Abstract
Background:
Preventive and modelling approaches to address the COVID-19 pandemic have been primarily based on the age or occupation, and often disregard the importance of the population contact structure and individual connectivity.
Methods:
We developed models that first incorporate the role of heterogeneity and connectivity and then can be expanded to make assumptions about demographic characteristics.
Results:
We demonstrate that variations in the number of connections of individuals within a population modify the impact of public health interventions such as vaccination approaches.
Conclusions:
The most effective vaccination strategy will vary depending on the underlying contact structure of individuals within a population and on timing of the interventions.
Keywords: COVID-19, SIR model, structure, heterogeneity, connectivity, vaccination strategies
Introduction
The Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), responsible for the coronavirus infectious disease COVID-19, has infected over 116 million and caused more than 2.5 million deaths globally. [1, 2] It continues to spread in most countries despite the wide range of preventive approaches that have been deployed.
In the absence of effective vaccines, country-level responses ranged from strict and prolonged lockdowns aimed at completely stopping transmission, to attempts to achieve natural herd immunity by allowing a substantial proportion of the population to get infected. [3] Even with new evidence suggesting that heterogeneity in age and social connectivity reduces the proportion of individuals that need to be infected to reach herd immunity, the human cost of this approach is unacceptable from a public health perspective and effective immunisation of the population remains the only viable approach to reach herd immunity. [4] Now that effective vaccines against SARS-CoV-2 have been approved, it is imperative to identify prioritisation strategies that maximise the impact of the limited doses available to reach herd immunity with the minimum loss of human life. [5]
Most countries have laid out vaccination strategies that prioritise healthcare workers, followed by those living in elderly care facilities, or essential workers (e.g., teachers, food industry), and then age groups from older to younger. [5–8] This prioritisation of older populations is based on disease severity and higher risk of death of COVID-19 among this group. However, another popular prioritisation strategy - that has been shown to be effective for influenza - is to begin with younger individuals to create an immune shelter around those most vulnerable. [9] Beubar et al. [10] evaluated these two strategies in different contexts and demonstrated that, in most scenarios, years of life lost were minimised by strategies that prioritised adults over 60 years, even as incidence was minimised when younger adults were prioritised. The exceptions to that rule were under low community transmission (basic reproductive number - R0 = 1.15 vs 1.5) or when the efficacy of the vaccine was significantly lower among older populations; in these cases, prioritising adults between 20–49 years resulted in the greatest reductions in mortality. A limitation of previous studies has been that they have mostly been based on age, and individual connectivity has not been considered. While dividing the population in age groups is a practical and actionable solution for vaccine prioritisation, it has the limitation of being a social construct with artificial cutoffs, which can sometimes mask the important role of the underlying contact structure.
Another challenge estimating the potential impact of COVID-19 preventive approaches has been that traditionally epidemiological models have depicted populations and epidemics as homogeneous [4], however the spread of COVID-19 has been characterized by a high variance in its reproductive number (Rt) which leads to a pattern of super-connected individuals being responsible for most infections. [11, 12] This represents a major deviation from how influenza, which has a smaller Rt variance, spreads and requires adjustments to traditional models. Hence, the aim of this study was to estimate the role of individual connectivity and contact heterogeneity on the COVID-19 epidemic, and to identify the vaccination strategies that minimise infection and death based on the underlying contact structure of the population.
Methods
Our model focuses on the interactions that infectious individuals have during the period when they are contagious. We considered a random graph in which connections between vertices (individuals) represent in-person interactions during the infectivity period (“risky interactions”). We first chose at random the interactions individuals will have if they become infected and then we spread the infection on top of the graph.
We compared two types of populations, a homogeneous one, where the number of contacts has lower variability among individuals (Erdős–Rényi graph -ER [13]) and another one that has a higher variance in the number of contacts of each individual (graph with power law degrees -PL [14]). See Supplementary Figure S1.
Interaction graphs
In the ER model [13] with Nvertices, any pair of vertices is connected with probability , independently on each other. The number of connections per individual is binomially distributed with mean Nq = e and variance Nq(1 − q).
We constructed a PL graph with exponent γ = 0.307 following Qiao et al. [14] This means that the probability that a vertex has k connections (i.e. has degree k) behaves like k−γ for large k.
Infection dynamics
After simulating the interactions, we simulated the spread of the infection using a SIR model. It is a continuous-time Markov chain where individuals can be in four compartments: Susceptible, Infected, Recovered or Dead. At time 0, we started with one infected individual chosen at random (the rest were all susceptible). When an individual gets infected, the duration of the infectivity period was exponentially distributed with parameter Tr = 1/10, meaning that its expected duration was 10 days. During this period, he could infect each of his contacts with probability Pi. To implement this in a Markovian way, we assigned to each of his neighbours an independent exponential variable of parameter Ti. If this variable was smaller than the infectivity period, the infected individual transmitted the disease to the corresponding neighbour. Using a classical result on the minimum of exponential variables, we chose Ti such that:
At the end of the infectious period, with probability Pd the focal individual died and with probability 1 − Pd he recovered. We assumed that recovered individuals could not be reinfected (See Supplementary Figure S2 and Supplementary Movies).
The process was simulated in R using a Gillespie algorithm.
Vaccination strategies
Vaccination corresponded to reducing the number of vertices with certain probability based on the effectiveness of the vaccine (∈). Each vaccination strategy corresponded to a way of subsampling D individuals:
“Uniform”: D individuals uniformly selected at random.
“Most connected”: D individuals selected in order from most connected (those with the highest degree) to less connected.
“Least connected” : D individuals selected in order from least connected to more connected.
We tested two variants of each of these strategies:
Strategy S: selecting only among the susceptible individuals.
Strategy SIR: selecting among the susceptible, infected or recovered (but not among the dead).
When vaccination was conducted at time 0, both strategies were equivalent.
As a simplifying assumption, vaccinated individuals did not contribute to disease spread. The vaccine had effectiveness ϵ = 0.9, meaning that among the D chosen individuals, only ∈D are deleted from the graph. Vaccination started after the cumulative proportion of infected individuals reached 0%, 10% or 30%. Most simulations were conducted with vaccine doses available for 25% of the population, but in an extended model the number of doses was doubled but the effectiveness (ϵ) was decreased from 0.9 to 0.5 to mimic the dose sparing strategies that have been applied in some countries.
Choice of the parameters
Since the simulations were computationally intensive and we wanted to test as many scenarios as possible, we chose a population size of N=20000. We assessed how this parameter affected the results by varying N from 5000 to 70000 (Supplementary Figure S2). Models seemed to converge at a sample size of approximately 10000, which is a good indicator that, qualitatively, a larger population should behave similarly to our small world simulations.
The average number of risky interactions was e=44 following a study in Utrecht (Netherlands), [15] where individuals reported the number of people with whom they had a conversation of at least 10 minutes during a week, which we consider is a proxy for e.
Originally, Pd = 0.01, thus the number of infected was directly correlated with the number of dead. In extended models that assumed that old individuals were less connected and more vulnerable than younger individuals, we assigned Pd = 0.07 to the 17% least connected and Pd = 0.005 for the rest. This percentage corresponds to the proportion of over 65 in Utrecht and the probabilities are in line with the case fatality rate of COVID-19 for different age groups. [16]
Repeatability and sample size
In a small number of simulations, the epidemic died out after infecting less than 0.5% of the population. These simulations were excluded from the results (because they are not consistent with the case of COVID-19). Results presented show the mean and dispersion of 30 independent realisations conditioned on infecting a macroscopic fraction of the population (>0.5%). We considered interventions significantly different if over 95% of the simulations did not overlap.
Results
Association of connectivity with time to infection
In the first model assessing the relationship between connectivity and time to infection, under the parameters described above (N=20000, e=44, Pi=0.05), we computed the basic reproduction number R0 as the expected number of cases directly generated by one infected individual if all their neighbours are susceptible, i.e. R0 = e.Pi = 2.2. The effective reproduction number Rt (estimated from the simulations using 14 days rolling windows) varied from 1.7 at the beginning of the outbreak to 0.9 towards the end.
In baseline simulations, highly connected individuals tended to get infected early, while less connected individuals get infected at random times throughout the epidemic. This was especially clear for heterogeneous contact structures, where the average number of risky connections per individual at early stages was significantly higher and decreased with time (Figure 1).
Figure 1.
Two different ways of modelling social interactions. Top panels represent the distribution of the number of risky interactions in the ER and the PL graphs with 20000 individuals. Panels in the middle show a realisation of the SIR process for each of the models. Bottom panels show the number of risky interactions individuals have, as a function of the order in which they are infected (dots show the average over 30 simulations).
Vaccination strategies based on contact structure.
In the simulations with doses available for 25% of the population, vaccinating the most connected among the susceptible always resulted in the smallest proportion of infected individuals independently of heterogeneity or timing of the intervention, although the effect was larger in the heterogeneous graph (Figure 2). In models where susceptibility status was not considered (SIR vaccination), the benefits of targeting the most connected individuals decreased when more people had been infected before the intervention. For both graphs, vaccinating uniformly performed similarly to vaccinating the most connected when the intervention started after 30% of the population had already been infected (Figure 2). Vaccinating the least connected or among the least connected resulted in the highest proportion of infections in every scenario.
Figure 2.
Proportion of infected and dead individuals for three vaccination strategies. The plot shows the proportion of infected at the end of the infection for 30 repetitions. The number of doses of the vaccine represents 25% of the population size (N = 20000). Error bars represent standard deviation. Different starting times are shown in the different panels (when 0, 10 and 30% of the individuals have been infected). On the top right panel, when vaccinating the most connected, the epidemic always died out quickly, before infecting at least 50 individuals, which is the minimum required to be considered a successful simulation (see Methods).
When the vaccine intervention was implemented early, when 10% of the population or less was infected, vaccinating the highly connected individuals first resulted in the greatest decline in the speed of propagation. However, when the intervention was implemented after 30% of the population had been infected, the decrease in the speed of propagation was much smaller (Figure 3).
Figure 3.
Effect of the vaccination strategy on the rank of infection. The dots represent averages over 30 simulations. The number of doses of the vaccine represents 25% of the population size (N= 20000). In these simulations we vaccinate individuals regardless of their status (S, I, R). Vertical bars indicate the time when the intervention is made.
Vaccination strategies, connectivity, and mortality
In models where number of connections was linked to fatality rate, in the homogeneous case (ER), uniform and most connected strategies performed similarly, preventing significantly more deaths than the control. Vaccinating the least connected (which were also the most vulnerable) was the best strategy. These results were independent of the time of the intervention. In a heterogeneous population (PL), the most connected (S and SIR) were the strategies that prevented more deaths, especially when the intervention started early. Uniform and least connected strategies performed equally well, preventing significantly less deaths than most connected S but slightly more deaths than the control. When vaccination started after 10 or 30% of the population had been infected, most connected S had an advantage over most connected SIR. (Figure 5).
Figure 5.
Proportion of deaths, for an effectiveness of 90% and for different values of tV (top panels vaccination starts at 10% and bottom panels at 30%) The horizontal lines in the middle of the boxes show the mean values among all simulations, the upper and lower edges of the boxes are the quantiles q0.25and q0.75 corresponding to 75% respectively 25%. The vertical lines reach until q0.25–1.5*(q0.75-q0.25) downwards and until q0.75+1.5*(q0.75-q0.25). The points represent outliers (i.e., simulations whose results are atypical). See Supplementary materials for a description of all strategies.
Dose sparing under different vaccination strategies
In the case of a homogeneous population (ER), strategies that complete the vaccination schedule increasing effectiveness and those that vaccinated more people with less effectiveness yielded similar results with respect to the total number of infected individuals. However, in the case of a heterogeneous population (PL), when vaccinating the most connected individuals, administering the two doses to fewer people was a more efficient strategy producing a smaller number of infected individuals. For the uniform strategy, the way of administering the doses made no difference. (Figure 6).
Figure 6.
Dose sparing. Two doses: 25% of the population receives two doses of the vaccine (effectiveness 90%). One dose: 50% of the population receives a single dose of the vaccine (effectiveness 50%). Error bars represent standard deviation. The time of the vaccination tV is when the cumulative number of infected individuals reaches 10%.
Discussion
In this study, we modelled different vaccination strategies based on connectivity and found that the heterogeneity that characterises COVID-19 spreading modified the effectiveness of different prioritisation approaches. In this heterogeneous context, interventions that prioritise more connected individuals performed better at preventing infection and deaths when compared with uniform strategies where vaccines are distributed at random. Conversely, strategies that prioritised the less connected individuals had the worst outcomes even when a higher fatality rate was assigned to the less connected. Most current strategies begin vaccination with highly connected individuals (healthcare workers, populations living in elderly care facilities) but then turn to vaccinate older and vulnerable populations which tend to be among the least connected [17]. If in fact more vulnerable individuals are less connected, current vaccination strategies that prioritise older and more vulnerable individuals over younger and more connected might be suboptimal to address the COVID-19 pandemic.
Another important result is the importance of intervention timing during the history of the epidemic. Especially under heterogeneous contact structures that are similar to the COVID-19 spread pattern, highly connected individuals tend to get infected very early and drive the early stages of the contingency. This finding highlights the importance of considering what proportion of the population has already been exposed to the virus and potentially developed some immunity when the intervention is implemented. We observed that in the absence of susceptibility-based targeting, interventions prioritising highly connected individuals were more effective early in the epidemic. Others have highlighted the importance of antibody testing to prevent infection and death, [10, 18] and our results further support this approach especially when targeting highly connected individuals late in an epidemic when a high proportion has already been infected. The proportion of healthcare workers with SARS-CoV-2 antibodies ranges between 2–50% in different settings, [19] based on our results, targeting the available doses based on previous immunity could maximise its impact especially in settings where seroprevalence is high.
A challenge of these connectivity-based strategies is that identifying the most connected individuals in real life is not easy. To address this, we tested two modified versions of the “most connected” vaccination strategy (see Supplementary Materials). The first one consists of randomly selecting an individual and then vaccinating a person connected to them (“neighbour”), which biases the selection towards the most connected. For the second one, we divided the population into two groups based on their number of connections (most and least connected) and then sampled among the most or least connected group. In the heterogeneous scenario, vaccinating the most connected remained the most effective approach; however, “neighbour” and “among the most connected” also performed better than “uniform”, supporting the targeting of highly-connected individuals as a promising strategy for COVID-19 vaccination. An operationalisation of vaccinating among the most connected would be to prioritise individuals with occupations that require face-to-face interactions, for example those in the service industry.
A criticism of targeting the most connected individuals has been that these tend to be young and less vulnerable [17], hence we also extended these models to assess their impact on mortality under the assumption that connectivity is inversely associated with fatality. In the homogeneous structure (ER), vaccinating the most vulnerable resulted in reduction in the number of deaths.
However, in heterogeneous structures, strategies targeting more connected individuals performed better than “uniform” and “least connected”, especially when targeting the susceptible. Uniform approaches have the advantage of being easier to implement and they perform significantly better than targeting less connected individuals, which resulted in the greatest number of infected. Some counties in the US have started implementing uniform approaches because they originally faced challenges distributing vaccines to older populations first, [20] our results support this vaccination approach over targeting less connected individuals.
Most of the vaccines that are available for COVID-19 were designed to be administered in two doses (and the effectiveness was shown to be over 90%). However, it has been suggested that governments should aim to give as many people as possible a single dose, instead of using half the vaccines currently available on second doses (i.e., dose sparing). [21] The effectiveness after a single dose has been estimated to be around 50%. In the case of a heterogeneous population, when vaccinating the most connected individuals, administering the two doses was a more effective strategy producing a smaller number of infected individuals. However, when the least connected were prioritised, dose sparing resulted in fewer infections, suggesting that if current strategies that prioritise less connected individuals continue, applying a single dose to more people would be the best approach (although still significantly worse than targeting the most connected or even distributing the vaccine at random).
A strength of this study is the middle-ground modelling approach between agent-based [22] and mean-field [23] models, that combines the dynamic nature of the first with the computational efficiency of the latter. This is achieved by focusing solely on the interactions that infectious individuals have during the period when they are infectious, a crucial part of the social dynamics of the propagation of the virus. While our method disregards some features of a real society, it allows us to capture the essential differences between homogeneous and heterogeneous contact structures. In this sense, a limitation of these models is that they do not provide quantitative estimates of the exact impact of different interventions, however they provide sound qualitative judgments that allow to rank different vaccination strategies based on number of infections and deaths. Similarly, the graphs used in this study were parameterised based on data from a single study from a European city and have a relatively small sample size compared to most urban areas, however modifying the average number of contacts or the sample size did not modify our findings.
In conclusion, the effectiveness of vaccination strategies depends on the heterogeneity of the contact structure and the specific infection dynamics, it is important to consider COVID-19 vaccine prioritisation based on individuals’ connectivity.
Supplementary Material
Movie S1. Propagation of the epidemics in an Erdős-Rényi graph. Vertices are coloured depending on the status of the individual they represent. Blue: susceptible, red: infected, green: recovered, black: dead. The edge connecting i and j is coloured in red when individual i infects individual j. The numbers that appear at the end are the number of individuals that were infected by each individual.
Movie S2. Propagation of the epidemics in a power-law degree distribution graph. Vertices are coloured depending on the status of the individual they represent. Blue: susceptible, red: infected, green: recovered, black: dead. The edge connecting i and j is coloured in red when individual i infects individual j. The numbers that appear at the end are the number of individuals that were infected by each individual.
Key messages:
The best vaccine prioritization strategies depend on the underlying contact structure and the timing when the intervention is implemented.
In general, for heterogeneous contact structures that mimic the COVID-19 spread, vaccinating the most connected individuals first was the most effective strategy to prevent infections and deaths.
These models can be used to identify the best potential vaccine prioritisation strategies for different settings.
Acknowledgements
We would like to thank Arno Siri-Jégousse, Fabian Freund, Ximena Escalera-Fanjul, Karla Galaviz, Gloria Soberon, Adithya Cattamanchi, April Mohanty and Anarina Murillo for fruitful discussions.
Funding
The authors acknowledge support of the Spanish Ministry of Science and Innovation to the EMBL partnership, the Centro de Excelencia Severo Ochoa and the CERCA Programme / Generalitat de Catalunya, the Indiana University Bloomington Mexico Gateway and Indiana University UITS Research Technologies computing resources. VMP also acknowledges support of the DGAPA-UNAM postdoctoral program. AGCS acknowledges support from the Mexican science council (CONACYT) grant A1-S-14615. IGC is supported by the National Heart, Lung, and Blood Institute (NHLBI) career development grants HL137338-03S1 and HL126146-02
Footnotes
Data availability
Code is available at: https://github.com/JulioNava31/The-role-of-connectivity-on-COVID-19-preventive-approaches.
Conflict of interest statement
Authors declare no competing interests.
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Associated Data
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Supplementary Materials
Movie S1. Propagation of the epidemics in an Erdős-Rényi graph. Vertices are coloured depending on the status of the individual they represent. Blue: susceptible, red: infected, green: recovered, black: dead. The edge connecting i and j is coloured in red when individual i infects individual j. The numbers that appear at the end are the number of individuals that were infected by each individual.
Movie S2. Propagation of the epidemics in a power-law degree distribution graph. Vertices are coloured depending on the status of the individual they represent. Blue: susceptible, red: infected, green: recovered, black: dead. The edge connecting i and j is coloured in red when individual i infects individual j. The numbers that appear at the end are the number of individuals that were infected by each individual.