The Mathematics of the Pandemic: How Does Epidemiological Dynamics Work?

It is impossible to understand the pandemic without understanding some mathematical concepts. This knowledge is necessary to plan and carry out efficient actions to control the pandemic, in addition to increasing adherence to individual prevention measures.

To start with, viral transmissibility is expressed by the reproduction rate indicator (R). The basic reproduction rate (R0), the initial transmissibility of the virus, assumes that no containment actions are in place. Sars-Cov-2, which causes covid-19, has an R0 that ranges from 2.5 to 5 depending on site and viral variant. But what does it mean? It is the number of transmissions from each case.

the mathematical calculation

The mathematical interpretation is one of exponential growth. If we consider an R0=5, knowing that each secondary case occurs after 5 days on average (serial interval), after 30 days, a single case becomes more than 15 thousand (considering R0=5).

The strength of exponentiality occurs for any value above 1. The Rt is the effective reproduction rate, that is, the transmissibility of the virus under containment measures. The objective is to reduce the transmissibility to below 1. What few people realize is that, as well as the exponential strength for growth with Rt above 1, with values ​​below 1, we have the strength of the exponential decrease.

What is being seen in Brazil is that there is no significant effort to maintain Rt below 1. One or two more weeks in this range would make a huge difference in the number of deaths.

Figure 1a: Example of exponential growth of cases for an R0=5.

Figure 1b: Example of exponential decay with Rt=0.5.

What are the factors influencing Rt?

Answering this question is the key to successful interventions. The Rt is the product of the following variables: duration of transmissibility; number of interactions of a transmitter per day; probability of transmission during an interaction; population susceptibility.

For example, considering that transmissibility lasts an average of 5 days and that transmitters have 10 interactions a day, the average transmission probability is 5% per interaction and that 80% of the population is susceptible, the Rt will be 2.

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If we can get everyone to reduce the number of interactions by 20% and have 80% of the population wear masks with 70% effectiveness, the Rt will drop to 0.9! And then we go into exponential decay.

In this last example, we assume that we don’t know who the transmitters are and use population measures to reduce social interactions and thus indirectly reduce transmitter interactions.

See, we know that the average viral prevalence is less than 1% (it can reach 3-4% at peaks). So, to reduce the interactions of 1% of the population, not knowing who they are, I have to intervene in the 100%, including the 99% who are not transmitters. It is effective and at the expense of intervention in large part by people who do not interfere with the account. Population actions should have the least possible side effects (ideally none), as they reach many people who do not affect the epidemiological dynamics.

In the case of covid-19, these are represented by 99% of the non-communicable population. Regarding the use of masks, for example, except for the discomfort of those who wear them, the negative impact is practically indifferent. Regarding interactions, it can be a problem for those who depend on them for income composition. This becomes a problem especially in places that lack mechanisms to minimize this impact, such as temporary government aid.

But then why not identify this 1%? If we can identify them quickly, early in the infection, it’s much easier to reduce interactions, isn’t that logical? But unfortunately it’s not that simple. For that, we would need to test them all frequently (it is a population measure, yes). There is the advantage that there are few negative impacts, but there is the cost of testing.

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Is Mass Testing Worth It?

Let’s reflect: what is the cost of reducing social interactions for the entire population? An individual who earns less will contribute less with taxes, which will be reverted less to collective benefits and will turn into a vicious circle. What is the cost of expanding the care network, with ward and ICU beds, mechanical ventilators, intubation medication, sedatives, oxygen, doctors, nurses and everything else that involves highly complex care? Don’t forget that the demand for all of this will grow at an exponential rate if the Rt goes above 1. So what is the cost of lost learning for children during the period that schools are closed? And what about a death?

Regarding the type of test, there are other relevant concepts: sensitivity (sens) and specificity (specif), the association of both generates accuracy — Positive Predictive Value (PPV) and Negative Predictive Value (NPV).

Simply put, sens is the ability to detect the disease (positive test) among cases of infection; specif is the ability to detect the absence of disease (negative test) among cases that are not infected. The PPV is the chance of infection given that a test is positive. The VPN is the chance of absence of illness, given that a test is negative.

Now the more complex part: the sens and specific are variable over time, according to the viral load kinetics. Test sensitivity peaks in the first few days of symptoms and progressively declines. In addition, it is important to have a clear definition of the evaluated outcome. It is one thing to assess sens and specific to detect disease, another is to assess transmissibility, which is the most interesting from the point of view of collective health.

For example, tests based on molecular biology, such as RT-PCR, tend to be positive even after the transmissibility period has passed, that is, specif is low for detecting transmissibility. This means that most reagent tests are from people who are no longer transmitting.

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Well, let’s go further. It is useless to have a test with 100% sens and 100% spec if it is not possible to perform it for all people with the disease and in a timely manner. A neglected variable is access, either by cost or by the ability to perform a high number of tests.

It is important to find an optimal balance between access and accuracy. There are a number of models and real-life data demonstrating the impact of applying mass antigen testing, proving that scalability and speed outweigh the drop in sensitivity. There are even concrete data that they are more accurate than PCR for detecting transmitters.

Life is all about choices. We hope that the managers who represent us understand these mathematical concepts and make the right choices.


Bernardo Almeida, columnist of TechWorld, is an infectious disease physician and Chief Medical Officer of Hilab health tech who developed Hilab, the first decentralized laboratory using remote laboratory tests. He is a physician specializing in infectology from the Federal University of Paraná, with residency in internal medicine, as well as internal medicine and infectology at the Hospital de Clinicas — UFPR. He is a master’s student at UFPR in internal medicine, area of ​​Infectious Diseases — Epidemiology of severe acute respiratory syndromes in adults. He has experience in the field of medicine with an emphasis on clinical medicine, infectious and parasitic diseases. He participates in research groups in the field of respiratory viruses.

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