) individual is isolated and, hence, is not contagious. On the contrary, an infected person that shows no symptoms ( A) is not isolated and continues to spread the disease. Figure 1 schematically explains the details of the model without the effects of vaccination. State transition times are assumed to follow an exponential distribution, where the transition rate is noted in the labels of the edges of the graph in Fig. 1. Let us note that a node in state S can become infected by those neighbors that are either in state E or state A. Thus, the transition rate S\(\rightarrow \) E depends on the contact time with those neighbors:

$$\begin{aligned} \Lambda _\nu = \frac{\epsilon ^{}_{E}\times \sum \limits _{\mu \in \eta (\nu )\cap E} w_{\mu ,\nu } +\epsilon ^{}_{A}\times \sum \limits _{\mu \in \eta (\nu )\cap A} w_{\mu ,\nu }}{\langle k^w \rangle }, \end{aligned}$$

where A and E are the sets of asymptomatic and exposed nodes, respectively; \(\epsilon ^{}_{E}\) and \(\epsilon ^{}_{A}\) are contagion rates that depend on whether the infectious agent is in the E or the A state, respectively, and \(\langle k^w \rangle \) is the mean weighted node degree. In the case of an unweighted network model, we assume \(w_{\mu ,\nu }=1\) for all \((\mu ,\nu )\in \mathcal {L}\). The values of all parameters, which are presented in Tables S.1 and S.2 of the Supplementary, were obtained either from the literature or the database openDataSUS [ 80], a Brazilian open site with hospitalization data related to COVID-19 and other infectious diseases.

In a contact network, state transition probabilities (Table S.2 in the Supplementary) are considered node properties. Indeed, we assume that \(\Theta (\nu ) = (\alpha _\nu ,\psi _\nu ,\tau _\nu )\), where \(\alpha _\nu \) is the probability of becoming symptomatic if exposed to the virus, \(\psi _\nu \) is the probability of being hospitalized, and \(\tau _\nu \) is the probability of dying (see Fig. 1).

The SEIHARDS model has to be modified in order to account for the effects of vaccination. As it is done in other works [ 81], we assume that the vaccine changes some of the parameters. In particular, the contagiousness (\(\epsilon ^{}_{A}\) and \(\epsilon ^{}_{E}\)), the probability of turning symptomatic (\(\alpha _\nu \)), the probability of hospitalization (\(\psi _\nu \)), and the probability of death (\(\tau _\nu \)) diminish (see Table S.3 in the Supplementary). [id=MI]A more detailed description of the SEIHARDS model can be found in Section S.1 of the Supplementary.

During the pandemic, vaccines became available only after COVID-19 had spread. For that reason, in our simulations we start the vaccination campaign when the dissemination of the disease reaches an apparent stationary state. As vaccines may not be available for everyone, we inoculate only a fraction f of the population, following different prioritization strategies. Only individuals who are recognized as "not infected" are included in the vaccination campaign. Thus, not only susceptible and recovered persons are inoculated, but also exposed and asymptomatic ones. For the sake of modeling simplicity, [id=MI]in most of our results, all individuals are vaccinated in a single day. It can happen that a fraction f of the population is not available for vaccination. In that case, individuals are vaccinated as soon as they recover, following the predefined priorities. [id=MI]Even if all individuals were vaccinated on the same day, as there are vaccines for only a fraction f of the population, a prioritization strategy would still be necessary. We also show that a gradual vaccination over time has little effect in our results.

3.3 Vaccine prioritization

In many countries, age (\(\textit{AG}\)) was used, at least partially, for vaccine prioritization [ 21, 22, 23]. Indeed, age can be used for a network-agnostic estimation of death probability (see the last line in Table S.2 of the Supplementary). If we want to incorporate information about the network of contacts, there are many other metrics that can be used, as explained in the following lines.

3.3.1 Node centrality measures

In the literature we find several ways of quantifying the relevance of a node. These metrics are known as centrality measures and their definitions depend on the problem. For example, we may be interested in ranking authors in a co-citation network [ 82] or in finding out which delinquents should be isolated to break a criminal network [ 41]. In this work, we test the most common centrality measures as proxies of the importance of vaccinating a given node: node degree (\(\textit{DE}\)), eccentricity (\(\textit{EC}\)), closeness (\(\textit{CL}\)), harmonic centrality (\(\textit{HH}\)), betweenness (\(\textit{BE}\)), eigenvalue centrality (\(\textit{EI}\)) [ 39], PageRank (\(\textit{PR}\)) [ 83], k-core (\(\textit{KC}\)) [ 84], and Laplacian centrality (\(\textit{LA}\)) [ 40].

Even though some of the aforementioned metrics can be adapted to account for edge weights, there are other centrality measures especially devised for weighted networks. Rêgo and dos Santos [ 42] studied co-authorship networks and defined a new metric, the so-called node utility, that accounts for the strength of the link between authors, both in absolute terms and in relation to other links. In order to account for some of the nodes' properties, we put forth a slight modification of the utility centrality:

$$\begin{aligned} \textit{UT}(\nu )= & \sum \limits _{\mu \in \eta (\nu )} \, \frac{w_{\nu ,\mu }f_\rightarrow (\Theta (\nu ))}{k^w_\nu }+\frac{w_{\nu ,\mu }f_\leftarrow (\Theta (\mu ))}{k^w_\mu }\nonumber \\ & +\frac{w_{\nu ,\mu }f_\rightarrow (\Theta (\nu ))}{k^w_\nu }\times \frac{w_{\nu ,\mu }f_\leftarrow (\Theta (\mu ))}{k^w_\mu }, \end{aligned}$$

where \(k^w_\mu \) is the weighted degree of node \(\mu \) and \(f_\rightarrow (\cdot )\) and \(f_\leftarrow (\cdot )\) are two conveniently chosen functions. In this work, we use two different choices. If we are interested in which nodes are more prone to be the cause of death of their neighbors, we use \(f_\rightarrow (\Theta (\nu )) = 1-\alpha _\nu \) and \(f_\leftarrow (\Theta (\mu )) = \tau _\mu \); that is, \(f_\rightarrow \) returns the probability of node \(\nu \) becoming asymptomatic and, hence, contagious for its neighbors, while \(f_\leftarrow \) returns the probability of death of each neighbor. When these functions are used, we say that we employ the altruistic version of \(\nu \)'s utility (\(\textit{AUT}\)). On the contrary, if we were more interested in the death of \(\nu \) itself, we would use \(f_\rightarrow (\Theta (\nu )) = \tau _\nu \) and \(f_\leftarrow (\Theta (\mu )) = 1-\alpha _\mu \). We call this choice the egoistic version of \(\nu \)'s utility, which, for the sake of simplicity, we denote by \(\textit{UT}\).

Inspired by Newton's gravitation law, de Andrade et al. [ 41] defined the gravity centrality for scalar node properties. As we deal with vectorial node attributes, we propose to modify the gravity as

$$\begin{aligned} \textit{GR}(\nu ) =\sum \limits _{\mu \ne \nu }\frac{f_\rightarrow (\Theta (\nu )) f_\leftarrow (\Theta (\mu ))}{(d_{\nu ,\mu }^w)^{1/s}}, \end{aligned}$$

where \(w_\chi \) is \(\chi \)'s weight and \(d_{\nu ,\mu }^w\) is the weighted shortest distance between \(\nu \) and \(\mu \). Observe that this measure takes the form of the gravitation law for \(s = 1/2\). Once again, we have an altruistic (\(\textit{AGR}\)) and an egoistic (\(\textit{GR}\)) version if the functions are chosen as in the altruistic and egoistic versions of the utility, respectively.

3.3.2 New metrics

As we have already mentioned, since age can be used as a proxy for an individual's death probability, it is a reasonable node centrality measure in the context of SARS-CoV-2. However, if we only look at a node's age, we are completely ignoring its neighborhood. Indeed, we do not expect an isolated person to be as vulnerable as a highly connected one. This intuition can be incorporated by computing a more accurate estimation of the node's own death probability: \(\textit{DD}(\nu ) = |\eta (\nu )|\alpha _\nu \psi _\nu \tau _\nu \).

We can also introduce other centrality measures based only on the properties of the node and its neighbors. For example, in many countries, elderly caregivers had a high vaccination probability, as they could potentially 'cause' the hospitalization of many persons vulnerable to the disease. We can try to capture this intuition by prioritizing those nodes whose neighbors have a high probability of being hospitalized. Under a simplifying independence assumption, we may write the probability of at least one neighbor being hospitalized as

$$\begin{aligned} HP(\nu )&= 1-\prod \limits _{\mu \in \eta (\nu )} \left( 1-\psi _\mu \right) . \end{aligned}$$

This metric does not consider the fact that \(\nu \) might not be 'responsible' for its neighbors' hospitalization. Thus, we may alter the metric by considering the probability of \(\nu \) becoming asymptomatic, i.e., \( HA(\nu ) = (1-\alpha _\nu )\times \textit{HP}(\nu )\).

If instead of looking at hospitalization probabilities, we focus on the death of the neighbors, we may define

$$\begin{aligned} DP(\nu ) = 1-\prod \limits _{\mu \in \eta (\nu )} \left( 1-\alpha _\mu \psi _\mu \tau _\mu \right) ,\qquad DA(\nu ) = (1-\alpha _\nu )\times \textit{DP}(\nu ). \end{aligned}$$

Observe that \(\alpha _\mu \psi _\mu \tau _\mu \) is the probability of \(\mu \) dying if it is exposed to the virus.

Although there are other centrality measures in the literature and many variations of those metrics presented here can be implemented, we think that we have included an already large number of alternatives.

4 Simulations

Networks with \(N=10000\) were generated as explained in Sect. 3.1.1. We ran 400 simulations for a total of 465 days, always starting with 10 randomly chosen nodes in the E state and all the remaining nodes in the S state. Results before the vaccination show that by the 100th day, all node state fractions stabilize. Thus, this will be taken as the starting day for the vaccination campaign (see Fig. S.1 of the Supplementary). We also evaluated the influence of the clustering coefficient and the link weights in the pre-vaccination period, finding that they have little influence on the evolution of the disease (for more details, see Section S.1 in the Supplementary).

In order to evaluate the effectiveness of different prioritization strategies, we used three performance metrics: the total hospitalization time \(T_H\), the fraction \(F_D\) of the population that dies as a consequence of the disease, and the fraction \(f_c\) of the population that needs to be vaccinated in order to extinguish the disease. In order to rank all strategies, and since the hospitalization time and the number of deaths depend on the fraction f of the population being vaccinated, we summarized these metrics by the area under the curves of \(T_H(f)\) and \(F_D(f)\), respectively. Besides ranking strategies for each of those three metrics, we also used the mean of the normalized values of the areas under the curves of \(T_H(f)\) and \(F_D(f)\), and \(f_c\). The normalization was computed such that the lowest value is 0 and the highest is 1.

Figure 2 shows the results for an unweighted network and two values of p, 0 and 1. Since we tested 37 prioritization strategies using centrality measures which take into account neither edge weights nor node attributes and 63 strategies that considered either of them, we only show a few significant results (the interested reader is referred to Section S.3 of the Supplementary material for more details). In particular, we show the best strategy for each metric and the best with respect to the mean of all metrics. For the sake of comparison, we also show results corresponding to random vaccination and age-dependent prioritization.

It can be observed in Fig. 2 that age prioritization is not a good strategy in general. For example, in order to extinguish the disease, \(\sim 90\)% of the population needs to be vaccinated, even higher than what is required when random selection (\(\textit{RN}\)) is applied. Moreover, this fact is true even as p and, hence, the network clustering changes. For PageRank, only \(\sim 30\)% of the population needs to be vaccinated. However, \(\textit{AG}\) appears as an efficient strategy to reduce the number of deaths when the vaccinated fraction is small (\(\lesssim 15\)%). These results can be explained by the much higher mortality among the elderly and the lower number of contacts in aging individuals, which is observed in the POLYMOD dataset. Figure 3 shows the results for a weighted contact network. Although there are some changes in relation to Fig. 2, it can be observed that the weighted PageRank and the unweighted PageRank obtained the best results on average for \(p=0\) and \(p=1\), respectively. As for PageRank, it has already been observed in the literature that it has a good capacity for forecasting diffusion in networks [ 85].

[id=MI]Results in Figs. 2- 3 correspond to the case where all individuals are vaccinated on the same day. In order to check whether this overly simplistic approach greatly affects our results, we also conducted simulations where vaccinations took place over time. In particular, a fraction f of the population (of size N) is vaccinated according to a Poisson process with a mean of \(f\times N/100\) vaccinations per day. Figures 4- 5 show the results for 200 network realizations. Comparison with Figs. 2- 3 exhibits no significant differences. Thus, in the remainder of the paper, we only present results for the simpler situation.

Considering the large number of analyzed strategies and the small number of them presented in Figs. 2- 3, we need another tool to better appreciate the relation between the three performance metrics. To that end, we turn to the Spearman correlation between strategy rankings. Figure 6 shows such correlation for both an unweighted (left) and a weighted network (right). As it may be intuitively expected, the number of deaths and the hospitalization time are highly correlated in all cases. The fraction of the population that needs to be vaccinated to extinguish the disease and the number of deaths exhibit the lowest correlation. This might also be expected from the explanation of the behavior of the age-dependent prioritization in Figs. 2- 3.

The influence of the network clustering can be better analyzed by observing Fig. 7, which presents the Spearman's correlation for each pair of values of probability p. As it can be readily seen, a high correlation is found in all cases, and, hence, the influence of p on vaccination strategy rankings is small. Since from the data we cannot know the actual contact network clustering, this observation is very important and allows us to extrapolate our results to real social networks.

Given that the computation cost of node centrality measures that take into account the edge weights is high, it is interesting to analyze whether incurring such cost is necessary. For that purpose, we ran all simulations twice, first considering all edge weights (\(M^{w}\)) and then setting all weights to unity in the computation of the centrality metrics ( M). For each strategy, we computed the relative difference (\(Z^{w}=(M^w-M)/M\)) of the mean results of 400 realizations, only for networks built with \(p = 0\). The values of \(Z^w\) are presented in Fig. 8, where positive values mean worse results for the weighted measures, since the objective is to minimize them. As can be observed, in general, there is little gain or even a loss of performance when actual edge weights are considered. A remarkable case is that of the closeness centrality, which exhibits \(\sim 150\)% loss.

We introduced two ways of computing the node centrality when its attributes are accounted for: altruistic (\(M^a\)) and egoistic (\(M^e\)). Figure 9 presents the relative difference (\(Z^a=(M^e-M^a)/M^a\)) in performance between using the egoistic and altruistic versions. As it can be observed, in the case of a weighted graph, the egoistic strategies are worse in almost all cases for all three performance metrics. However, in the case of an unweighted network, no significant performance differences were found, with the exception of two gravity centrality measures.

5 Conclusion

We began this work by stating that our purpose is twofold. On one hand, we want to assess how convenient it was to prioritize vaccination based on individuals' age. On the other hand, we look for optimal vaccine prioritization strategies.

We found that age-based prioritization is a quasi-optimal strategy if the goal is to minimize the number of deaths and the supply of vaccines is so small that the immunization coverage is smaller than 15%. However, if the goal is to eradicate the disease, age-based vaccination is even worse than random immunization. Furthermore, we have shown that the minimization of the death toll in a given time period and disease eradication are somewhat opposite goals, as evidenced by Fig. 6. We must note that there is a catch to this analysis. Indeed, age-based prioritization showed fair results in relation to the death toll for the given period of time simulated. Since only disease eradication can avert an increasing number of deaths, in the long run, age-based vaccination will not fare well in that metric either.

As for the optimal strategies, it was shown that centrality measures that considered weight on the nodes and the network structure were very effective for vaccination against COVID-19 on average and that PageRank has great effectiveness against the spread. This strategy was able to reduce mortality by more than 60% and the total time hospitalized by 66%. Moreover, it was necessary to vaccinate less than 40% of the network to inhibit the proliferation of the virus. Moreover, it was shown that using edges' weights had no significant gain; nevertheless, it increases computational cost. Taking an altruistic approach to nodes' attributes proved to be a strong strategy against the virus. Finally, the classification of the best metrics proved to be quite stable with the increase in network clustering, which means that it is not so necessary to estimate this value in a real contagion network.

We must note that, in order to evaluate the effectiveness of different prioritization strategies, we needed to build realistic contact networks. Such network generation was based on the POLYMOD surveys [ 37], which provide data on the number of connections and the contact time between different age groups. An important fact taken from the POLYMOD data is that the elderly have fewer connections than other age groups, and hence, they do not act as important disease spreaders. This is partly the reason why age-based vaccine prioritization is inefficient for disease eradication.

One of the limitations of our work is that, despite a lot of data being available in the POLYMOD surveys, the extent to which they can be applied to different countries and cultures is unknown. [id=MI]Moreover, the practical application of the proposed centrality metrics as vaccination strategies depends on the construction of a real network of physical contacts. One possibility of building such a network would be to use cell phone trace data. We also left out one important characteristic of a pandemic outbreak: hospital overcrowding. We intend to tackle these issues in future research in this area.

Data Availability

No datasets were generated or analyzed during the current study.

References

1. Cărăus̨su, D.-N., Gherghina, S.C., Lupu, D., Simionescu, L.N.: Covid-19 vaccination and stock markets volatility around the world evidence from wavelet analysis. Appl. Econ. Lett. 5, 1-14 (2024)

2. Venkatesh, A., Rao, M.A.: Mathematical model for COVID-19 pandemic with implementation of intervention strategies and cost-effectiveness analysis. Res. Control Optim. 14, 100345 (2024)

Google Scholar

3. Bartsch, S.M., O'Shea, K.J., John, D.C., Strych, U., Bottazzi, M.E., Martinez, M.F., Ciciriello, A., Chin, K.L., Weatherwax, C., Velmurugan, K., et al.: The potential epidemiologic, clinical, and economic value of a universal coronavirus vaccine: a modelling study. EClinicalMedicine 68 (2024)

4. Gunder, J.R., Ballenger, L.: COVID-19 vaccine scarcity and prioritization. Available at SSRN 3908231 (2021)

5. World Health Organization, et al.: WHO SAGE roadmap for prioritizing uses of COVID-19 vaccines in the context of limited supply: an approach to inform planning and subsequent recommendations based on epidemiological setting and vaccine supply scenarios, first issued 20 October 2020, latest update 16 July 2021. Technical report, World Health Organization (2021)

6. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. In: Proceedings of the Royal Society of London. Series A 115(772), 700-721 (1927)

7. Cruz, M.P., Santos, E., Cervantes, M.A.V., Juárez, M.L.: COVID-19, a worldwide public health emergency. Rev. Clin. Esp. 221(1), 55-61 (2021)

Google Scholar

8. Pereira, M.D., Oliveira, L.C., Costa, C.F.T., Oliveira Bezerra, C.M., Pereira, M.D., Santos, C.K.A., Dantas, E.H.M.: A pandemia de COVID-19, o isolamento social, consequências na saúde mental e estratégias de enfrentamento: uma revisão integrativa. Res. Soc. Dev. 9(7), 652974548 (2020)

9. Gonçalves, M.: Análise de estratégias de vacinação para COVID-19 baseada em redes complexas. Master's thesis, Universidade Federal do Ceará, Centro de Ciências, Departamento de Estatística e Matemática Aplicada, Programa de Pós-Graduação em Modelagem e Métodos Quantitativos (2024)

10. Mossong, J., Niel Hens, Jit, M., Beutels, P., Auranen, K., Mikolajczyk, R., Massari, M., Salmaso, S., Tomba, G.S., Wallinga, J., Heijne, J., Malgorzata Sadkowska-Todys, Rosinska, M., W. John Edmunds: POLYMOD social contact data. Zenodo (2017). https://zenodo.org/record/3746312

11. Eikenberry, S.E., Mancuso, M., Iboi, E., Phan, T., Eikenberry, K., Kuang, Y., Kostelich, E., Gumel, A.B.: To mask or not to mask: modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic. Infect. Dis. Modell. 5, 293-308 (2020)

Google Scholar

12. Newman, M.: Networks, 2nd edn. Oxford University Press, Oxford (2018)

Book Google Scholar

13. Cohen, R., Havlin, S., ben-Avraham, D.: Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91(24), 247901 (2003)

14. Lipsitch, M., Cohen, T., Cooper, B., Robins, J.M., Ma, S., James, L., Gopalakrishna, G., Chew, S.K., Tan, C.C., Samore, M.H.: Transmission dynamics and control of severe acute respiratory syndrome. Science 300(5627), 1966-1970 (2003)

15. Eames, K.T., Keeling, M.J.: Contact tracing and disease control. Proc. R. Soc. Lond. Ser. B Biol. Sci. 270(1533), 2565-2571 (2003)

16. Habiba, Yu, Y., Berger-Wolf, T.Y., Saia, J.: Finding spread blockers in dynamic networks. In: Advances in Social Network Mining and Analysis: Second International Workshop, SNAKDD 2008, Las Vegas, NV, USA, August 24-27, 2008, pp. 55-76 (2010). Springer

17. Chen, J., Hoops, S., Marathe, A., Mortveit, H., Lewis, B., Venkatramanan, S., Haddadan, A., Bhattacharya, P., Adiga, A., Vullikanti, A., Srinivasan, A., Wilson, M.L., Ehrlich, G., Fenster, M., Eubank, S., Barrett, C., Marathe, M.: Effective social network-based allocation of COVID-19 vaccines. In: Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. KDD '22, pp. 4675-4683. ACM, ??? (2022)

18. Mossong, J., Hens, N., Jit, M., Beutels, P., Auranen, K., Mikolajczyk, R., Massari, M., Salmaso, S., Tomba, G.S., Wallinga, J., Heijne, J., Sadkowska-Todys, M., Rosinska, M., Edmunds, W.J.: Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Med. 5(3), 74 (2008)

19. Manzo, G., Rijt, A.: Halting SARS-CoV-2 by targeting high-contact individuals. J. Artifi. Soc. Soc. Simul. 23(4) (2020)

20. Instituto Brasileiro de Geografia e Estatística: Censo 2022. https://www.ibge.gov.br/estatisticas/sociais/trabalho/22827-censo-demografico-2022.html. Accessed 19 July 2024 (2022)

21. Ministério da Saúde de Brasil: SRAG 2020 - Banco de Dados de Síndrome Respiratória Aguda Grave - incluindo dados da COVID-19. https://opendatasus.saude.gov.br/dataset/srag-2020. Accessed 05 Dec 2023 (2020)

22. Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Comput. Netw. 30, 107-117 (1998)

Google Scholar

Download references

Funding

The Article Processing Charge (APC) for the publication of this research was funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) (ROR identifier: 00x0ma614).

Author information

Author notes

Miguel Gonçalves, Pablo Ignacio Fierens, Leandro Chaves Rêgo contributed equally to this work.

Authors and Affiliations

1. Departamento de Física, Universidade Federal do Ceará, Bloco 922, Campus do Pici, CEP: 60440-554, Fortaleza, Ceará, Brazil

Miguel Gonçalves

2. Instituto Tecnológico de Buenos Aires (ITBA), Iguazú 341, C1437ETC, Ciudad Autónoma de Buenos Aires, CABA, Argentina

Pablo Ignacio Fierens

3. National Scientific and Technical Research Council (CONICET), Godoy Cruz 2290, C1425FQB, Ciudad Autónoma de Buenos Aires, CABA, Argentina

Pablo Ignacio Fierens

4. Departamento de Estatística e Matemática Aplicada, Universidade Federal do Ceará, Bloco 910, Campus do Pici, CEP: 60440-554, Fortaleza, Ceará, Brazil

Pablo Ignacio Fierens & Leandro Chaves Rêgo

5. Programa de Pós-Graduação em Engenharia de Produção, Universidade Federal de Pernambuco, Av. Prof. Moraes Rego, 1235, Cidade Universitária, CEP: 50.740-550, Recife, Pernambuco, Brazil

Leandro Chaves Rêgo

Authors

1. Miguel Gonçalves

2. Pablo Ignacio Fierens

3. Leandro Chaves Rêgo

Contributions

M.G. was in charge of the software and prepared all figures. P.I.F. and L.C.R. oriented and supervised the research. All authors discussed results and proposed new ideas during the research process. Furthermore, all authors were involved in writing the manuscript text.

Corresponding author

Correspondence to Pablo Ignacio Fierens.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Reprints and permissions

About this article

Cite this article

Gonçalves, M., Fierens, P.I. & Rêgo, L.C. A new take on optimal vaccine prioritization: when age-based strategy is efficient?. Int J Data Sci Anal 22, 95 (2026). https://doi.org/10.1007/s41060-026-01053-5
Download citation

Received:

Accepted:

Published:

Version of record:

DOI: https://doi.org/10.1007/s41060-026-01053-5

Share this article

Anyone you share the following link with will be able to read this content:

Get shareable link

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

# data[].attributes.attachment_link.title
A new take on optimal vaccine prioritization: when age-based strategy is efficient? - International Journal of Data Science and Analytics

# data[].attributes.attachment_link.description
The rapid spread, high mortality rate, and hospital overcrowding made the SARS-CoV-2 virus a significant challenge for humanity. The emergent situation urgently called for investments in vaccine development as a critical measure to mitigate this issue. However, the financial costs and time required for vaccine development posed considerable obstacles to achieving comprehensive vaccination coverage.

# data[].attributes.body
Stratified Configuration Model for Simulating Networks

# data[].attributes.attachment_link.authors[].name
Carlos Miguel Moreira GonÃ§alves & Pablo Ignacio Fierens & Leandro RÃªgo

# data[].attributes.attachment_link.title
Stratified Configuration Model for Simulating Networks

# data[].attributes.attachment_link.description
Conventional network generation models exhibit limitations in capturing complex structural patterns influenced by nodal and link attributes. While recent models have partially addressed this issue, si

# data[].attributes.body
Sustainable Microextraction for Trace Determination of Aflatoxin M1 in Domestic Waters by High-Performance Liquid Chromatography with Fluorescence Detection (HPLC-FLD): A Computationally Optimized Design

# data[].attributes.body
An explainable machine learning framework for identifying left bundle branch block mechanisms via vectorcardiographic features

# data[].attributes.body
Automatic search for transiting planets in TESS-SPOC FFIs with RAVEN: over 100 newly validated planets and over 2000 vetted candidates

# data[].attributes.body
Long-term activity cycles in planetary M stars observed with SOPHIE

# data[].attributes.attachment_link.title
Chitosan nanogels for the improvement of benznidazole activity - PubMed

# data[].attributes.attachment_link.description
Chagas disease, caused by Trypanosoma cruzi, remains a major neglected disease with limited therapeutic options. Benznidazole (Bzn) is the first-line treatment, despite low aqueous solubility, poor bioavailability and significant side effects, which reduce its efficacy, particularly during the chron …

# data[].attributes.attachment_caption
Exploring the role of sialidases in Galectin-1-associated resistance to cancer therapies

# data[].attributes.attachment_link.description
Please confirm you are a human by completing the captcha challenge below.

# data[].attributes.attachment_caption
A minimal stochastic model captures collective T cell behavior in one-dimensional microchannels

# data[].attributes.headline
Lucas Wiebke; Joaquín R. Colonnello &  Daniel Parisi

# data[].attributes.subject
Lucas Wiebke; Joaquín R. Colonnello &  Daniel Parisi

# data[].attributes.body
,

Joaquín R. Colonnello

# data[].attributes.attachment_link.description
Please confirm you are a human by completing the captcha challenge below.

# data[].attributes.attachment_caption
Anonymous leadership and stochastic resonance in collectives of self-propelled robots

# data[].attributes.attachment_caption
Occupants’ willingness to share information for improved comfort and energy efficiency in offices

# data[].attributes.attachment_link.description
Please confirm you are a human by completing the captcha challenge below.

# data[].attributes.attachment_link.title
Iter-T: ITERative Test suite generation for automated program repair

# data[].attributes.attachment_link.description
Test-based automated program repair (TB-APR) techniques automatically fix buggy programs by relying on a failing test suite. This test suite serves a dual purpose: pinpointing bugs and evaluating the validity of potential patches. However, the effectiveness of TB-APR techniques in generating correct patches is highly dependent on the test suite utilized. The primary shortcoming of TB-APR techniques arises from the intrinsic incompleteness of test suites, resulting in a significant drawback: overfitting, i.e., the generation of ‘overfitted patches’, patches that pass the given test suites but fail to repair the subject program correctly regarding its more general intended behavior. <p xmlns:mml=“http://www.w3.org/1998/Math/MathML” xmlns:xlink=“http://www.w3.org/1999/xlink”>To address this challenge, we present a novel technique designed to enhance the effectiveness of TB-APR methods by automatically generating test suites tailored for program repair. Unlike prior TB-APR techniques, it is rooted in the recognition that edge cases that invalidate overfitted patches play a pivotal role in guiding the repair process away from incorrect solutions. This technique leverages formal specifications and bounded verification to evaluate candidate patches and transforms the counterexamples (CEs) obtained from verifying candidate patches into tests for program repair. The efficacy of iteratively using such CEs as tests for TB-APR is substantiated by Iter-T our implementation of this technique for Java programs and JML specifications, evaluated on a benchmark of 717 buggy Java programs drawn from the APR literature. By progressively constructing test suites exclusively from CEs of overfitted candidate patches, Iter-T increases the odds of fixing a bug by about 58% compared to the originally provided test suites. Moreover, in cases where a TB-APR tool repairs a program using its original suite, employing CEs alone as test suites reduces the median time required to generate a correct patch by 42%. Remarkably, the generated CEs-based test suites are very small, accomplishing these results with only 2.4 tests on average.</p>

# data[].attributes.attachment_caption
AggrescanAI: Prediction of Aggregation-Prone Regions Using Contextualized Embeddings

# data[].attributes.headline
Alvaro Navarro; Santiago Palacios; Thierry Galmarini & Cristina Marino-Buslje

# data[].attributes.subject
Alvaro Navarro; Santiago Palacios; Thierry Galmarini & Cristina Marino-Buslje

# data[].attributes.body
,

Santiago Palacios 2

,

Thierry Galmarini

# data[].attributes.attachment_link.description
Please confirm you are a human by completing the captcha challenge below.

# data[].attributes.attachment_link.authors[].name
Ramiro Plüss and Hernán Villota and Patricio Orio

# data[].attributes.attachment_link.title
Hemispheric-Specific Coupling Improves Modeling of Functional Connectivity Using Wilson–Cowan Dynamics

# data[].attributes.attachment_link.description
Large-scale neural mass models have been widely used to simulate resting-state brain activity from structural connectivity. In this work, we extend a well-established Wilson–Cowan framework by introducing a novel hemispheric-specific coupling scheme that…

# data[].attributes.attachment_caption
Who is who? Competing coalitions in climate-AFOLU politics in Argentina

# data[].attributes.attachment_link.title
Pressure dynamics in the bottleneck flow of self-propelled particles

# data[].attributes.attachment_link.description
Intermittency and clogging statistics have been extensively investigated in a variety of systems -- from granular materials to pedestrian evacuations. Here, the authors experimentally study the evolution of mechanical pressure in a two-dimensional system of self-propelled agents flowing through a bottleneck, revealing a strong correlation between pressure dynamics and the system state (clogged vs unclogged).#ClearMotivation #SoftMatterSpotlight

# data[].attributes.attachment_link.title
The Role of Basal Ganglia Theta Oscillations in Predicting the Onset of Levodopa-Induced Dyskinesias - PubMed

# data[].attributes.attachment_link.description
Our results provide insights into the sequential nature of beta, theta, and gamma oscillatory changes. Theta activity may serve as a key biomarker for adaptive DBS. © 2025 International Parkinson and Movement Disorder Society.

# data[].attributes.attachment_link.title
Biomonitoring of an urban lake (Soldati Lake, Argentina): physicochemical analysis of water and ecotoxicological tests using a native species - Ecotoxicology

# data[].attributes.attachment_link.description
Ecotoxicology - This work aimed to evaluate the water quality of Soldati Lake (City of Buenos Aires, Argentina) through physicochemical and ecotoxicological studies. Two samplings were performed:…

# data[].attributes.attachment_caption
Stimulated Reservoir Volume (SRV) Estimation in the Vaca Muerta Formation Using a Geomechanical Permeability Evolution Law and Microseismic Monitoring Correlation

# data[].attributes.attachment_caption
Selective Lithium Uptake and Competition with Cobalt into Biobased Aerogels Loaded with Hydrogen–Manganese Oxide

# data[].attributes.attachment_link.title
Demographics of Close-In TESS Exoplanets Orbiting FGK Main-sequence Stars

# data[].attributes.attachment_link.description
Understanding the demographics of close-in planets is crucial for insights into exoplanet formation and evolution. We present a detailed analysis of occurrence rates for close-in (0.5-16 day) planets with radii between 2 and 20$\,R_{\oplus}$ around FGK main-sequence stars. Our study uses a comprehensive sample from four years of TESS Science Processing Operations Center full-frame image data cross-matched with Gaia, analysed through our rigorous detection, vetting, and validation pipeline. Using high-confidence planet candidates, we apply a hierarchical Bayesian model to determine occurrence rates in the two-dimensional orbital period-radius plane. Our results are presented using 10-by-10 bins across the period-radius parameter space, offering unprecedented resolution and statistical precision. We find an overall occurrence rate of $9.4^{+0.7}_{-0.6}\%$. When using identical binning, our occurrence rate posteriors distributions align with Kepler’s but have a magnitude smaller uncertainties on average. For hot Jupiters, we estimate the overall occurrence rate of $0.39^{+0.03}_{-0.02}\%$. This value is consistent with the previous Kepler FGK-type result within $1σ$. We find an overall occurrence rate of Neptunian desert planets of $0.08\pm0.01\%$, to our knowledge the first such determination. Additionally, in a volume-limited Gaia subsample within 100 pc in the same parameter region, we measure an overall planet occurrence rate of $15.4^{+1.6}_{-1.5}\%$ and a hot Jupiter occurrence rate of $0.42^{+0.16}_{-0.12}\%$. Our results establishes an improved foundation for constraining theoretical models of exoplanet populations.

# data[].attributes.attachment_caption
Glucuronoxylans from Moringa oleifera and other wood species obtained by alkaline extraction: Modulation of uronic acid content depending on the dosage of sodium borohydride

# data[].attributes.attachment_link.description
Please confirm you are a human by completing the captcha challenge below.

# data[].attributes.attachment_link.title
Photostability and phototoxicity studies of a promising UVA filter for sunscreens: 1-(2-hydroxyphenyl)-3-(4-methoxyphenyl)-1,3-propanedione - PubMed

# data[].attributes.attachment_link.description
A few years ago, the 2-hydroxy-4´-methoxy dibenzoylmethane (DBM11) molecule was synthesized and studied as a replacement for avobenzone (AVB, butyl methoxydibenzoylmethane) because of the latter’s instability under UVA radiation. These studies, conducted in absolute ethanol with an estimated UV irra …

# data[].attributes.headline
Milton Márquez Cadena, Alejandro Villarreal & Alberto Ramos

# data[].attributes.subject
Milton Márquez Cadena, Alejandro Villarreal & Alberto Ramos

# data[].attributes.attachment_link.title
Persistent DNA methylation and down-regulation of homeostatic genes in astrocytes after pilocarpine-induced status epilepticus: implications for epileptogenesis - PubMed

# data[].attributes.attachment_link.description
Epilepsy is a debilitating neurological disorder characterized by recurrent seizures, affecting millions of patients worldwide. Retrospective studies in temporal lobe epilepsy (TLE) patients have shown a high incidence of an initial precipitating event (IPE) in early childhood, followed by a silent …

# data[].attributes.body
Evaluation of Δ²-1,3,4-thiadiazolines derived from thiosemicarbazones for antichagasic activity: Insights into structural limitations

# data[].attributes.attachment_link.description
Please confirm you are a human by completing the captcha challenge below.

# data[].attributes.attachment_link.readable_text
The .gov means it's official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you're on a federal government site.

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

doi: 10.1007/s10646-026-03033-8.

Affiliations

¹ Departamento de Química Biológica. Laboratorio de Evaluación Ecotoxicológica del Agua Invertebrados Nativos y otros Modelos (EEAINM), Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina.

² Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales (IQUIBICEN), CONICET - Universidad de Buenos Aires, Buenos Aires, Argentina.

³ Departamento de Química Inorgánica. Laboratorio de Análisis de Trazas y Materiales, Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina.

⁴ Instituto Tecnológico de Buenos Aires, ITBA, Buenos Aires, Argentina.

⁵ Departamento de Química Biológica. Laboratorio de Enzimología, Estrés Oxidativo y Metabolismo, Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina.

⁶ Departamento de Química Biológica. Laboratorio de Evaluación Ecotoxicológica del Agua Invertebrados Nativos y otros Modelos (EEAINM), Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina. kristoffgisela@gmail.com.

⁷ Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales (IQUIBICEN), CONICET - Universidad de Buenos Aires, Buenos Aires, Argentina. kristoffgisela@gmail.com.

doi: 10.1007/s10646-026-03033-8.

Affiliations

¹ Departamento de Química Biológica. Laboratorio de Evaluación Ecotoxicológica del Agua Invertebrados Nativos y otros Modelos (EEAINM), Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina.

² Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales (IQUIBICEN), CONICET - Universidad de Buenos Aires, Buenos Aires, Argentina.

³ Departamento de Química Inorgánica. Laboratorio de Análisis de Trazas y Materiales, Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina.

⁴ Instituto Tecnológico de Buenos Aires, ITBA, Buenos Aires, Argentina.

⁵ Departamento de Química Biológica. Laboratorio de Enzimología, Estrés Oxidativo y Metabolismo, Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina.

⁶ Departamento de Química Biológica. Laboratorio de Evaluación Ecotoxicológica del Agua Invertebrados Nativos y otros Modelos (EEAINM), Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales, Buenos Aires, Argentina. kristoffgisela@gmail.com.

⁷ Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales (IQUIBICEN), CONICET - Universidad de Buenos Aires, Buenos Aires, Argentina. kristoffgisela@gmail.com.

No abstract available

Keywords: Biochemical parameters; Biomonitoring; Embryotoxicity; Gastropods; Hemocytes.

PubMed Disclaimer

Conflict of interest statement

Declarations. Conflict of interests: The authors declare no competing interests. Ethics approval: The use of B. straminea was approved by the "Hygiene and Safety Service, Faculty of Exact and Natural Sciences, University of Buenos Aires" (Resol. 1722/2003; 1401/2018).

# data[].attributes.attachment_link.keywords[]
national center for biotechnology information

# data[].attributes.attachment_link.title
Biomonitoring of an urban lake (Soldati Lake, Argentina): physicochemical analysis of water and ecotoxicological tests using a native species - PubMed

# data[].attributes.attachment_link.description
Biomonitoring of an urban lake (Soldati Lake, Argentina): physicochemical analysis of water and ecotoxicological tests using a native species

# data[].attributes.attachment_link.title
Monitoring diaphragmatic effort during diaphragm neurostimulation-assisted ventilation - Critical Care

# data[].attributes.attachment_link.description
Rationale Diaphragm neurostimulation-assisted ventilation (DNAV) can improve cardiopulmonary function during passive mechanical ventilation. However, this technique requires a reliable method to monitor and titrate diaphragmatic loading to avoid both insufficient and excessive diaphragmatic stimulation. Objective To establish whether the reduction in airway pressure-time product (ΔPTPaw) obtained during diaphragm neurostimulation in assist control volume-controlled mode accurately quantifies the magnitude of respiratory muscle effort elicited by neurostimulation. Methods This was a secondary analysis of the STIMULUS trial. Diaphragm neurostimulation was titrated across four levels targeting progressive occlusion pressures of 0, − 5, − 10, and − 15 cm H₂O at two PEEP levels. At each level, airway, esophageal, and gastric pressures were recorded to compute transdiaphragmatic pressure-time product (PTPdi), respiratory muscles pressure-time product (PTPmus), and ΔPTPaw, defined as the difference in airway pressure-time product between non-stimulated and stimulated breaths. Linear mixed-effects models, Bland–Altman analyses, and receiver operating characteristic (ROC) curves were used to assess agreement and discriminative ability. Measurements and main results Twelve patients contributed 494 high-quality respiratory cycles (63% of recorded cycles). Valid Pdi data were available in nine patients. Increasing neurostimulation was associated with higher PTPdi and PTPmus and a corresponding reduction in PTPaw. ΔPTPaw was correlated with both PTPdi (R² = 0.82) and PTPmus (R² = 0.92), with good agreement observed (limits: − 4 to 44 cm H₂O·s/min for PTPdi, and − 5 to 39 cm H₂O·s/min for PTPmus). ΔPTPaw demonstrated excellent discrimination for inadequate (area under receiver operating characteristic curve, AUROC ≥ 0.94) and excessive (AUROC ≥ 0.86) diaphragmatic effort. Conclusions ΔPTPaw is a reliable, non-invasive surrogate for monitoring diaphragm loading during DNAV under assist-controlled volume-controlled mode and may guide neurostimulation titration in mechanically ventilated patients.