Q&A: Current Global Access to the COVID-19 Vaccine – American University

It has been more than three years since the first COVID-19 vaccine received emergency use authorization by the US Food and Drug Administration in December 2020. Since that time, the US has seen the emergence and approval of several other COVID-19 vaccines and distributed hundreds of millions of doses to populations across the nation.

But what about other countries? What is the status of COVID-19 vaccine access around the world? To gain greater clarity on the status of global vaccine access, we asked SIS professor Nina Yamanis a few questions.

The purpose of the analyses presented here was to evaluate distinct vaccine uptake strategies in the context of COVID-19. We aim to achieve qualitative results by exploring counterfactual scenarios driven by vaccine uptake. Therefore, we exploited the unfolding of the Belgian COVID-19 crisis with the induction of SARS-COV-2 in February 2020 and the emerging Alpha, Delta, and Omicron (BA.1 and BA.2) VOCs. Each simulation spans the first two years of the COVID-19 pandemic, running from March 2020 to February 2022. It includes age-specific uptake of first, second, and booster doses of adenovirus and mRNA-based vaccines. The uptake scenarios being examined vary between August 2021 and February 2022, which is the period our results primarily focus on.

We extended a previously published stochastic transmission model for SARS-CoV-2 in Belgium by Abrams et al.[27], by including COVID-19 vaccination, emergence of different VOCs and waning immunity. Our transmission model is a discrete-time age-structured compartmental model with a chain-binomial transition process between various disease compartments that can be categorised into susceptible, exposed, infected, recovered, and death states. Overall, after exposure to the pathogen and acquiring infection, an individual becomes infectious after a latent period and moves to a pre-symptomatic state. Subsequently, individuals develop symptoms or remain asymptomatic, before recovering. Symptomatic infections start mild and have an age-specific probability of progressing to serious illness, implying hospitalisation with or without admission to the ICU. We also account for disease-related mortality of hospitalised cases. The original model formulation is duplicated into a two-strain compartmental structure (see Fig.1), and transitions between multiple copies of the two-strain model (see Fig.2) allowed for waning immunity against infection and severe disease. Further elaboration on the construction of the model, specifically based on multiple substructures, is presented in the subsequent paragraphs.

The model structure proposed by Abrams et al.[27] including ten 10-year age groups has been adapted to a two-strain version with a common susceptible class and a duplication of all infection-related health states. Our model structure (see Fig.1) enabled co-circulation of two variants at the same time with distinct properties with respect to susceptibility, latent period, disease severity, hospital length of stay, mortality, and vaccine-related protection. To cover the newly emerging Delta VOC, we re-used the health states of the dominated original strain in the simulation after book keeping all states. A similar transition was made with the Omicron VOC when the Alpha VOC was fully dominated by the Delta VOC. More information about model dynamics and parameters is provided in the Supplementary Information. Our model operates starting from March 1st, 2020, and accounts for the emergence of new pathogen strains and the administration of various vaccine doses. In the early stages, these factors are represented in small amounts, with heterogeneity and randomness playing critical roles. Even slight variations can become amplified over time. Later on, after COVID-19 vaccination is introduced and attains high coverage, the size of the remaining susceptible population becomes small. This makes the stochastic nature of infection and subsequent processes like hospitalisation and death increasingly significant, especially since the model is calibrated using age-specific incidence data for each of these stages. These elements highlight the importance of the stochastic nature of our compartmental model in accurately reflecting and predicting evolving dynamics.

Health states and transitions in the two-strain transmission model. The model structure is described in the main text and model parameters are listed in the Supplementary Information

We used the reported social contact rates of 42 Belgian CoMix survey waves between April2020 and March2022[28, 29] as proxy for effective contacts that allow disease transmission according to the social contact hypothesis[30]. CoMix has been designed as a collection of surveys in which a panel of participants retrospectively reports all social contacts made from 5:00 AM on the day preceding the survey up to 5:00 AM on the day of the survey. A contact was defined as an in-person conversation of at least a few words or a skin-to-skin contact[28]. Changes in transmission that are not directly attributable to changes in contact behaviour are captured in age-specific proportionality factors. They represent, for example, changes in compliance to (social distancing) measures, seasonality effects, and shifts in the location-specific contact intensity (e.g., contacts inside are more risky than contacts outside). For each wave, we estimated age-specific proportionality factors to translate social contact rates into transmission rates that capture age-specific susceptibility and infection-related risk behaviour associated with social contacts[31].

The introduction and presence of VOCs in the model population are taken into account in the parameter estimation process based on the baseline genomic surveillance of SARS-CoV-2 in Belgium by the National Reference Laboratory[32]. To simulate the replacement of the original strain in 2021, we aggregated all Alpha, Beta, and Gamma VOC samples that were identified, which we refer to hereafter as Alpha VOC infections, when estimating the penetration of the VOCs into the Belgian population. We attributed the growth advantage of the Alpha VOC completely to transmissibility and ignored the potential effect of immune escape. We assumed that there was no change in the probability of hospital admission for the (aggregated) Alpha VOC. Conflicting post hoc observations have been reported on the severity of this VOC[33]. Therefore, we have chosen to highlight the significant role that increased transmissibility potential plays in hospitalisations and mortality, regardless of any direct effect of the variant on severity.

For the Delta VOC, we account for increased transmissibility and adopted an adjusted hazard ratio for hospitalisation of 2.26 relative to the Alpha VOC based on a cohort study conducted in the UK[34]. Due to the lack of age-specific information to align the reported 95% confidence interval of [1.32;3.89] with our age-specific model design, we opted to use the estimated mean value without considering parameter uncertainty. This adjusted hazard ratio was essential to match the reported incidence of hospitalisations with genomic surveillance data on the Delta VOC[32].

With the emergence of the Omicron VOC, studies[35, 36] indicated a change in the incubation period and the serial interval, which contributes to its transmission advantage. This had a large impact on the estimated reproduction number and the effect of restrictive measures. As such, we included a VOC-specific latent period in our transmission model, which was inferred specifically for the Omicron VOC during the calibration process. Furthermore, Omicron-specific hazard ratios for hospitalisation were pivotal to capture the trends observed in 2022. We adopted age-specific hazard ratios for hospital attendance with the Omicron VOC compared to the Delta VOC, from a cohort study in the UK[37]. More specifically, we used for our 10-year age bands: 1, 0.89, 0.67, 0.57, 0.54, 0.42, 0.32, 0.42, 0.49 and 0.49. The simulation period covers both Omicron sub-lineages BA.1 and BA.2, the latter became dominant in Belgium on February 28th, 2022. The differences in transmission for BA.2 are absorbed in the wave-specific proportionality parameters for February/March 2022.

All the levels of protection adopted are summarised in Table1. We used a leaky vaccine implementation approach in which vaccination with 74% effectiveness implies that for a vaccinated individual the probability of acquiring infection is 74% lower compared to a non-vaccinated individual of the same age. Vaccine-induced immunity against infection is implemented as a step function in terms of protection against infection 21days after the first dose of vaccine. Protection induced by second and booster vaccine doses is assumed to be fully achieved 7days after vaccine administration (i.e., depending on the maximal effectiveness of the vaccine as reported in Table1). We consider the differences between mRNA- and adenovirus-based vaccines in how they induce immunity and in terms of protection. We assumed that vaccinated individuals who acquire infection are at a lower risk of hospital admission with COVID-19 and all booster doses in Belgium are mRNA-based vaccines. Given our model structure, reported protection levels against hospital admission were applied as protection against severe disease, which ultimately leads to hospital admission. Vaccinated individuals (with or without a booster) who acquire infection do not have a lower risk of transmitting the disease. This assumption is challenged in the sensitivity analysis.

Vaccine-induced protection and waning immunity have been included through duplication of the two-strain compartmental structure with uptake-based and time-specific transitions (see Fig.2). This model structure allowed us to explicitly keep track of vaccine type and dose-specific vaccine uptake and to differentiate protection against infection and severe disease between vaccine type and number of doses. The duplicated two-strain compartmental structure also allowed differential waning immunity against infection and severe disease.

We integrated waning immunity into our model by establishing a series of steps transitioning from complete protection to a state of diminishing immunity over an average period of 90 days. In the framework of the compartmental model, the waning rate is defined as the fraction of individuals transitioning from full protection per time unit, which inversely correlates with the average protection duration. Consequently, we incorporated submodels for diminishing vaccine-induced immunity, featuring levels of reduced protection as detailed in Table1. Initially, infection-induced immunity offers 100% protection, assuming individuals in the Recovered state are not susceptible to reinfection. Therefore, our model accounts for the decrease in infection-induced immunity by moving individuals from the Recovered to the Susceptible compartment within a submodel, which still affords a degree of protection against future infections. We assumed the effect of a booster dose independent of the immunity state upon vaccination, i.e., with or without prior infection or a specific vaccine scheme. We accounted for waning immunity after the booster dose with a dedicated submodel and an average transition time of 90 days. Note that even with waning immunity, vaccinated individuals maintain partial protection against subsequent infection and severe disease upon infection. VOC-specific protection levels for the booster dose have been derived from the literature (see Table1).

Vaccine uptake in the model is based on age-specific data at the national level reported by the Belgian Scientific Institute for Public Health, Sciensano[38]. By August 2021, on average 90% of the population aged over 20 years had completed their two-dose regimen with mRNA or adenovirus-based vaccines. On the contrary, about 10% of the 0-19-year-olds received two doses of an mRNA vaccine at that time. It is important to note that in August 2021, mRNA vaccines were only authorized for use in children aged 12 years and older by the European Medicine Agencys Committee for Medicinal Products for Human Use. Subsequently, in 2022, the authorisation was extended to younger children, initially to those aged 6 years and older, and later to infants as young as 6 months of age. The decision to administer booster doses at the end of 2021 was based on the evaluation by the European Medicines Agency that indicated an increase in antibody levels following a booster dose administered about 6 months after the second dose in individuals aged 18 to 55 years. Based on this evidence, first booster doses were recommended in Belgium for people 18 years and older at least 6 months after the second dose.

Full details on the type- and dose-specific vaccine uptake by age we included in the model is presented in Fig.S2. We did not explicitly account for risk-group vaccination, since our model structure did not facilitate more subpopulations with differential risk and potentially a more severe episode of COVID-19 disease once infected (i.e., a higher probability of hospitalisation and/or a higher probability of death, if hospitalised). In our analysis, we primarily considered age as the main determinant of risk and severity. The reported uptake of Pfizer-BioNtech (Comirnaty) and Moderna (Spikevax) vaccines are aggregated into one mRNA vaccine type. The relatively low number of reported Johnson & Johnson (Ad26.COV2.S) and Curevac (CV07050101) vaccines were aggregated in the model with the adeno-based AstraZeneca vaccine (ChAdOx1 or Vaxzevria) based on similarities in protection and waning immunity. Third doses (i.e. first booster dose) are included in the transmission model as a separate submodel with all health-related compartments. A comprehensive summary of vaccine uptake we included in our model is depicted in Fig.3, which presents also the scenarios discussed in the subsequent sections of the Methods.

Overview of the duplicated two-strain model structure to account for vaccine type- and dose-specific immunity against infection and severe disease in combination with differential waning immunity over time. The grey boxes embody the transmission structure included in Fig.1 while only the Susceptible and Recovered are shown here (with (R_i) representing (R_a) and (R_b)). More information on the waning states is included in Table1

We used Bayesian methods to fit our transmission model to multiple data sources, including daily hospital admissions and bed occupancy, early seroprevalence, genomic surveillance, and mortality data. In order to capture the full extent of the intrinsic variability of the model, we relied on Markov Chain Monte Carlo (MCMC) sampling with 60 chains in the calibration procedure. An adaptive Metropolis-within-Gibbs algorithm was used as MCMC sampler, and parameter priors were based on permutations of previously converged calibration results. The model parameters related to hospital incidence and VOC prevalence were estimated by gradually extending the time horizon over consecutive calibration runs for the stochastic model. The absence of age-specific data on daily hospital discharges and transitions between general wards and ICU hampered a likelihood approach to accommodate hospital occupancy in general and in the ICU. Therefore, the fitting of the model was performed using a multi-step procedure. First, all transmission-related model parameters were estimated while calibrating the model to the observed incidence data on hospitalisation, early seroprevalence and genomic surveillance as described above. Next, all parameters related to hospital and ICU occupancy (including discharge rates) were estimated based on minimising a least squares criterion for the distance between the observed and generated loads. Finally, the estimated mortality-related parameters are inferred again using a likelihood-based approach, distinguishing whether a hospital discharge was due to mortality or recovery. This multi-step procedure has been performed multiple times, of which the final iteration is described in TableS2. Finally, we selected the 40 best performing MCMC chains of the last step to derive parameter estimates for our simulation study.

We used hospital admissions with COVID-19 as a primary source of information to capture the burden of disease. During the development of the model, we observed that around 10%-20% of the admissions with the Alpha and Delta VOCs were primarily due to other pathologies, but patients who tested positive when admitted were transferred to the COVID-19 wards and counted in the COVID-19 hospital load. With the Omicron VOC, the difference between admissions with COVID-19 and for COVID-19 increased even more. Given our focus on hospital capacity, hence occupancy, hospital admissions with COVID-19 were most informative in combination with reported estimates for hospital stay.

We estimated a transmission advantage of the Alpha VOC compared to the original strain of 32% (95% CrI: 24-39%). For the Delta VOC, the transmission advantage compared to the Alpha strain was estimated to be 87% (95% CrI: 71-106%). For Omicron, we estimated an almost instant transition from the exposed to the pre-symptomatic infectious health state (which is in line with the shorter serial interval we referred to previously) and a transmission advantage compared to the Delta VOC of 35% (95% CrI: 9-70%). A comprehensive overview of the model parameters is presented in TableS3 of the Supplementary Information.

The baseline scenario consisted of all estimated parameters during the calibration of the compartmental model and fitted the national trends of SARS-CoV-2 pandemic in Belgium. This includes, for example, the emergence and dominance of the Omicron variant from December 2021 and the observed decrease in hospital admissions and deaths at that time. The full model output from March 2020 is presented in Fig.S4. The transmission model was based on bi-weekly social contact survey data, which allows for including adjusted behaviour over time in the model. The survey data represented changing contact rates, while the estimated proportionality factors captured differences in, for example, contact intensity, susceptibility and infectivity. These factors were age-specific and part of the parameter estimation process (see Fig.S5).

To estimate the burden of disease, we included the loss of QALYs from a published study on the model-based cost-effectiveness of SARS-CoV-2 vaccination along with physical distancing in the United Kingdom[22]. Disease morbidity estimates were obtained by multiplying the model-based incidence of mild infections, and hospitalised and ICU admitted patients with the QALY loss values in Table2. Disease-related mortality based on the quality-adjusted life expectancy[39] is obtained by combining the age-specific model estimations on mortality with the Belgian life expectancy for 2019 reported by Statbel[40] and the latest age-specific Belgian population norms based on EQ-5D-5L[25].

We explored retrospective counterfactual scenarios based on vaccine uptake in the presence or absence of the Omicron VOC. None of these scenarios explicitly included the importation of infected cases as a result of international travel except for the introduction of VOCs. We started from the final calibration of the model and the reported vaccine uptake scheme and explored proportionally increased uptake of two doses in 511-year-old children and first booster doses in adults over 18-years. Vaccine uptake levels and timing could be explored more in detail with additional objectives and trade-offs, although this analysis aims to provide a basis for predominantly qualitative interpretations. We allow for stochastic variation in the transmission process by running each of the 40 estimated model parameter sets 10 times, hence incorporating 400 model realisations in the final comparison. The number of realisations was determined through a process of model exploration and consideration of the trade-off between model realisations and computational feasibility due to model complexity.

To explore the impact of the uptake of the COVID-19 vaccine, we evaluated an adjustment of the uptake of the first booster dose in adults and an increase in the level of childhood vaccination. First, we changed the uptake of the first booster dose so that it matches the age-specific two-dose uptake levels by 1 March 2022 (see Fig.3). That is, we assumed that all those eligible for a first booster dose effectively received an mRNA booster dose. The reported first booster dose uptake in the Belgian adult population was 76%, so the additional uptake in the scenario analysis was rather limited. Secondly, we defined a scenario in which we arbitrarily included only 60% of the reported uptake of the first booster dose. The 40% reduction is applied uniformly across all age groups. A third adoption scenario focused on children aged 5-11 years in July-August of 2021. Vaccination in this age class was was not licensed at the time in Belgium, although we explore possible outcomes if 5- to 17-year-old children had been vaccinated simultaneously. More specifically, we aligned the uptake of the mRNA vaccine for children aged 5-9 and 10-11 years with the reported uptake of children aged 12-15 and 16-17 years, respectively. This approach required scaling the reported uptake to match the number of age bins in each group. For example, for each reported first dose in 12-15-year-old children (i.e. 4 age bins), we included (frac{5}{4}) dose within the 5-9-year-olds (i.e. 5 age bins) at the given point in time. The time between two consecutive doses is assumed to be three weeks, and the resulting vaccine uptake is presented in Fig.3. The final uptake of two doses of vaccine is approximately 80% in the group of 10-19 years and 40% in the group of 0-9 years (which corresponds to 80% in the group of 5-9 years).

Reported and scenario-based uptake of COVID-19 vaccines over time in adults above the age of 20y (top) and 0-19-year-old children (bottom). The uptake is presented in terms of the absolute number of doses (left axis) and as a percentage of the target group (right axis)

To assess the impact of the Omicron VOC on the epidemic trajectory, we performed simulations of our three adjusted vaccine uptake scenarios without the presence of the Omicron VOC, while keeping all other model parameters constant.

In our main analysis, we adopted a conservative approach that assumed no infectiousness-related protection from the COVID-19 vaccine, which potentially underestimates the effect of the intervention. As such, we opt to minimise the risk of overestimating the intervention-related benefits for this exploratory analysis. However, household studies conducted in Denmark[41] and the UK[42] have reported a 31%-45% decrease in the risk of SARS-CoV-2 transmission among vaccinated individuals. Therefore, as a sensitivity analysis taking into account these findings, we performed a comprehensive model calibration assuming a 30% reduction in infectiousness for vaccinated individuals and exploring the effect on the vaccine scenarios.

As robustness analyses, we performed the full model calibration with invariant proportionality factors across different consecutive CoMix waves. A single set of age-specific parameters was not possible as the link between observed contact rates and disease transmission was not constant throughout 20202022 due to differences in contact intensity, duration, and location, among other things. Therefore, we aggregated the CoMix waves into five groups based on the distancing measures that were in place, the (school) holiday periods, and the model fit. For the time period without CoMix data (i.e. for March and SeptemberNovember 2020), we still required time-specific q-factors. Details are provided in Supplementary TableS1.

Part of this epidemiological mathematical modelling study was carried out to inform the Belgian government and the general public about COVID-19 trends and possible interventions. This study is based on data sources from the Belgian Institute of Public Health (Sciensano) in combination with published estimates and data sets (e.g., CoMix). Funding agencies did not have a role in study design, data collection, data analysis, data interpretation, reporting, or in the writing of this manuscript. Data preparation and statistical analyses were performed using R (version 4.2.2; R Foundation for Statistical Computing, Vienna, Austria) on MacOS 12.5 and using R (version 4.0.2) on Rocky Linux 8.8 on the VSC cluster.

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The study showed that the bivalent vaccine was better at neutralizing more recent viral variants, such as omicron and its subvariants

Credit: Fbio Rodrigues-Pozzebom/Agncia Brasil

A major bivalent COVID-19 vaccine induces production of neutralizing antibodies against the coronavirus that circulated at the start of the pandemic as well as subvariants of omicron, albeit less abundantly, according to a Brazilian studyreportedin theJournal of Medical Virology.

The study confirmed the vaccines effectiveness and its importance to control of the disease, while also showing that, more than three years after the first application of a COVID-19 vaccine in Brazil, the vaccination model should be similar to that adopted for influenza, with frequent adjustments to the formulation to prioritize more recent variants.

This was the first research project conducted to evaluate the immunity induced by the Pfizer-BioNTech bivalent vaccine (COMIRNATY Original/Omicron BA.4-5) in a group of Brazilian subjects. The scientists investigated the antibody neutralization response against different variants of SARS-CoV-2 using serum samples from 93 healthy volunteers (31 males and 62 females) aged between 16 and 84 years and living in Barreiras, Bahia state. Some of the volunteers had previously been given three or four doses of monovalent vaccines based only on the original strain of the virus first identified in Wuhan, China, such as Coronavac (Butantan Institute/Sinovac), Covishield (Oxford/AstraZeneca), or those of Janssen and Pfizer.

Others were also given as an extra booster the bivalent vaccine containing components of the original strain as well as omicron subvariants BA.4 and BA.5.

Serum samples collected from the volunteers were submitted to antibody neutralization assays using different strains of SARS-CoV-2: the original strain from the start of the pandemic; omicron (BA.1), predominant in 2021; and omicron subvariants FE.1.2 and BQ.1.1, predominant in Brazil more recently.

The study was funded by FAPESP (projects20/052047,20/064091,20/089435,21/056611,22/119811 and23/019250), and by the Brazilian Ministry of Science, Technology and Innovation (MCTI).

The study showed that the bivalent vaccine administered as a booster reinforced the immune response and was more effective in neutralizing omicron and its subvariants than in volunteers given only four shots of a monovalent vaccine. However, its main focus was still the original strain that predominated at the start of the pandemic, and the resulting competition limited medium- to long-term immunity against more recent variants, which are now more important epidemiologically.

This was expected because immune memory is based on cells capable of recognizing fractions of the virus and is reinforced by the number of contacts with the contaminant. The immune system will naturally react more against what it already knows, and the participants given the bivalent vaccine had already taken three or four doses of a monovalent vaccine, saidJaime Henrique Amorim, last author of the article. Amorim is a professor at the Federal University of Western Bahia (UFOB) and a visiting researcher at the University of So Paulos Biomedical Sciences Institute (ICB-USP).

Model for the future

Controlling a virus with the high transmission capacity of SARS-CoV-2 requires equally high vaccine coverage, saidLus Carlos de Souza Ferreira,head of ICB-USPs Vaccine Development Laboratory and a co-author of the article. The results of the study show that bivalent vaccines are effective to achieve immunity against subvariants of omicron and that their administration has been fundamental to control novel variants.

According to the researchers, another conclusion to be drawn from the findings is that future planning of vaccination policy should take into account the fact that the immune response induced by existing vaccines is mainly to the original strain, which has ceased circulating since 2020, and vaccines should have their formulation adjusted so that they no longer include these components.

Forthcoming doses should be designed to combat the variants that are circulating now, instead of those that have disappeared, so that immunity is updated and reinforced in accordance with the current epidemiological situation, as it already is in the case of influenza vaccines, Amorim said.

The joint first authors of the article are Milena Silva Souza and Jssica Pires Farias, researchers at UFOB. The other co-authors are affiliated with institutions in Brazil and the United States.

About So Paulo Research Foundation (FAPESP)

The So Paulo Research Foundation (FAPESP) is a public institution with the mission of supporting scientific research in all fields of knowledge by awarding scholarships, fellowships and grants to investigators linked with higher education and research institutions in the State of So Paulo, Brazil. FAPESP is aware that the very best research can only be done by working with the best researchers internationally. Therefore, it has established partnerships with funding agencies, higher education, private companies, and research organizations in other countries known for the quality of their research and has been encouraging scientists funded by its grants to further develop their international collaboration. You can learn more about FAPESP atwww.fapesp.br/enand visit FAPESP news agency atwww.agencia.fapesp.br/ento keep updated with the latest scientific breakthroughs FAPESP helps achieve through its many programs, awards and research centers. You may also subscribe to FAPESP news agency athttp://agencia.fapesp.br/subscribe.

Journal of Medical Virology

Neutralizing antibody response after immunization with a COVID-19 bivalent vaccine: Insights to the future

29-Jan-2024

Disclaimer: AAAS and EurekAlert! are not responsible for the accuracy of news releases posted to EurekAlert! by contributing institutions or for the use of any information through the EurekAlert system.

For years, researchers have been working on vaccines that aim to prevent viral infections by strengthening immune defenses at viruses doorway to the body: the nose.

A small study recently published in PNAS presents a similar, if lower-tech, idea. Coating the inside of the nose with the over-the-counter antibiotic ointment Neosporin seems to trigger an immune response that may help the body repel respiratory viruses like those that cause COVID-19 and the flu, the study suggests.

The research raises the idea that Neosporin could serve as an extra layer of protection against respiratory illnesses, on top of existing tools like vaccines and masks, says study co-author Akiko Iwasaki, an immunobiologist at the Yale School of Medicine and one of the U.S. leading nasal vaccine researchers.

The study builds upon some of Iwasakis prior researchwhich has shown that similar antibiotics can trigger potentially protective immune changes in the bodybut its still preliminary, she cautions. For the new study, her team had 12 people apply Neosporin inside their nostrils twice a day for a week, while another seven people used Vaseline for comparison. At several points during the study, the researchers swabbed the participants noses and ran PCR tests to see what was going on inside.

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They found that Neosporinand specifically one of its active ingredients, the antibiotic neomycin sulfateseems to stimulate receptors in the nose that are fooled into thinking theres a viral infection and in turn create a barrier thats put up against any virus, Iwasaki explains. In theory, she says, that means it could protect against a range of different infections.

Right now, though, thats just a theory. For this study, Iwasakis team didnt take the next step of testing whether that immune response actually prevents people from getting infected when theyre exposed to virusesin part because its ethically questionable to intentionally expose people to pathogens for research. (They did, however, demonstrate that rodents whose noses were coated with neomycin were protected from the virus that causes COVID-19.)

On its website, the maker of Neosporin says that the product has not "been tested or formulated to prevent against COVID-19 or any other virus," and also note that they do not advise putting the product inside the eyes, nose, or mouth.

Dr. James Crowe, who directs the Vanderbilt Vaccine Center and was not involved in the research, says the study is intriguing, but hed need to see more human data before he gets excited. Im skeptical it would be strongly effective in people, Crowe says. If you have a modest effect on the virus, is that enough to really benefit you clinically?

It is somewhat counterintuitive to think that an antibiotic, which kills bacteria, could do anything to protect people from viruses. Its not that the antibiotic has a direct effect against viruses, Iwasaki explains. Instead, it seems that neomycin, when applied topically, provoke changes in the body that help it fight off virusesessentially, triggering a natural antiviral effect.

So should you smear Neosporin in your nose next time a COVID-19 wave hits? Not so fast, says Dr. Benjamin Bleier, who specializes in nasal disorders at Massachusetts Eye and Ear and has studied nasal immunity.

Read More: COVID-Cautious Americans Feel Abandoned

Bleier, who was not involved in the new study, calls the research very well done, but says there are questions that need to be answered before it hits clinical prime time. First, could the body develop tolerance or resistance to neomycin if the antibiotic were regularly used in this way? (Antibiotic resistance is a growing concern, and overusing or inappropriately prescribing antibiotics is a contributor to the problem.) Second, could the average person apply neomycin deeply and thoroughly enough for meaningful protection? And finally, could this approach damage the delicate inner nose or have other side effects over time? (Even in the small study, one of the people who used intranasal Neosporin dropped out due to minor side effects, apparently related to a drug allergy.)

Its great science, but theres still a long way to go before we should put it in our noses, agrees Dr. Sean Liu, an infectious disease physician at New Yorks Mount Sinai health system who was also not involved in the study.

Iwasaki agrees that more research is necessary. She says the next step is testing higher doses of neomycin, since Neosporin contains a fairly small amount that may not be enough to provide robust protection for humans. To gather more data, she says, researchers could track people going about their normal livesexcept that some apply neomycin to their noses and some apply Vaselineand see if one group gets sick less often than the other, though that would require a lot of time and people.

Despite the difficulties, Liu says theres good reason for further study. Finding new uses for affordable, widely accessible medications is good for public health, and any progress toward neutralizing viruses is welcome. If the approach is proven to work, it could also be useful to have a tool that's effective against a broad range of viruses and could potentially be paired with other drugs to strengthen its efficacy, Crowe adds.

Plus, Iwasaki says, additional disease-prevention tools could help people who are especially vulnerable to respiratory diseasessuch as those who are immunocompromisedand need additional protection to feel safe. If further research proves promising, Iwasaki says, she could imagine neomycin serving as an additional disease-fighting tool when people are in particularly germy places, like a crowded party or an airport.

Let us emphasize that the spectral matrix collocation approach based on the SCPSK may not yield convergence on a long time interval ([t_a,t_b]). One remedy is to use a large number of bases on the long domains accordingly to reach the desired level of accuracy. Another approach is to divide the given interval into a sequence of subintervals and employ the proposed collocation scheme on each subinterval consequently.

Towards this end, we split the time interval ([t_a,t_b]) into (Nge 1) subdomains in the forms

$$begin{aligned} K_n:=[t_{n},t_{n+1}],quad n=0,1,ldots , N-1. end{aligned}$$

Here, we have (t_0:=t_a) and (t_N:=t_b). The uniform time step is taken as (h=t_{n+1}-t_n=(t_b-t_a)/N). Note that by selecting (N=1), we turn back to the traditional spectral collocation method on the whole domain ([t_a,t_b]). Therefore, on each subinterval (K_n) we take the approximate solution of the modelEq. (1) to be in the formEq. (25) as

$$begin{aligned} x^n_{mathcal {J}}(t):=sum limits _{j=0}^{mathcal {J}} omega ^n_j,mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^n_mathcal {J},quad tin K_n, end{aligned}$$

(29)

where we utilized the notations

$$begin{aligned} varvec{W}_{mathcal {J}}^n:=left[ omega ^n_0quad omega ^n_1quad ldots quad omega ^n_{mathcal {J}}right] ^T,quad varvec{U}_{mathcal {J}}(t):=left[ mathbb {U}_0(t)quad mathbb {U}_1(t)quad ldots quad mathbb {U}_J(t)right] , end{aligned}$$

as the vector of unknown coefficients and the vector of SCPSK bases respectively. Once we get the all local approximate solutions for (n=0,1,ldots ,N-1), the global approximate solution on the given (large) interval ([t_a,t_b]) will be constructed in the form

$$begin{aligned} x_{mathcal {J}}(t)=sum limits _{n=0}^{N-1} c_n(t),x^n_{mathcal {J}}(t),quad c_n(t):= left{ begin{array}{ll} 0, &{} tnotin K_n,\ 1, &{} tin K_n.\ end{array}right. end{aligned}$$

In order to collocate a set of ((mathcal {J}+1)) linear equations to be obtained later at some suitable points, we consider the roots of (mathbb {U}_{mathcal {J}+1}(t)) on the subinterval (K_n). By modifying the points giveninEq. (17), we take the collocation nodes as

$$begin{aligned} t_{nu ,n}=frac{1}{2}left( t_n+t_{n+1}+h,cos left( frac{nu ,pi }{mathcal {J}+2}right) right) ,quad nu =1,2,ldots ,mathcal {J}+1. end{aligned}$$

(30)

At the end, we note that in the proposed splitting approach, the given initial conditions of the underlying model problem are prescribed on the first subinterval (K_0). Once the approximate solution on (K_0=[t_0,t_1]) is determined, we utilize it to assign the initial conditions on the next time interval (K_1). To do so, it is sufficient to evaluate the obtained approximation at (t_1). We repeat this idea on the next subintervals in order until we arrive at the last subinterval (K_{N-1}). Below, we illustrate the main steps of our matrix collocation algorithm on an arbitrary subinterval (K_n) for (n=0,1,ldots ,N-1).

Our chief aim is to solve the nonlinear COVID-19 systemEq. (1) efficiently by using the spectral method based on SCPSK basis. Towards this end, we first need to get rid of the nonlinearity of the model. This can be done by employing the Bellmans quasilinearization method (QLM)[39]. Thus we will get more advantages in terms of running time, especially for large values of J in comparison to the performance of directly applied collocation methods to nonlinear models, see cf.[40,41,42]. By combining the idea of QLM and the splitting of the domain we will obtain more gains in terms of accuracy for the approximate solutions of nonlinear modelEq. (1). Let us first describe the technique of QLM. For more information, we may refer the readers to the above-mentioned works.

By reformulating the original COVID-19 modelEq. (1) in a compact form we get

$$begin{aligned} frac{d}{dt} varvec{z}(t)=varvec{G}(t,varvec{z}(t)), end{aligned}$$

(31)

where

$$begin{aligned} varvec{z}(t)=left[ begin{array}{c} S(t)\ S_v(t)\ I(t)\ I_v(t)\ R(t)\ R_v(t)\ J(t)\ J_v(t) end{array}right] ,quad varvec{G}(t,varvec{z}(t))=left[ begin{array}{c} g_1(t)\ g_2(t)\ g_3(t)\ g_4(t)\ g_5(t)\ g_6(t)\ g_7(t)\ g_8(t) end{array}right] = left[ begin{array}{c} Lambda - beta S(I+I_v)- (lambda +mu ) S+ theta _1 R\ -beta ' S_v(I+ I_v)+ theta _2R_v+ lambda S- (delta +mu ) S_v\ beta S(I+ I_v)- (gamma _1+alpha _1+mu ) I \ beta ' S_v(I+ I_v)- (gamma _2+alpha _2+mu )I_v\ gamma _1 I-(theta _1+mu ) R+ eta _1 J \ gamma _2 I_v- (theta _2+mu ) R_v+ eta _2J_v+ delta S_v\ alpha _1 I- (eta _1+mu _1)J\ alpha _2I_v- (eta _2+mu _2) J_v end{array}right] . end{aligned}$$

To begin the QLM process, we assume (varvec{z}_0(t)) is available as an initial rough approximation for the solution (varvec{z}(t)) of the COVID-19 systemEq. (31). Through an iterative manner, the QLM procedure reads as follows

$$begin{aligned} frac{d}{dt}varvec{z}_{s}(t)approx varvec{G}(t,varvec{z}_{s-1}(t))+varvec{G}_{varvec{z}}(t,varvec{z}_{s-1}(t)),left( varvec{z}_{s}(t)-varvec{z}_{s-1}(t)right) ,quad s=1,2,ldots . end{aligned}$$

Here, the notation (varvec{G}_{varvec{z}}) stands for the Jacobian matrix of the COVID-19 systemEq. (31), which is of size 8 by 8. By performing some calculations we reach the linearized equivalent model form as

$$begin{aligned} frac{d}{dt}varvec{z}_{s}(t)+varvec{M}_{s-1}(t),varvec{z}_{s}(t)=varvec{r}_{s-1}(t),qquad s=1,2,ldots , end{aligned}$$

(32)

where (varvec{M}_{s-1}(t):=varvec{J}(S_{s-1}(t), (S_v)_{s-1}(t), I_{s-1}(t), (I_v)_{s-1}(t))) as the Jacobian matrix (varvec{J}) previously constructed inEq. (7). Also we have

$$begin{aligned} varvec{z}_{s}(t)= left[ begin{array}{c} S_{s-1}(t)\ (S_v)_{s-1}(t)\ I_{s-1}(t)\ (I_v)_{s-1}(t)\ R_{s-1}(t)\ (R_v)_{s-1}(t)\ J_{s-1}(t)\ (J_v)_{s-1}(t) end{array}right] ,quad varvec{r}_{s-1}(t)= left[ begin{array}{c} Lambda +beta ,S_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ beta ',(S_v)_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ -beta ,S_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ -beta ',(S_v)_{s-1}(t)Big (I_{s-1}(t)+(I_v)_{s-1}(t)Big )\ 0\ 0\ 0\ 0 end{array}right] . end{aligned}$$

Along with the systemEq. (32) the initial conditions

$$begin{aligned} varvec{z}_{s}(0)=left[ begin{array}{cccccccc} S_0&S_{v0}&I_0&I_{v0}&R_0&R_{v0}&J_0&J_{v0} end{array}right] ^T, end{aligned}$$

(33)

are given due toEq. (2). We now are able to solve the family of linearized initial-value problemsEqs. (32)-(33) numerically by our proposed matrix collocation method on an arbitrary (long) domain ([t_a,t_b]). For this purpose and for clarity of exposition, we restrict our illustrations to a local subinterval (K_n) for (n=0,1,ldots ,N-1).

In view ofEq. (29) by utilizing only ((mathcal {J}+1)) SCPSK basis functions, we assume that the eight solutions of systemEq. (32) can be represented in terms ofEq. (29). Thus, we take these solutions at iteration (sge 1) as

$$begin{aligned} left{ begin{array}{l} S^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,1},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},1},quad (S_v)^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,2},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},2},\ I^{n}_{mathcal {J},s}(t),=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,3},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},3},quad (I_v)^{n}_{mathcal {J},s}(t)~=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,4},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},4},\ R^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,5},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},5},quad (R_v)^{n}_{mathcal {J},s}(t)=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,6},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},6},\ J^{n}_{mathcal {J},s}(t),=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,7},mathbb {U}_j(t)=varvec{U}_{J}(t),varvec{W}^{n,s}_{mathcal {J},7},quad (J_v)^{n}_{mathcal {J},s}(t),=sum _{j=0}^{mathcal {J}}omega ^{n,s}_{j,8},mathbb {U}_j(t)=varvec{U}_{mathcal {J}}(t),varvec{W}^{n,s}_{mathcal {J},8},\ end{array}right. end{aligned}$$

(34)

for (tin K_n). Moreover, by (varvec{W}^{n,s}_{mathcal {J},i}= left[ begin{array}{cccc} omega ^{n,s}_{0,i}&omega ^{n,s}_{1,i}&dots&omega ^{n,s}_{mathcal {J},i} end{array}right] ^T) we denote the vectors of unknowns for (1le ile 8) at the iteration (sge 1). Also, the vector of SCPSK basis, i.e., (varvec{U}_mathcal {J}(t)) is defined inEq. (29). We next provide a decomposition for (varvec{U}_mathcal {J}(t)) given by

$$begin{aligned} varvec{U}_mathcal {J}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J}. end{aligned}$$

(35)

Here, the vector (varvec{Q}_mathcal {J}(t)) including the powers of ((t-t_n)) introduced by

$$begin{aligned} varvec{Q}_mathcal {J}(t)=left[ 1quad t-t_nquad (t-t_n)^{2}quad ldots quad (t-t_n)^{mathcal {J}}right] . end{aligned}$$

The next object is the matrix (varvec{F}_mathcal {J}=(f_{i,j})_{i,j=0}^{mathcal {J}}) of size ((mathcal {J}+1)times (mathcal {J}+1)). The entries of the latter matrix are given inEq. (15). One can also show that (det (varvec{F}_mathcal {J})ne 0) and it is a triangular matrix. It follows that

$$begin{aligned} f_{i,j}:= left{ begin{array}{ll} o_{i,j}, &{} textrm{if}~ ile j,\ 0, &{} textrm{if}~ i> j. end{array}right. end{aligned}$$

We then insert the obtained term (varvec{U}_mathcal {J}(t)) inEq. (35) intoEq. (34). The resulting expansions are

$$begin{aligned} left{ begin{array}{l} S^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},1},quad (S_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},2},\ I^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{F}_J,varvec{W}^{n,s}_{mathcal {J},3},quad (I_v)^{n}_{mathcal {J},s}(t)~=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},4},\ R^{n}_{mathcal {J},s}(t) =varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},5},quad (R_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},6},\ J^{n}_{mathcal {J},s}(t), =varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},7},quad (J_v)^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},8}, end{array}right. tin K_n. end{aligned}$$

(36)

We then proceed by nothing that the derivative of the vector (varvec{Q}_mathcal {J}(t)) can be stated in terms of itself. A vivid calculation reveals that

$$begin{aligned} dot{varvec{Q}}_{mathcal {J}}(t)=varvec{Q}_{mathcal {J}}(t),varvec{D}_mathcal {J},quad varvec{D}_mathcal {J}=left[ begin{array}{lllll} 0 &{} 1 &{} 0 &{}ldots &{} 0\ 0 &{} 0 &{} 2 &{}ldots &{} 0\ vdots &{} vdots &{} ddots &{}vdots &{} vdots \ 0 &{} 0 &{} 0 &{}ddots &{} mathcal {J}\ 0 &{} 0 &{} 0 &{} ldots &{} 0 end{array}right] _{(mathcal {J}+1)times (mathcal {J}+1)}. end{aligned}$$

(37)

From this relation, we are able to derive a matrix forms of the derivatives of the unknown solutions inEq. (36).

$$begin{aligned} left{ begin{array}{l} dot{S}^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},1},quad (dot{S}_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},2},\ dot{I}^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},3},quad (dot{I}_v)^{n}_{mathcal {J},s}(t)~=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},4},\ dot{R}^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},5},quad (dot{R}_v)^{n}_{mathcal {J},s}(t)=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},6},\ dot{J}^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},7},quad (dot{J}_v)^{n}_{mathcal {J},s}(t),=varvec{Q}_mathcal {J}(t),varvec{D}_mathcal {J},varvec{F}_mathcal {J},varvec{W}^{n,s}_{mathcal {J},8}, end{array}right. tin K_n. end{aligned}$$

(38)

The exact solutions of the linearized systemEq. (32) can be written in a vectorized form as

$$begin{aligned} varvec{z}_s(t)approx varvec{z}^n_{mathcal {J},s}(t):= left[ begin{array}{l} S^{n}_{mathcal {J},s}(t)\ (S_v)^{n}_{mathcal {J},s}(t)\ I^{n}_{mathcal {J},s}(t)\ (I_v)^{n}_{mathcal {J},s}(t)\ R^{n}_{mathcal {J},s}(t)\ (R_v)^{n}_{mathcal {J},s}(t)\ J^{n}_{mathcal {J},s}(t)\ (J_v)^{n}_{mathcal {J},s}(t) end{array}right] ,quad dot{varvec{z}}_s(t)approx frac{d}{dt}varvec{z}^n_{mathcal {J},s}(t):= left[ begin{array}{l} dot{S}^{n}_{mathcal {J},s}(t)\ (dot{S}_v)^{n}_{mathcal {J},s}(t)\ dot{I}^{n}_{mathcal {J},s}(t)\ (dot{I}_v)^{n}_{mathcal {J},s}(t)\ dot{R}^{n}_{mathcal {J},s}(t)\ (dot{R}_v)^{n}_{mathcal {J},s}(t)\ dot{J}^{n}_{mathcal {J},s}(t)\ (dot{J}_v)^{n}_{mathcal {J},s}(t) end{array}right] . end{aligned}$$

(39)

We next introduce the following block diagonal matrices of dimensions (8(mathcal {J}+1)times 8(mathcal {J}+1)) as

$$begin{aligned} widehat{varvec{Q}}(t){} & {} =mathrm {{textbf {Diag}}} left( begin{array}{cccccccc} varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t)&varvec{Q}_mathcal {J}(t) end{array}right) ,\ widehat{varvec{D}}{} & {} =mathrm {{textbf {Diag}}} left( begin{array}{cccccccc} varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J}&varvec{D}_mathcal {J} end{array}right) ,\ widehat{varvec{F}}{} & {} =mathrm {{textbf {Diag}}} left( begin{array}{cccccccc} varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J}&varvec{F}_mathcal {J} end{array}right) . end{aligned}$$

By the aid of the former definitions, the matrix formats of (varvec{z}^n_{mathcal {J},s}(t)) and (dot{varvec{z}}^n_{mathcal {J},s}(t)) will rewrite concisely as

$$begin{aligned} varvec{z}^n_{mathcal {J},s}(t)=widehat{varvec{Q}}(t),widehat{varvec{F}},varvec{W}^n,quad dot{varvec{z}}^n_{mathcal {J},s}(t)=widehat{varvec{Q}}(t),widehat{varvec{F}},widehat{varvec{D}},varvec{W}^n. end{aligned}$$

(40)

Here, (varvec{W}^n) is the successive vector of eight previously defined vector of unknowns

$$begin{aligned} varvec{W}^n=left[ begin{array}{cccc} varvec{W}^{n,s}_{mathcal {J},1}&varvec{W}^{n,s}_{mathcal {J},2}&ldots&varvec{W}^{n,s}_{mathcal {J},8} end{array}right] ^T. end{aligned}$$

We now can collocate the linearized Eq.(32) at the zeros of SCPSK given inEq. (17) on the subdomain (K_n). We get

$$begin{aligned} frac{d}{dt}varvec{z}_{s}(t_{nu ,n})+varvec{M}_{s-1}(t_{nu ,n}),varvec{z}_{s}(t_{nu ,n})=varvec{r}_{s-1}(t_{nu ,n}),qquad nu =1,2,ldots ,mathcal {J}, end{aligned}$$

(41)

for (s=1,2,ldots). Denote the coefficient matrix by (widehat{varvec{M}}^n_{s-1}) and the right-hand-side vector as (widehat{varvec{R}}^n_{s-1}). These are defined by

$$begin{aligned} widehat{varvec{M}}^n_{s-1}= left[ begin{array}{cccc} varvec{M}_{s-1}(t_{0,n})&{}textbf{0}&{}ldots &{}textbf{0}\ textbf{0}&{}varvec{M}_{s-1}(t_{1,n})&{}ldots &{}textbf{0}\ vdots &{}vdots &{}ddots &{}vdots \ textbf{0}&{}textbf{0}&{}ldots &{}varvec{M}_{s-1}(t_{mathcal {J},n}) end{array}right] ,quad widehat{varvec{R}}^n_{s-1}= left[ begin{array}{c} varvec{r}_{s-1}(t_{0,n})\ varvec{r}_{s-1}(t_{1,n})\ vdots \ varvec{r}_{s-1}(t_{mathcal {J},n}) end{array}right] . end{aligned}$$

Let us define further the vectors of unknowns as

$$begin{aligned} dot{varvec{Z}}^n_s= left[ begin{array}{c} dot{varvec{z}}_{s}(t_{0,n})\ dot{varvec{z}}_{s}(t_{1,n})\ vdots \ dot{varvec{z}}_{s}(t_{mathcal {J},n}) end{array}right] ,quad varvec{Z}^n_s= left[ begin{array}{c} dot{varvec{z}}_{s}(t_{0,n})\ dot{varvec{z}}_{s}(t_{1,n})\ vdots \ dot{varvec{z}}_{s}(t_{mathcal {J},n}) end{array}right] . end{aligned}$$

Consequently, the system of Eq.(41) can be stated briefly as

$$begin{aligned} dot{varvec{Z}}^{n}_{s}+widehat{varvec{M}}^n_{s-1},varvec{Z}^n_s=widehat{varvec{R}}^n_{s-1},quad n=0,1,ldots ,N-1, end{aligned}$$

(42)

and with (s=1,2,ldots). Before we talk about the fundamental matrix equation, we need to state two vectors (varvec{Z}^n_s) and (dot{varvec{Z}}^{n}_{s})inEq. (42) in the matrix representation forms. The proof is easy by just considering the definitions of the involved matrices and vectors inEq. (40).

If two vectors (varvec{z}^n_{mathcal {J},s}(t)) and (dot{varvec{z}}^n_{mathcal {J},s}(t)) inEq. (40) computed at the collocation pointsEq. (30), we arrive at the next matrix forms

$$begin{aligned} varvec{Z}^n_s=bar{widehat{varvec{Q}}},widehat{varvec{F}},varvec{W}^n,qquad dot{varvec{Z}}^n_s=bar{widehat{varvec{Q}}},widehat{varvec{F}},widehat{varvec{D}},varvec{W}^n, end{aligned}$$

(43)

where the matrix (bar{widehat{varvec{Q}}}) is given by

$$begin{aligned} bar{widehat{varvec{Q}}}=[widehat{varvec{Q}}(t_{0,n})quad widehat{varvec{Q}}(t_{1,n})quad ldots quad widehat{varvec{Q}}(t_{mathcal {J},n}) ]^T. end{aligned}$$

Moreover, two matrices (widehat{varvec{Q}}, widehat{varvec{F}}) are defined inEq. (40). Similarly, the vector (varvec{W}^n) is given inEq. (40).

By turning to relationEq. (40) we substitute the derived matrix formats into it. Precisely speaking, after replacing (varvec{Z}^n_s) and (dot{varvec{Z}}^n_s) we gain the so-called fundamental matrix equation (FME) of the form

$$begin{aligned} varvec{B}_n,varvec{W}^n=widehat{varvec{R}}^n_{s-1}, quad textrm{or}quad left[ varvec{B}_n;widehat{varvec{R}}^n_{s-1}right] ,quad sge 1,~0le nle N-1, end{aligned}$$

(44)

where

$$begin{aligned} varvec{B}_n:=bar{widehat{varvec{Q}}},widehat{varvec{F}}+widehat{varvec{M}}^n_{s-1},bar{widehat{varvec{Q}}},widehat{varvec{F}},widehat{varvec{D}}. end{aligned}$$

To complete the process of QLM-SCPSK approach, it is necessary to implement the initial conditionsinEq. (2) and add them intoEq. (44). So, the next task is to constitute the matrix representation ofEq. (2). Let us approach (trightarrow 0) in the first relation ofEq. (40). It gives us

$$begin{aligned} varvec{B}_{0,n},varvec{W}^n=widehat{varvec{R}}^n_{s-1,0},qquad varvec{B}_{0,n}:=widehat{varvec{Q}}(0),widehat{varvec{F}},quad widehat{varvec{R}}^n_{s-1,0}=left[ begin{array}{cccccccc} S_0&S_{v0}&I_0&I_{v0}&R_0&R_{v0}&J_0&J_{v0} end{array}right] ^T. end{aligned}$$

We then replace eight rows of the augmented matrix ([varvec{B}_n;widehat{varvec{R}}^n_{s-1}]) by the already obtained row matrix ([varvec{B}_{0,n};widehat{varvec{R}}^n_{s-1,0}]). Denote the modified FME by

$$begin{aligned} check{varvec{B}_{n}},varvec{W}^n=check{textbf{R}}^n_{s-1},quad textrm{or} quad left[ check{varvec{B}_{n}};check{textbf{R}}^n_{s-1}right] . end{aligned}$$

(45)

This implies that the solution of the modelEq. (1) is obtainable on each subdomain (K_n) by iterating (n=0,1,ldots ,N-1). On (K_0) as the first subdomain, the given initial conditionsinEq. (2) will be used to find the corresponding approximations for the systemEq. (1). Hence, this approximate solutions on (K_0) evaluated at the starting point of (K_1) will be utilized for the initial conditions on (K_1). By repeating this process we acquire all approximations on all (K_n) for (0le nle N-1).

Generally, finding the true solutions of the COVID-19 systemEq. (1) is not possible practically. In this case, the residual error functions (REFs) help us to measure the quality of approximations obtained by the QLM-SCPSK technique. Once we calculate the eight approximations by the illustrated method, we substitute them into the model systemEq. (1). In fact, the REFs are defined as the difference between the left-hand side and the right-hand side of the considered equation. On the subdomain (K_n) we set the REFs as

$$begin{aligned}{} & {} mathbb {R}_{1,mathcal {J}}^{n}(t):=left| dot{S}^{n}_{mathcal {J},s}(t)-Lambda +beta S^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+(lambda +mu ) S^{n}_{mathcal {J},s}(t)- theta _1 R^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{2,mathcal {J}}^{n}(t):=left| (dot{S_v})^{n}_{mathcal {J},s}(t)+beta ' (S_v)^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)- theta _2(R_v)^{n}_{mathcal {J},s}(t)- lambda S^{n}_{mathcal {J},s}(t)+ (delta +mu ) (S_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{3,mathcal {J}}^{n}(t):=left| dot{I}^{n}_{mathcal {J},s}(t)-beta S^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+ (gamma _1+alpha _1+mu ) I^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{4,mathcal {J}}^{n}(t):=left| (dot{I_v})^{n}_{mathcal {J},s}(t)- beta ' (S_v)^{n}_{mathcal {J},s}(t)L^n_{mathcal {J},s}(t)+ (gamma _2+alpha _2+mu )(I_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{5,mathcal {J}}^{n}(t):=left| dot{R}^{n}_{mathcal {J},s}(t) - gamma _1 I^{n}_{mathcal {J},s}(t)+(theta _1+mu ) R^{n}_{mathcal {J},s}(t)- eta _1 J^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{6,mathcal {J}}^{n}(t):=left| (dot{R_v})^{n}_{mathcal {J},s}(t)- gamma _2 (I_v)^{n}_{mathcal {J},s}(t)+ (theta _2+mu ) (R_v)^{n}_{mathcal {J},s}(t)- eta _2(J_v)^{n}_{mathcal {J},s}(t)- delta (S_v)^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{7,mathcal {J}}^{n}(t):=left| dot{J}^{n}_{mathcal {J},s}(t) -alpha _1 I^{n}_{mathcal {J},s}(t)+ (eta _1+mu _1)J^{n}_{mathcal {J},s}(t)right| cong 0, nonumber \{} & {} mathbb {R}_{8,mathcal {J}}^{n}(t):=left| (dot{J_v})^{n}_{mathcal {J},s}(t)- alpha _2(I_v)^{n}_{mathcal {J},s}(t) +(eta _2+mu _2) (J_v)^{n}_{mathcal {J},s}(t)right| cong 0, end{aligned}$$

(46)

for a fixed iteration number s and we have defined (L^n_{mathcal {J},s}:=I^{n}_{mathcal {J},s}(t)+ (I_v)^{n}_{mathcal {J},s}(t)) for brevity.

Analogously, at the fixed iteration s, the numerical order of convergence associated with the obtained REFs can be defined in the infinity norm. These are given by

$$begin{aligned} L^{infty }_{ell }equiv L^{infty }_{ell }(mathcal {J}):=max _{0le nle N-1}left( max _{tin K_n},left| mathbb {R}_{ell ,mathcal {J}}^{n}(t)right| right) ,quad ell =1,2,ldots ,8. end{aligned}$$

Therefore, the convergence order (Co) for each solution is defined by

$$begin{aligned} textrm{Co}_{mathcal {J}}^{ell }:=log _2left( frac{L^{infty }_{ell }(mathcal {J})}{L^{infty }_{ell }(2mathcal {J})}right) ,quad ell =1,2,ldots ,8. end{aligned}$$

(47)

In a recent study published in the journal JAMA Network Open, researchers investigated whether the coronavirus disease 2019 (COVID-19) vaccine for adolescents between the ages of 12 and 15 years, which was approved in May 2021, was associated with changes in the incidence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and related hospitalizations among the pediatric population in California, United States (U.S.).

Study:COVID-19 Vaccination and Incidence of Pediatric SARS-CoV-2 Infection and Hospitalization. Image Credit:Prostock-studio/Shutterstock.com

The spread and severity of the COVID-19 pandemic have been successfully controlled due to the rapid development of vaccines against SARS-CoV-2 and concerted efforts worldwide to vaccinate adult and at-risk populations.

The messenger ribonucleic acid (mRNA) vaccines against SARS-CoV-2, developed largely by Moderna and Pfizer BioNTech, were widely used for the adult populations.

Since children and adolescents were not found to be at a high risk of severe COVID-19, developing vaccines for the younger populations was of secondary priority during the peak periods of the pandemic.

However, in May 2021, the first mRNA COVID-19 vaccine for adolescents between 12 and 15 years was approved. In the subsequent months, vaccines for children between the ages of five and 11 years and six months and five years were also approved.

Although these vaccines are safe, vaccine hesitancy because of parental concerns about safety and adverse effects, and perceptions of reduced severity of the infection among children have resulted in low vaccine uptake among the younger populations.

In the present study, the researchers examined whether the COVID-19 vaccine for adolescents impacted the incidence of SARS-CoV-2 infection and hospitalizations among the pediatric population in California.

A better understanding of the impact of the vaccine in lowering incidence rates, reducing the severity of the disease, and mitigating the need for hospitalization is essential in formulating future public health policies on booster doses and developing vaccines against emerging SARS-CoV-2 variants.

The researchers analyzed deidentified data for close to four million pediatric COVID-19 cases and over 12,000 hospitalizations from California.

The outcomes associated with COVID-19 vaccination, including the incidence of SARS-CoV-2 infections and hospitalizations, were analyzed for each county and state according to the vaccine introduction phases for the three age groups.

The deidentified data contained age, county of residence, and hospitalization status information. A polymerase chain reaction (PCR) test was required to confirm COVID-19.

For the statistical analyses, the researchers grouped the cases based on county of residence, as well as age groups according to vaccine eligibility.

Furthermore, the data for each age group was also divided into periods of vaccine ineligibility and eligibility, and the outcomes were evaluated from a month after the vaccination until the analysis of the data or the beginning of the vaccine eligibility period for the next age group.

The results suggested that the COVID-19 vaccine effectively limited the transmission of SARS-CoV-2 among the pediatric population in California.

The analysis found that close to 380,000 COVID-19 cases and 273 hospitalizations among children between the ages of six months and 15 years were averted in four to seven months after the availability of the vaccine. These numbers represent 26% of the cases in the pediatric population.

The researchers stated that their results among the pediatric population were similar to those from various U.S. and Israeli studies reporting the effectiveness of the COVID-19 vaccine in averting a substantial number of COVID-19 cases among the adult population.

The positive impact of the COVID-19 vaccine was found to be the highest among children between the ages of 12 and 15 years.

Among children ages six months to five years, the reduction in the number of COVID-19 cases was not found to be significant. However, the researchers believe this could be because of low transmission rates of the variant circulating during the evaluation period for that age group.

Notably, despite the vaccination coverage being just above half (53.5%) among adolescents between the ages of 12 and 15 years and even lower among children below 12, a total of 376,085 cases of COVID-19 were averted in California.

These findings highlight the effectiveness of the COVID-19 vaccine in lowering the incidence and severity of SARS-CoV-2 infections and limiting the transmission of the virus among children and adolescents.

To conclude, the study found that despite just over 50% vaccination coverage, the COVID-19 mRNA vaccine approved for use among adolescents and children in the U.S. averted close to 400,000 cases among the pediatric population.

These results highlighted the importance of the COVID-19 vaccine in protecting individuals of all age groups against SARS-CoV-2 infections. Furthermore, these findings also support future public health decisions to administer booster doses.

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National Eye Institute / Wikimedia Commons

The incidence of uveitis in the year after COVID-19 was 17% among nearly 474,000 Korean adults with a history of the inflammatory eye condition, according to areport in JAMA Ophthalmology.

Researchers at the Hanyang University College of Medicine in Seoul mined theKorean National Health Insurance Service and Korea Disease Control and Prevention Agency databases for information on473,934 patients diagnosed as having uveitis from January 2015 to February 2021.

The patients had previously had uveitis and had received at least one dose of an mRNA (Pfizer/BioNTech or Moderna) or adenovirus vectorbased (AstraZeneca or Johnson & Johnson) COVID-19 vaccine. The average patient age was 58.9 years, 51.3% were women, and none tested positive for COVID-19 during the study period.

Uveitisis a potentially serious inflammation of the eye's middle layer of tissue that can cause symptoms such as pain, redness, and blurry vision.

The incidence of uveitis was 8.6% at 3 months, 12.5% at 6 months, and 16.8% at 1 year. The odds of uveitis were increased among recipients of all four vaccines, including Pfizer (hazard ratio [HR], 1.68), Moderna (HR, 1.51),AstraZeneca (HR, 1.60), andJohnson & Johnson(HR, 2.07). The risk was highest in the first 30 days after vaccination and peaked between the first and second doses (HR, 1.64).

These results emphasize the importance of vigilance and monitoring for uveitis in the context of vaccinations, including COVID-19 vaccinations, particularly in individuals with a history of uveitis.

"Although uveitis following vaccination is rare, our findings support an increased risk after COVID-19 vaccination, particularly in the early postvaccination period," the study authors wrote. "These results emphasize the importance of vigilance and monitoring for uveitis in the context of vaccinations, including COVID-19 vaccinations, particularly in individuals with a history of uveitis."

In a relatedcommentary, Anika Kumar and Nisha Acharya, MD, said it's important to weigh the risk of uveitis with that of remaining unvaccinated against COVID-19. "Indeed, other investigations of postvaccine NIU [noninfectious uveitis] that similarly identified increased risks of NIU after vaccination noted that effect sizes were small and attributable risks were low; thus, the findings should not preclude individuals from receiving a vaccination," they wrote.

Following the availability of COVID-19 vaccines, there was a noticeable surge in negativity surrounding vaccines on Twitter. Researchers at the ESCMID Global Congress (formerly ECCMID) in Barcelona, Spain (held from 27th to 30th April) made this observation.

The analysis revealed that spikes in negative tweets coincided with announcements from governments and healthcare authorities regarding vaccination. To address this issue, the researchers propose adopting a fresh approach to discussing vaccinesone that avoids using the term anti-vaxxers.

Dr. Guillermo Rodriguez-Nava, lead researcher from Stanford University School of Medicine, Stanford, USA, emphasized the significance of vaccines as one of humanitys greatest achievements. Vaccines have the potential to eradicate dangerous diseases like smallpox, prevent deaths from illnesses with 100% mortality rates (such as rabies), and even protect against cancers caused by HPV.

Despite these benefits, opposition to vaccine use has grown in recent years. Negative voices have already had consequences, with measles re-emerging in countries where it was once considered eradicated. This situation not only affects children who cannot decide for themselves but also impacts immunocompromised patients who cannot receive vaccinations.

Dr Rodriguez-Nava and colleagues analyzed COVID-19 vaccine-related posts on Twitter. They used open-source software (the Snscrape library in Python) to download tweets with the hashtag vaccine from 1 January 2018 to 31 December 2022. Cutting-edge AI methods were then employed for sentiment analysis, classifying tweets as either positive or negative. Additionally, they created a counterfactual scenario to understand how tweet patterns would have looked if COVID-19 vaccines hadnt been introduced in December 2020.

The results showed that both before and after vaccine availability, negative sentiment tweets dominated. For instance, one negative tweet read: The EU Commission should immediately terminate contracts for new doses of fake #vaccines against #COVID19 and demand the return of the 2.5bn paid so far. Everyone who lied that #vaccines prevent the spread of the virus must be held accountable.

In contrast, positive tweets celebrated vaccination milestones. For example: Two-month shots! #vaccines are always a reason to celebrate in our house. #VaccinesWork.

Since the introduction of COVID-19 vaccines, the number of vaccine-related tweets has increased significantly10,201 more per month on average than expected if vaccination hadnt started. Negativity also rose, with approximately 12,420 negative sentiment tweets per month after 11 December 202027% more than expected without vaccination.

The proportion of positive tweets decreased slightly (from 20.3% to 18.8%), while negative tweets increased (from 79.6% to 81.1%) after COVID vaccine introduction.

Notably, negative activity spiked during vaccination announcements. For instance, April 2021 saw the highest number of negative tweetsthe same month the White House announced COVID-19 vaccine eligibility for all people aged 16 and older.

Interestingly, the lowest number of negative tweets occurred in April 2022, the month Elon Musk acquired Twitter. While the exact reason is unknown, it may be related to seasonal patterns (higher negativity during winter) or users focusing on platform changes under new ownership.

In summary, negative sentiments about vaccines were already prevalent on social media before COVID-19 vaccines arrived. Their introduction significantly amplified negative sentiments on X (formerly Twitter) regarding vaccines.