Analysing vaccine efficacy evaluated in phase 3 clinical trials carried out during outbreaks.

In this paper we examine several definitions of vaccine efficacy (VE) that we found in the literature, for diseases that express themselves in outbreaks, that is, when the force of infection grows in time, reaches a maximum and then vanishes. The fact that the disease occurs in outbreaks results in several problems that we analyse. We propose a mathematical model that allows the calculation of VE for several scenarios. Vaccine trials usually needs a large number of volunteers that must be enrolled. Ideally, all volunteers should be enrolled in approximately the same time, but this is generally impossible for logistic reasons and they are enrolled in a fashion that can be replaced by a continuous density function (for example, a Gaussian function). The outbreak can also be replaced by a continuous density function, and the use of these density functions simplifies the calculations. Assuming, for example Gaussian functions, one of the problems one can immediately notice is that the peak of the two curves do not occur at the same time. The model allows us to conclude: First, the calculated vaccine efficacy decreases when the force of infection increases; Second, the calculated vaccine efficacy decreases when the gap between the peak in the force of infection and the peak in the enrollment rate increases; Third, different trial protocols can be simulated with this model; different vaccine efficacy definitions can be calculated and in our simulations, all result are approximately the same. The final, and perhaps most important conclusion of our model, is that vaccine efficacy calculated during outbreaks must be carefully examined and the best way we can suggest to overcome this problem is to stratify the enrolled volunteer's in a cohort-by-cohort basis and do the survival analysis for each cohort, or apply the Cox proportional hazards model for each cohort.

Analysing vaccine efficacy evaluated in phase 3 clinical trials carried out during outbreaks

In this paper we examine several definitions of vaccine efficacy (VE) that we found in the literature, for diseases that express themselves in outbreaks, that is, when the force of infection grows in time, reaches a maximum and then vanishes. The fact that the disease occurs in outbreaks results in several problems that we analyse. We propose a mathematical model that allows the calculation of VE for several scenarios. Vaccine trials usually needs a large number of volunteers that must be enrolled. Ideally, all volunteers should be enrolled in approximately the same time, but this is generally impossible for logistic reasons and they are enrolled in a fashion that can be replaced by a continuous density function (for example, a Gaussian function). The outbreak can also be replaced by a continuous density function, and the use of these density functions simplifies the calculations. Assuming, for example Gaussian functions, one of the problems one can immediately notice is that the peak of the two curves do not occur at the same time. The model allows us to conclude: First, the calculated vaccine efficacy decreases when the force of infection increases; Second, the calculated vaccine efficacy decreases when the gap between the peak in the force of infection and the peak in the enrollment rate increases; Third, different trial protocols can be simulated with this model; different vaccine efficacy definitions can be calculated and in our simulations, all result are approximately the same. The final, and perhaps most important conclusion of our model, is that vaccine efficacy calculated during outbreaks must be carefully examined and the best way we can suggest to overcome this problem is to stratify the enrolled volunteer's in a cohort-by-cohort basis and do the survival analysis for each cohort, or apply the Cox proportional hazards model for each cohort

Time-dependent vaccine efficacy estimation quantified by a mathematical model

In this paper we calculate the variation of the estimated vaccine efficacy (VE) due to the time-dependent force of infection resulting from the difference between the moment the Clinical Trial (CT) begins and the peak in the outbreak intensity. Using a simple mathematical model we tested the hypothesis that the time difference between the moment the CT begins and the peak in the outbreak intensity determines substantially different values for VE. We exemplify the method with the case of the VE efficacy estimation for one of the vaccines against the new coronavirus SARS-CoV-2.

Modelling the impact of delaying vaccination against SARS-CoV-2 assuming unlimited vaccine supply.

BACKGROUND: At the moment we have more than 177 million cases and 3.8 million deaths (as of June 2021) around the world and vaccination represents the only hope to control the pandemic. Imperfections in planning vaccine acquisition and difficulties in implementing distribution among the population, however, have hampered the control of the virus so far. METHODS: We propose a new mathematical model to estimate the impact of vaccination delay against the 2019 coronavirus disease (COVID-19) on the number of cases and deaths due to the disease in Brazil. We apply the model to Brazil as a whole and to the State of Sao Paulo, the most affected by COVID-19 in Brazil. We simulated the model for the populations of the State of Sao Paulo and Brazil as a whole, varying the scenarios related to vaccine efficacy and compliance from the populations. RESULTS: The model projects that, in the absence of vaccination, almost 170 thousand deaths and more than 350 thousand deaths will occur by the end of 2021 for Sao Paulo and Brazil, respectively. If in contrast, Sao Paulo and Brazil had enough vaccine supply and so started a vaccination campaign in January with the maximum vaccination rate, compliance and efficacy, they could have averted more than 112 thousand deaths and 127 thousand deaths, respectively. In addition, for each month of delay the number of deaths increases monotonically in a logarithmic fashion, for both the State of Sao Paulo and Brazil as a whole. CONCLUSIONS: Our model shows that the current delay in the vaccination schedules that is observed in many countries has serious consequences in terms of mortality by the disease and should serve as an alert to health authorities to speed the process up such that the highest number of people to be immunized is reached in the shortest period of time.

Estimating the effects of reopening of schools on the course of the epidemic of COVID-19.

In this paper, we present a method to estimate the risk of reopening of schools illustrated with the case of the State of São Paulo, Brazil. The model showed that, although no death of children would result from the reopening of the schools in the three cities analysed, the risk of asymptomatic and symptomatic cases and secondary cases among teachers, school staff and relatives of the children is not negligible. Although the epidemic hit different regions with different intensities, our model shows that, for regions where the incidence profile is similar to the cities analysed, the risk of reopening of schools is still too high. This in spite of the fact that incidences in these cities were declining in the period of the time considered. Therefore, although we cannot extend the result to the entire country, the overall conclusion is valid for regions with a declining incidence and it is even more valid for regions where incidence is increasing. We assumed a very conservative level of infection transmissibility of children of just 10% as that of adults. In spite of the very low level of transmissibility is assumed, the number of secondary cases caused by infected children among teachers, school staff and relatives varied from 2 to 85. It is, therefore, too soon to have any degree of confidence that reopening of schools before the advent of a vaccine is the right decision to take. The purpose of our model and simulations is to provide a method to estimate the risk of school reopening, although we are sure it could be applied as a guide to public health strategies.

Modelling the impact of contact tracing of symptomatic individuals on the COVID-19 epidemic.

OBJECTIVES: With the declining numbers of coronavirus disease 2019 (COVID-19) cases in the state of São Paulo, Brazil, social distancing measures have gradually been lifted. However, the risk of a surge in the number of cases cannot be overlooked. Even with the adoption of nonpharmaceutical interventions, such as restrictions on mass gatherings, wearing of masks, and complete or partial closure of schools, other public health measures may help control the epidemic. We aimed to evaluate the impact of the contact tracing of symptomatic individuals on the COVID-19 epidemic regardless of the use of diagnostic testing. METHODS: We developed a mathematical model that includes isolation of symptomatic individuals and tracing of contacts to assess the effects of the contact tracing of symptomatic individuals on the COVID-19 epidemic in the state of São Paulo. RESULTS: For a selection efficacy (proportion of isolated contacts who are infected) of 80%, cases and deaths may be reduced by 80% after 60 days when 5000 symptomatic individuals are isolated per day, each of them together with 10 contacts. On the other hand, for a selection efficacy of 20%, the number of cases and deaths may be reduced by approximately 40% and 50%, respectively, compared with the scenario in which no contact-tracing strategy is implemented. CONCLUSION: Contact tracing of symptomatic individuals may potentially be an alternative strategy when the number of diagnostic tests available is not sufficient for massive testing.

Modelling the impact of delaying vaccination against SARS-CoV-2 assuming unlimited vaccine supply

Background At the moment we have more than 177 million cases and 3.8 million deaths (as of June 2021) around the world and vaccination represents the only hope to control the pandemic. Imperfections in planning vaccine acquisition and difficulties in implementing distribution among the population, however, have hampered the control of the virus so far. Methods We propose a new mathematical model to estimate the impact of vaccination delay against the 2019 coronavirus disease (COVID-19) on the number of cases and deaths due to the disease in Brazil. We apply the model to Brazil as a whole and to the State of Sao Paulo, the most affected by COVID-19 in Brazil. We simulated the model for the populations of the State of Sao Paulo and Brazil as a whole, varying the scenarios related to vaccine efficacy and compliance from the populations. Results The model projects that, in the absence of vaccination, almost 170 thousand deaths and more than 350 thousand deaths will occur by the end of 2021 for Sao Paulo and Brazil, respectively. If in contrast, Sao Paulo and Brazil had enough vaccine supply and so started a vaccination campaign in January with the maximum vaccination rate, compliance and efficacy, they could have averted more than 112 thousand deaths and 127 thousand deaths, respectively. In addition, for each month of delay the number of deaths increases monotonically in a logarithmic fashion, for both the State of Sao Paulo and Brazil as a whole. Conclusions Our model shows that the current delay in the vaccination schedules that is observed in many countries has serious consequences in terms of mortality by the disease and should serve as an alert to health authorities to speed the process up such that the highest number of people to be immunized is reached in the shortest period of time.

Estimating the effects of reopening schools on the course of the epidemic of COVID-19

In this paper we present a method do estimate the risk of reopening schools illustrated with the case of the State of São Paulo, Brazil. The model showed that, although no death of children would result from the reopening of the schools in the three cities analyzed, the risk of asymptomatic and symptomatic cases and secondary cases among teacher, school staff and relatives of the children is not negligible. Although the epidemic hit different regions with different intensity, our model shows that, for regions where the incidence profile is similar to the cities analysed, the risk of reopening schools is still too high. This in spite of the fact that incidence in these cities were declining in the period of time considered. Therefore, although we cannot extend the result for the entire country, the overall conclusion is valid for regions with declining incidence and it is even more valid for regions where incidence is increasing. We assumed a very conservative level of infection transmissibility of children of just 10% as that of adults. In spite of this very low level of transmissibility assumed, the number of secondary cases caused by infected children among teachers, school staff a relatives varied from 2 to 85. It is therefore too soon to have any degree of confidence that reopening school before the advent of a vaccine is the right decision to take. The purpose of our model and simulations is to provide a method to estimate the risk of schools reopening, although we are sure it could be applied as a guide to public health strategies.

Modelling the impact of contact tracing of symptomatic individuals on the COVID-19 epidemic

OBJECTIVES: With the declining numbers of coronavirus disease 2019 (COVID-19) cases in the state of São Paulo, Brazil, social distancing measures have gradually been lifted. However, the risk of a surge in the number of cases cannot be overlooked. Even with the adoption of nonpharmaceutical interventions, such as restrictions on mass gatherings, wearing of masks, and complete or partial closure of schools, other public health measures may help control the epidemic. We aimed to evaluate the impact of the contact tracing of symptomatic individuals on the COVID-19 epidemic regardless of the use of diagnostic testing. METHODS: We developed a mathematical model that includes isolation of symptomatic individuals and tracing of contacts to assess the effects of the contact tracing of symptomatic individuals on the COVID-19 epidemic in the state of São Paulo. RESULTS: For a selection efficacy (proportion of isolated contacts who are infected) of 80%, cases and deaths may be reduced by 80% after 60 days when 5000 symptomatic individuals are isolated per day, each of them together with 10 contacts. On the other hand, for a selection efficacy of 20%, the number of cases and deaths may be reduced by approximately 40% and 50%, respectively, compared with the scenario in which no contact-tracing strategy is implemented. CONCLUSION: Contact tracing of symptomatic individuals may potentially be an alternative strategy when the number of diagnostic tests available is not sufficient for massive testing.

Modelling the test, trace and quarantine strategy to control the COVID-19 epidemic in the state of São Paulo, Brazil

Testing for detecting the infection by SARS-CoV-2 is the bridge between the lockdown and the opening of society. In this paper we modelled and simulated a test-trace-and-quarantine strategy to control the COVID-19 outbreak in the State of São Paulo, Brasil. The State of São Paulo failed to adopt an effective social distancing strategy, reaching at most 59% in late March and started to relax the measures in late June, dropping to 41% in 08 August. Therefore, São Paulo relies heavily on a massive testing strategy in the attempt to control the epidemic. Two alternative strategies combined with economic evaluations were simulated. One strategy included indiscriminately testing the entire population of the State, reaching more than 40 million people at a maximum cost of 2.25 billion USD, that would reduce the total number of cases by the end of 2020 by 90%. The second strategy investigated testing only symptomatic cases and their immediate contacts – this strategy reached a maximum cost of 150 million USD but also reduced the number of cases by 90%. The conclusion is that if the State of São Paulo had decided to adopt the simulated strategy on April the 1st, it would have been possible to reduce the total number of cases by 90% at a cost of 2.25 billion US dollars for the indiscriminate strategy but at a much smaller cost of 125 million US dollars for the selective testing of symptomatic cases and their contacts.

Modelling the impact of contact tracing of symptomatic individuals on the COVID-19 epidemic

OBJECTIVES: With the declining numbers of coronavirus disease 2019 (COVID-19) cases in the state of São Paulo, Brazil, social distancing measures have gradually been lifted. However, the risk of a surge in the number of cases cannot be overlooked. Even with the adoption of nonpharmaceutical interventions, such as restrictions on mass gatherings, wearing of masks, and complete or partial closure of schools, other public health measures may help control the epidemic. We aimed to evaluate the impact of the contact tracing of symptomatic individuals on the COVID-19 epidemic regardless of the use of diagnostic testing. METHODS: We developed a mathematical model that includes isolation of symptomatic individuals and tracing of contacts to assess the effects of the contact tracing of symptomatic individuals on the COVID-19 epidemic in the state of São Paulo. RESULTS: For a selection efficacy (proportion of isolated contacts who are infected) of 80%, cases and deaths may be reduced by 80% after 60 days when 5000 symptomatic individuals are isolated per day, each of them together with 10 contacts. On the other hand, for a selection efficacy of 20%, the number of cases and deaths may be reduced by approximately 40% and 50%, respectively, compared with the scenario in which no contact-tracing strategy is implemented. CONCLUSION: Contact tracing of symptomatic individuals may potentially be an alternative strategy when the number of diagnostic tests available is not sufficient for massive testing.

Two complementary model-based methods for calculating the risk of international spreading of a novel virus from the outbreak epicentre. The case of COVID-19.

We present two complementary model-based methods for calculating the risk of international spread of the novel coronavirus SARS-CoV-2 from the outbreak epicentre. One model aims to calculate the number of cases that would be exported from an endemic country to disease-free regions by travellers. The second model calculates the probability that an infected traveller will generate at least one secondary autochthonous case in the visited country. Although this paper focuses on the data from China, our methods can be adapted to calculate the risk of importation and subsequent outbreaks. We found an average R0 = 5.31 (ranging from 4.08 to 7.91) and a risk of spreading of 0.75 latent individuals per 1000 travellers. In addition, one infective traveller would be able to generate at least one secondary autochthonous case in the visited country with a probability of 23%.

Modelling an optimum vaccination strategy against ZIKA virus for outbreak use.

We present a model to optimise a vaccination campaign aiming to prevent or to curb a Zika virus outbreak. We show that the optimum vaccination strategy to reduce the number of cases by a mass vaccination campaign should start when the Aedes mosquitoes' density reaches the threshold of 1.5 mosquitoes per humans, the moment the reproduction number crosses one. The maximum time it is advisable to wait for the introduction of a vaccination campaign is when the first ZIKV case is identified, although this would not be as effective to minimise the number of infections as when the mosquitoes' density crosses the critical threshold. This suboptimum strategy, however, would still curb the outbreak. In both cases, the catch up strategy should aim to vaccinate at least 25% of the target population during a concentrated effort of 1 month immediately after identifying the threshold. This is the time taken to accumulate the herd immunity threshold of 56.5%. These calculations were done based on theoretical assumptions that vaccine implementation would be feasible within a very short time frame.

Is vaccinating monkeys against yellow fever the ultimate solution for the Brazilian recurrent epizootics?

Vaccinating monkeys against yellow fever (YF) has been a common practice in the beginning of the 17D vaccine development. Although it may seem strange at first sight, vaccinating monkeys as a public health strategy is, we think, feasible and theoretically could eliminate the infection among non-human primates, interrupting the virus circulation (or significantly reducing it) and therefore reducing the risk of spilling over to the human population. We propose a series of studies that could demonstrate (or not) the efficacy and feasibility of vaccinating non-human primates YF reservoirs living in green areas of urban centres to cut off or curb the virus circulation that recurrently spill over to the human population. Therefore, vaccinating monkeys in relatively small green areas of the urban centres is perhaps the ultimate solution for the Brazilian recurrent YF epizootics.

A Note on the Risk of Infections Invading Unaffected Regions.

We present two probabilistic models to estimate the risk of introducing infectious diseases into previously unaffected countries/regions by infective travellers. We analyse two distinct situations, one dealing with a directly transmitted infection (measles in Italy in 2017) and one dealing with a vector-borne infection (Zika virus in Rio de Janeiro, which may happen in the future). To calculate the risk in the first scenario, we used a simple, nonhomogeneous birth process. The second model proposed in this paper provides a way to calculate the probability that local mosquitoes become infected by the arrival of a single infective traveller during his/her infectiousness period. The result of the risk of measles invasion of Italy was of 93% and the result of the risk of Zika virus invasion of Rio de Janeiro was of 22%.

The risk of urban yellow fever resurgence in Aedes-infested American cities.

Aedes aegypti, historically known as yellow fever (YF) mosquito, transmits a great number of other viruses such as Dengue, West Nile, Chikungunya, Zika, Mayaro and perhaps Oropouche, among others. Well established in Africa and Asia, Aedes mosquitoes are now increasingly invading large parts of the American continent, and hence the risk of urban YF resurgence in the American cities should because of great concern to public health authorities. Although no new urban cycle of YF was reported in the Americas since the end of an Aedes eradication programme in the late 1950s, the high number of non-vaccinated individuals that visit endemic areas, that is, South American jungles where the sylvatic cycle of YF is transmitted by canopy mosquitoes, and return to Aedes-infested urban areas, increases the risk of resurgence of the urban cycle of YF. We present a method to estimate the risk of urban YF resurgence in dengue-endemic cities. This method consists in (1) to estimate the number of Aedes mosquitoes that explains a given dengue outbreak in a given region; (2) calculate the force of infection caused by the introduction of one infective individual per unit area in the endemic area under study; (3) using the above estimates, calculate the probability of at least one autochthonous YF case per unit area produced by one single viraemic traveller per unit area arriving from a YF endemic or epidemic sylvatic region at the city studied. We demonstrate that, provided the relative vector competence, here defined as the capacity to being infected and disseminate the virus, of Ae. aegypti is greater than 0.7 (with respect to dengue), one infected traveller can introduce urban YF in a dengue endemic area.

Estimating the probability of dengue virus introduction and secondary autochthonous cases in Europe.

Given the speed of air travel, diseases even with a short viremia such as dengue can be easily exported to dengue naïve areas within 24 hours. We set out to estimate the risk of dengue virus introductions via travelers into Europe and number of secondary autochthonous cases as a result of the introduction. We applied mathematical modeling to estimate the number of dengue-viremic air passengers from 16 dengue-endemic countries to 27 European countries, taking into account the incidence of dengue in the exporting countries, travel volume and the probability of being viremic at the time of travel. Our models estimate a range from zero to 167 air passengers who are dengue-viremic at the time of travel from dengue endemic countries to each of the 27 receiving countries in one year. Germany receives the highest number of imported dengue-viremic air passengers followed by France and the United Kingdom. Our findings estimate 10 autochthonous secondary asymptomatic and symptomatic dengue infections, caused by the expected 124 infected travelers who arrived in Italy in 2012. The risk of onward transmission in Europe is reassuringly low, except where Aedes aegypti is present.

The effect of the infection within the individual host on its propagation in the population.

We consider nested or multiscale models to study the effect of the temporal evolution of the disease within the host in the population dynamics of the disease, for one and two infectious agents. We assumed a coupling between the within-host infection rate and the between-host transmission rate. The age of infection within each individual in a population affects the probability of transmission of the disease to a susceptible host and this will affect the temporal evolution of the disease in the host population. To analyze the infection within the host, we consider bacterial-like and viral-like infections. In the model for two infectious agents, we found that, when strain 2 has a basic reproduction number R 02 greater than the basic reproduction number R 01 of strain 1, strain 2 replaces strain 1 in the population. However, if R 02 > R 01 but the values are closer, the replacement does not occur immediately and both strains can coexist for a long time. We applied the model to a scenario in which patients infected with the hepatitis C virus (HCV) are cleared of HCV when super-infected with the hepatitis A virus (HAV). We compared the time for the replacement of HCV by HAV in the population considering instantaneous and non-instantaneous replacement within the individuals. The model developed can be generalized for more than two infectious agents.

Estimating the prevalence of infectious diseases from under-reported age-dependent compulsorily notification databases.

BACKGROUND: National or local laws, norms or regulations (sometimes and in some countries) require medical providers to report notifiable diseases to public health authorities. Reporting, however, is almost always incomplete. This is due to a variety of reasons, ranging from not recognizing the diseased to failures in the technical or administrative steps leading to the final official register in the disease notification system. The reported fraction varies from 9 to 99% and is strongly associated with the disease being reported. METHODS: In this paper we propose a method to approximately estimate the full prevalence (and any other variable or parameter related to transmission intensity) of infectious diseases. The model assumes incomplete notification of incidence and allows the estimation of the non-notified number of infections and it is illustrated by the case of hepatitis C in Brazil. The method has the advantage that it can be corrected iteratively by comparing its findings with empirical results. RESULTS: The application of the model for the case of hepatitis C in Brazil resulted in a prevalence of notified cases that varied between 163,902 and 169,382 cases; a prevalence of non-notified cases that varied between 1,433,638 and 1,446,771; and a total prevalence of infections that varied between 1,597,540 and 1,616,153 cases. CONCLUSIONS: We conclude that the model proposed can be useful for estimation of the actual magnitude of endemic states of infectious diseases, particularly for those where the number of notified cases is only the tip of the iceberg. In addition, the method can be applied to other situations, such as the well-known underreported incidence of criminality (for example rape), among others.

Estimating the size of Aedes aegypti populations from dengue incidence data: Implications for the risk of yellow fever outbreaks.

In this paper we present a model to estimate the density of aedes mosquitoes in a community affected by dengue. The method consists in fitting a continuous function to the incidence of dengue infections, from which the density of infected mosquitoes is derived straightforwardly. Further derivations allow the calculation of the latent and susceptible mosquitoes' densities, the sum of the three equals the total mosquitoes' density. The method is illustrated with the case of the risk of urban yellow fever resurgence in dengue infested areas but the same procedures apply for other aedes-transmitted infections like Zika and chikungunya viruses.