Effects of migration rates and vaccination on the spread of yellow fever in Latin American communities.

Objective: To assess how relevant the flow of people between communities is, compared to vaccination and type of vector, on the spread and potential outbreaks of yellow fever in a disease-free host community. Methods: Using a SEIRV-SEI model for humans and vectors, we applied numerical simulations to the scenarios: (1) migration from an endemic community to a disease-free host community, comparing the performance of Haemagogus janthinomys and Aedes aegypti as vectors; (2) migration through a transit community located on a migratory route, where the disease is endemic, to a disease-free one; and (3) effects of different vaccination rates in the host community, considering the vaccination of migrants upon arrival. Results: Results show no remarkable differences between scenarios 1 and 2. The type of vector and vaccination coverage in the host community are more relevant for the occurrence of outbreaks than migration rates, with H. janthinomys being more effective than A. aegypti. Conclusions: With vaccination being more determinant for a potential outbreak than migration rates, vaccinating migrants on arrival may be one of the most effective measures against yellow fever. Furthermore, H. janthinomys is a more competent vector than A. aegypti at similar densities, but the presence of A. aegypti is a warning to maintain vaccination above recommended levels.

Apparent prevalence and risk factors for bovine tuberculosis in the state of Paraná, Brazil: an assessment after 18 years since the beginning of the Brazilian program.

Bovine tuberculosis (bTB) impacts considerably animal production and one health worldwide. To describe the prevalence, risk factors, and spatial pattern of the disease in the state of Paraná, Brazil, a cross-sectional study was conducted from September 2018 to February 2019. The area was divided into seven regions. Within each region, farms were randomly selected, and a predetermined number of cows was selected and tested by a comparative cervical tuberculin test. 17,210 animals were tested across 1757 farms. Herd prevalence of bTB-infected herds in Paraná was 2.5% [1.87-3.00%]. It has varied from 0.8 to 3.98% among seven regions, with clustering being detected in the west, central, and northeast areas. Animal prevalence was 0.35% [0.21-0.59%] and has varied from 0.08 to 0.6% among the pre-set regions. No major shifts in the prevalence of bTB were detected since 2007. Large-sized herds, dairy production, and feeding with whey were detected to be correlated with the presence of bTB. Exclusively among dairy herds, veterinary assistance from cooperatives, possession of self-owned equipment to cool milk, and feeding with whey were correlated with the disease. Considering these results, it is recommended that the state of Paraná seek to implement a surveillance system for the detection of bTB-infected herds transforming them into free ones, if possible, incorporating elements of risk-based surveillance. Health education is also recommended to inform farmers about the risks of introducing animals without testing and of feeding raw whey to calves.

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.

Seroprevalence and risk factors for bovine brucellosis in the state of Paraná, Brazil: an analysis after 18 years of ongoing control measures.

Seroprevalence and risk factors of bovine brucellosis (Brucella abortus) in herds and cattle were estimated by a cross-sectional study in the state of Paraná, Brazil. The state was divided into seven regions and a random, two-stage sampling was performed on properties and cattle from each region between 2018 and 2019. Serum samples were collected from 11,592 cows over 24 months from 1,757 properties and a questionnaire was applied to identify potential risk factors. As recommended by the National Program for the Control and Eradication of Animal Brucellosis and Tuberculosis (PNCEBT), serological testing for the detection of anti-Brucella antibodies included the buffered plate agglutination test (screening test) and the fluorescence polarization assay (confirmatory test). The seroprevalence of bovine brucellosis on properties and in cattle was 4.87% (95% confidence interval [CI]: 3.98-5.93%) and 2.24% (95% CI: 1.47-3.41%), respectively. Multiple logistic regression analysis identified larger herd size and failure to test for brucellosis as risk factors for the presence of anti-B. abortus antibodies. These results demonstrate no change in the prevalence when comparing initial studies conducted in 2002. Given our findings, it is recommended that policies for brucellosis control include a widespread vaccination program for higher prevalence areas and eradication approach to lower prevalence areas. All steps related to correct immunization of the herds should be verified and improved by training and education. Health education action must be carried out informing farmers about the risks of introducing animals not tested for brucellosis into their herds and the benefits of testing their herds regularly.

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%.

The Optimal Age of Vaccination Against Dengue with an Age-Dependent Biting Rate with Application to Brazil.

In this paper we introduce a single serotype transmission model, including an age-dependent mosquito biting rate, to find the optimal vaccination age against dengue in Brazil with Dengvaxia. The optimal vaccination age and minimal lifetime expected risk of hospitalisation are found by adapting a method due to Hethcote (Math Biosci 89:29-52). Any number and combination of the four dengue serotypes DENv1-4 is considered. Successful vaccination against a serotype corresponds to a silent infection. The effects of antibody-dependent enhancement (ADE) and permanent cross-immunity after two heterologous infections are studied. ADE is assumed to imply risk-free primary infections, while permanent cross-immunity implies risk-free tertiary and quaternary infections. Data from trials of Dengvaxia indicate vaccine efficacy to be age and serostatus dependent and vaccination of seronegative individuals to induce an increased risk of hospitalisation. Some of the scenarios are therefore reconsidered taking these findings into account. The optimal vaccination age is compared to that achievable under the current age restriction of the vaccine. If vaccination is not considered to induce risk, optimal vaccination ages are very low. The assumption of ADE generally leads to a higher optimal vaccination age in this case. For a single serotype vaccination is not recommended in the case of ADE. Permanent cross-immunity results in a slightly lower optimal vaccination age. If vaccination induces a risk, the optimal vaccination ages are much higher, particularly for permanent cross-immunity. ADE has no effect on the optimal vaccination age when permanent cross-immunity is considered; otherwise, it leads to a slight increase in optimal vaccination age.

The impact of dry ageing vacuum-packed pork on the viability of Toxoplasma gondii tissue cysts.

The present study evaluated the viability of Toxoplasma gondii tissue cysts in dry-aged pork loins (m. longissimus) after 14, 21 and 28 days under controlled temperature (0 °C ±â€¯1 °C). The pigs (n = 9) were orally inoculated with 3,000 T. gondii oocysts. The right loin of each pig was aged for a predetermined period, and the left loin was kept unprocessed as a control. Two experiments were performed. In Experiment 1, the loins of three pigs were aged for 14 days and then bioassayed in both cats and mice. In Experiment 2, the loins of six pigs were bioassayed only in mice, and the ageing periods were 14, 21, and 28 days. Toxoplasma gondii tissue cysts remained viable in loins aged up to 14 days, as confirmed by bioassays in cats and mice. Viable T. gondii was not recovered by bioassays in mice from loins that were aged for 21 or 28 days. These results demonstrate that T. gondii remained viable in vacuum-packed dry-aged pork loins for 14 days at controlled temperature but not for 21 days or longer.

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 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.

The risk of dengue for non-immune foreign visitors to the 2016 summer olympic games in Rio de Janeiro, Brazil.

BACKGROUND: Rio de Janeiro in Brazil will host the Summer Olympic Games in 2016. About 400,000 non-immune foreign tourists are expected to attend the games. As Brazil is the country with the highest number of dengue cases worldwide, concern about the risk of dengue for travelers is justified. METHODS: A mathematical model to calculate the risk of developing dengue for foreign tourists attending the Olympic Games in Rio de Janeiro in 2016 is proposed. A system of differential equation models the spread of dengue amongst the resident population and a stochastic approximation is used to assess the risk to tourists. Historical reported dengue time series in Rio de Janeiro for the years 2000-2015 is used to find out the time dependent force of infection, which is then used to estimate the potential risks to a large tourist cohort. The worst outbreak of dengue occurred in 2012 and this and the other years in the history of Dengue in Rio are used to discuss potential risks to tourists amongst visitors to the forthcoming Rio Olympics. RESULTS: The individual risk to be infected by dengue is very much dependent on the ratio asymptomatic/symptomatic considered but independently of this the worst month of August in the period studied in terms of dengue transmission, occurred in 2007. CONCLUSIONS: If dengue returns in 2016 with the pattern observed in the worst month of August in history (2007), the expected number of symptomatic and asymptomatic dengue cases among tourists will be 23 and 206 cases, respectively. This worst case scenario would have an incidence of 5.75 (symptomatic) and 51.5 (asymptomatic) per 100,000 individuals.

Estimating the Size of the HCV Infection Prevalence: A Modeling Approach Using the Incidence of Cases Reported to an Official Notification System.

In this paper we propose two methods to give a first rough estimate of the actual number of hepatitis C virus (HCV)-infected individuals (prevalence) taking into account the notification rate of newly diagnosed infections (incidence of notification) and the size of the liver transplantation waiting list (LTWL) of patients with liver failure due to chronic HCV infection. Both approaches, when applied to the Brazilian HCV situation converge to the same results, that is, the methods proposed reproduce both the prevalence of reported cases and the LTWL with reasonable accuracy. We use two methods to calculate the prevalence of HCV that, as a first, and very crude approximation, assumes that the actual prevalence of HCV in Brazil is proportional to the reported incidence to the official notification system with a constant denoted [Formula: see text]. In the paper we discuss the limitations and advantages of this assumption. With the two methods we calculated [Formula: see text], which reproduces both the reported incidence and the size of the LTWL. With the value of [Formula: see text] we calculated the prevalence I(a) (the integral of which resulted in 1.6 million people living with the infection in Brazil, most of whom unidentified). Other variables related to HCV infection (e.g., the distribution of the proportion of people aged a who got infected n years ago) can be easily calculated from this model. These new variables can then be measured and the model can be recursively updated, improving its accuracy.

Modeling Importations and Exportations of Infectious Diseases via Travelers.

This paper is an attempt to estimate the risk of infection importation and exportation by travelers. Two countries are considered: one disease-free country and one visited or source country with a running endemic or epidemic infectious disease. Two models are considered. In the first model (disease importation), susceptible individuals travel from their disease-free home country to the endemic country and come back after some weeks. The risk of infection spreading in their home country is then estimated supposing the visitors are submitted to the same force of infection as the local population but do not contribute to it. In the second model (disease exportation), it is calculated the probability that an individual from the endemic (or epidemic) country travels to a disease-free country in the condition of latent infected and eventually introduces the infection there. The input of both models is the force of infection at the visited/source country, assumed known. The models are deterministic, but a preliminary stochastic formulation is presented as an appendix. The models are exemplified with two distinct real situations: the risk of dengue importation from Thailand to Europe and the risk of Ebola exportation from Liberia to the USA.

Potential for international spread of wild poliovirus via travelers.

BACKGROUND: The endgame of polio eradication is hampered by the international spread of poliovirus via travelers. In response to ongoing importations of poliovirus into polio-free countries, on 5 May 2014, WHO's Director-General declared the international spread of wild poliovirus a public health emergency of international concern. Our objective was to develop a mathematical model to estimate the international spread of polio infections. METHODS: Our model took into account polio endemicity in polio-infected countries, population size, polio immunization coverage rates, infectious period, the asymptomatic-to-symptomatic ratio, and also the probability of a traveler being infectious at the time of travel. We applied our model to three scenarios: (1) number of exportations of both symptomatic and asymptomatic polio infections out of currently polio-infected countries, (2) the risk of spread of poliovirus to Saudi Arabia via Hajj pilgrims, and (3) the importation risk of poliovirus into India. RESULTS: Our model estimated 665 polio exportations (>99 % of which were asymptomatic) from nine polio-infected countries in 2014, of which 78.3 % originated from Pakistan. Our model also estimated 21 importations of poliovirus into Saudi Arabia via Hajj pilgrims and 20 poliovirus infections imported to India in the same year. CONCLUSION: The extent of importations of asymptomatic and symptomatic polio infections is substantial. For countries that are vulnerable to polio outbreaks due to poor national polio immunization coverage rates, our newly developed model may help guide policy-makers to decide whether imposing an entry requirement in terms of proof of vaccination against polio would be justified.

Epidemiology and pathology of avian malaria in penguins undergoing rehabilitation in Brazil.

Seabird rehabilitation is a valuable strategy to mitigate the impacts of oil pollution and other anthropogenic factors, and can significantly contribute to the conservation of penguins. However, infectious diseases such as avian malaria (Plasmodium spp.) can hamper the success of rehabilitation efforts. We combined morphological and molecular diagnostic methods to investigate the epidemiology and pathology of Plasmodium in Magellanic penguins (Spheniscus magellanicus) at rehabilitation centers along 2500 km of the coastline of Brazil. True prevalence of malarial parasites was estimated between 6.6% and 13.5%. We identified five species, three of which had not been described infecting penguins (P. cathemerium, P. nucleophilum, P. unalis); an additional five distinct Plasmodium lineages were also distinguished, and albeit unidentified these clearly correspond to species that also have not yet been reported in penguins. Our results indicate that the diversity of plasmodia that may infect these birds is greater than previously recognised. Considering the well-defined seasonality observed in this study, it is clear that rehabilitation centers could benefit by narrowing their preventative efforts on penguins maintained or admitted during the Austral spring-summer, particularly by preventing mosquitoes from coming into contact with penguins.

A public health risk assessment for yellow fever vaccination: a model exemplified by an outbreak in the state of São Paulo, Brazil.

We propose a method to analyse the 2009 outbreak in the region of Botucatu in the state of São Paulo (SP), Brazil, when 28 yellow fever (YF) cases were confirmed, including 11 deaths. At the time of the outbreak, the Secretary of Health of the State of São Paulo vaccinated one million people, causing the death of five individuals, an unprecedented number of YF vaccine-induced fatalities. We apply a mathematical model described previously to optimise the proportion of people who should be vaccinated to minimise the total number of deaths. The model was used to calculate the optimum proportion that should be vaccinated in the remaining, vaccine-free regions of SP, considering the risk of vaccine-induced fatalities and the risk of YF outbreaks in these regions.

A comparative analysis of the relative efficacy of vector-control strategies against dengue fever.

Dengue is considered one of the most important vector-borne infection, affecting almost half of the world population with 50 to 100 million cases every year. In this paper, we present one of the simplest models that can encapsulate all the important variables related to vector control of dengue fever. The model considers the human population, the adult mosquito population and the population of immature stages, which includes eggs, larvae and pupae. The model also considers the vertical transmission of dengue in the mosquitoes and the seasonal variation in the mosquito population. From this basic model describing the dynamics of dengue infection, we deduce thresholds for avoiding the introduction of the disease and for the elimination of the disease. In particular, we deduce a Basic Reproduction Number for dengue that includes parameters related to the immature stages of the mosquito. By neglecting seasonal variation, we calculate the equilibrium values of the model's variables. We also present a sensitivity analysis of the impact of four vector-control strategies on the Basic Reproduction Number, on the Force of Infection and on the human prevalence of dengue. Each of the strategies was studied separately from the others. The analysis presented allows us to conclude that of the available vector control strategies, adulticide application is the most effective, followed by the reduction of the exposure to mosquito bites, locating and destroying breeding places and, finally, larvicides. Current vector-control methods are concentrated on mechanical destruction of mosquitoes' breeding places. Our results suggest that reducing the contact between vector and hosts (biting rates) is as efficient as the logistically difficult but very efficient adult mosquito's control.

Predictive criteria to study the pathogenesis of malaria-associated ALI/ARDS in mice.

Malaria-associated acute lung injury/acute respiratory distress syndrome (ALI/ARDS) often results in morbidity and mortality. Murine models to study malaria-associated ALI/ARDS have been described; we still lack a method of distinguishing which mice will develop ALI/ARDS before death. This work aimed to characterize malaria-associated ALI/ARDS in a murine model and to demonstrate the first method to predict whether mice are suffering from ALI/ARDS before death. DBA/2 mice infected with Plasmodium berghei ANKA developing ALI/ARDS or hyperparasitemia (HP) were compared using histopathology, PaO2 measurement, pulmonary X-ray, breathing capacity, lung permeability, and serum vascular endothelial growth factor (VEGF) levels according to either the day of death or the suggested predictive criteria. We proposed a model to predict malaria-associated ALI/ARDS using breathing patterns (enhanced pause and frequency respiration) and parasitemia as predictive criteria from mice whose cause of death was known to retrospectively diagnose the sacrificed mice as likely to die of ALI/ARDS as early as 7 days after infection. Using this method, we showed increased VEGF levels and increased lung permeability in mice predicted to die of ALI/ARDS. This proposed method for accurately identifying mice suffering from ALI/ARDS before death will enable the use of this model to study the pathogenesis of this disease.

Risk of symptomatic dengue for foreign visitors to the 2014 FIFA World Cup in Brazil.

Brazil will host the FIFA World Cup™, the biggest single-event competition in the world, from June 12-July 13 2014 in 12 cities. This event will draw an estimated 600,000 international visitors. Brazil is endemic for dengue. Hence, attendees of the 2014 event are theoretically at risk for dengue. We calculated the risk of dengue acquisition to non-immune international travellers to Brazil, depending on the football match schedules, considering locations and dates of such matches for June and July 2014. We estimated the average per-capita risk and expected number of dengue cases for each host-city and each game schedule chosen based on reported dengue cases to the Brazilian Ministry of Health for the period between 2010-2013. On the average, the expected number of cases among the 600,000 foreigner tourists during the World Cup is 33, varying from 3-59. Such risk estimates will not only benefit individual travellers for adequate pre-travel preparations, but also provide valuable information for public health professionals and policy makers worldwide. Furthermore, estimates of dengue cases in international travellers during the World Cup can help to anticipate the theoretical risk for exportation of dengue into currently non-infected areas.

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.