Publicação: 11 de June de 2020
Despite not being error-free, mathematical models are one of the parameters used to understand the evolution of the pandemic
Numerous mathematical models are being produced to predict the future of COVID-19 worldwide. More than estimating how the disease will spread, the number of infected and the percentage of deaths and hospitalizations, these tools allow simulating numerous scenarios and, thus, testing the effectiveness of interventions that can be adopted by health authorities to reduce contagion, such as closing schools, canceling public events and restricting travel. Despite not being error-free, mathematical models are consolidated as efficient methods in the projection and understanding of the pandemic.
Enhanced over 250 years, mathematical models of infectious diseases, despite uncertainties, guide actions to combat COVID-19. An example of its importance in conducting measures against the pandemic was the change in strategy in the United Kingdom following projections made by Imperial College, which built a mathematical model to estimate the evolution and impacts of COVID-19 in almost 200 countries. Dr. Neil Ferguson, head of the mathematical model program at Imperial College, acknowledged that we can live in a very different world than we have known for a year or more. In a report published on May 8, experts at the institution stated that the COVID-19 outbreak in Brazil is still in its infancy, and the disease is out of control in Brazilian lands. Imperial College experts considered the problem of underreporting to perform their calculations, arriving at much greater contagion data than those reported by the authorities.
Professor at the Federal University of Piauí (UFPI) Dr. Jefferson Cruz dos Santos Leite, PhD in applied mathematics with an emphasis on mathematical models in epidemiology, adds that since always and especially today, doing scientific research and applying it to reality without thinking about numbers, data, statistics and equations is very unlikely. According to him, mathematical models are a very important tool to guide researchers from the most diverse areas, allowing the projection and analysis of different scenarios, contributing with elements for decision making on how to solve public health problems. “The models applied to epidemics make it possible, for example, to predict the speed at which diseases spread through a territory or how they can affect certain populations. This is the case of the spread of the COVID-19 pandemic in Brazil, which integrate studies developed by the Working Group of the Federal University of Piauí in partnership with several collaborators from other federal universities”, he points out. For him, more than estimating how the disease will spread, the number of infected and the proportion of deaths and hospitalizations, these tools allow to simulate countless scenarios and, thus, test the effectiveness of interventions that can be adopted by health authorities to reduce contagion.
Another example is the study done in Singapore using complex mathematical models. In April, the country’s University of Technology and Design developed a model that had predicted the end of the epidemic in Brazil between June and August. However, according to the latest update made by the group of researchers, which considered the advancement of COVID-19 and the increase in the number of deaths, the virus should remain a problem until November 11. The survey also says that Brazil is experiencing the peak of the disease and that it should last until the end of July. Global expectations have also changed: the tool had predicted the end of the world pandemic by December 1. It has now shifted to January 05 2021. According to the Singapore researchers, Spain must be affected by the disease until August 15 while the date for the United Kingdom is September 30 and Italy, October 23.
Mathematical models can be useful tools, but they should not be overestimated, especially for long-term projections or subtle characteristics, such as the exact date or number of infections. Dr. Samir Bhatt, professor of geostatistics at the Department of Epidemiology in Infectious Diseases at Imperial College London, recognizes that long-term predictions are really not possible. “The future is uncertain and nobody should interpret predictions from two months onwards as if they were the truth. What is done is to provide ways to see what could happen. Scientists need to be transparent with what these predictions mean and how they should be interpreted. If the scenario presented does not happen, fine, that in itself gives us a lot of information in our understanding. In this way, no one should consider the models as Delphic Oracles, but in the same way they are not useless, they are great to fill our knowledge when there is a lack of information”, he explains.
Unlike pandemics previously faced by humanity, COVID-19 occurs in a world based and relatively organized in data. In this scenario, the exact sciences are fundamental to fight the disease through mathematical models. These models use data to predict the spread of the virus and that can help authorities define their actions. “As a planned experiment, we understood the COVID-19 infection process perfectly, as we knew exactly who would infect and etc. and simulated it in the future. Leaving aside the arguments against Laplace’s demon, even if that model was perfect, if the government suddenly launches or withdraws a policy, the prediction would not be wrong, despite being a perfect representation of reality at this time”, adds Dr. Bhatt. For him, the secret is communication, that is, telling people which models may or may not mean what they say.
Although useful, mathematical models have limitations
Mathematical models, while useful, also have limitations. For Professor Bhatt, all models are wrong, but some are useful and others are very technical. “The model is just that, it will never be reality. So, it’s up to the researcher to explain them to different audiences, from the lay reader to the mathematician. It is very difficult to know if you are really adapting to the data signal or just noise”, he says. Still according to him, this is a subject that exists in all fields of statistics and machine learning. It is up to the researcher to ensure that his model is not over adjusted.
Professor Leite recalls that a mathematical model consists of a set of equations that represent quantitatively the hypotheses that were used in the construction of the model, which are based on the real system. Such equations are solved according to some values known or predicted by the real model and can be tested by comparing with the known or predicted data with the measurements performed in the real world. The mathematical equations of a model interpret the hypotheses from a quantitative point of view, allowing us to deduce consequences and show where the details are that should be accepted or rejected. “When we talk about models that simulate reality, we are trying to interpret phenomena that also depend on human behavior, as in the case of COVID-19 and, therefore, any change in this behavior will cause changes in the model. And, in fact, this is the biggest problem. For example, we make predictions of the spread of the virus with various levels of social isolation and these levels are very variable and difficult to predict because each person has a way of dealing with these situations”, he points out.
Regarding regional particularities, Professor Leite points out that when we look at them, we realize that many of the problems of lack of data or underreporting can be minimized if a more homogeneous behavior of data and people is observed. “The generalization of models is a very dangerous aspect when forecasting and analyzing data. The calculation of the number of reproductions of the disease[1] (R0), for example, is easier to be estimated in a city than in a country, since for different R₀, there will be different predictions of the model and this can be quite sensitive. The periods can be different, the impacts can be very different”, he says.
Alternative epidemiological model
Several caveats must be taken into account when using mathematical models, but is there an alternative epidemiological model that may be more appropriate? The professor of geostatistics at Imperial College London emphasizes that like everything in science, we all build on each other, and the more models, the more different approaches, the better to build and support evidence for a conclusion. “However, a major embargo is that not all models are made equal, and these models need due diligence and must be viewed with the expertise of infectious disease epidemiology, as linear regression with little depth should not be considered an alternative” , justifies Dr. Bhatt.
For Professor Leite, the most appropriate model is the one that best portrays the previous facts and that best fits COVID-19’s data. “I have the impression that there is no universal model, it depends on each region and the government’s prevention measures”. Models that include subjectivities in some parameters, such as fuzzy logic, for example, may be more accurate in some situations. But it is always up to the researcher to find the best model according to the available data”, he observes.
Asked if the contribution of mathematical models is still qualitative, testing which is the best combination of these strategies, the professor at Imperial College London believes that the answer is a little of each. “Qualitative models allow us to have an idea about specific situations, but they are predominantly quantitative. We use models to test hypotheses while estimating statistical significance. This is where uncertainty is key: we need to consider as many sources as possible and factor them into our decision-making”, details Professor Bhatt.
For Professor Leite, the quantitative method is conclusive, and aims to quantify a problem and understand its dimension. In short, this type of research provides numerical information about the behavior of the object studied. Qualitative research focuses on understanding the behavior of the object studied, instead of measuring. “I believe that the combination of the two is the ideal”, he stresses when he emphasizes that, with the development of new mathematical tools and also in the health area, mathematical models began to be much more than a quantitative analysis of the data. “We started to see a very large interdisciplinarity in areas that lead us to transform models into both qualitative and quantitative tools. The variables today can be seen in several ways and have different interpretations, it is up to the specialists to interpret them in the best possible way”, he completes. For him, mathematics is very precise, if the estimated parameters and the tools are used correctly, there will certainly be a correctly defined scenario. “But it is up to the experts to interpret whether these are plausible and how we should act to avoid or get past them”, he reinforces.
Mathematical modeling in the face of the rapidly advancing epidemic
Mathematical modeling is a tool that aggregates all the sciences necessary for the de ion and prediction of the problem studied. All tools to combat the spread of diseases and the development of medications, for example, are developed considering mathematical models to validate the experiment. Dr. Leite recalls that mathematics has always been necessary, but, without a doubt, this pandemic was a trigger that caused all the exposure of the need for mathematical models for solving problems, both everyday and the most complex. “Science should always be the pillar of governments that are minimally concerned with the social and economic development of their country or state”, says the professor.
Several studies using mathematical models have emerged around the world, but will any of them be able to answer the question of the moment: when will the pandemic end? For Professor Bhatt, from Imperial College, the answer is no! According to him, nothing can answer that question. “Even the assumptions of mass immunization are questioned, given that reinfection may be possible. The model can give an indication, but it will almost certainly be wrong, as well as any other long-term projection”, he stresses. In his opinion, this question is much bigger than it seems – which country, in what circumstances, how many will we let die, what will be the effect on the global dynamic – and, when someone thinks better about it, it seems more and more that it will be impossible to answer without asking for many difficult-to-answer questions. “That way, I don’t think modeling will tell us when the end will be, but it can help us understand what the end will be like”, concludes Dr. Bhatt.
Finally, Professor Leite weighs that the vaccine may take about a year and a half to reach the market and, therefore, we have to be prepared with the new normal until an effective vaccine or treatment appears. He warns that the peak predictions are only a first point of attention, since as long as there is no vaccine we will have many sick people and several peaks, of course smaller, should appear. “I reiterate everyone’s concern for the future of the world as it is today and that we must be aware that it will never be as it was before. We have to adapt and learn from this new world to come. Our relationships in terms of social behavior will change and, perhaps, it will be a great opportunity to be more human. The development of a society only happens if we are focused on people”, he concludes.
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