The Mathematics Of The Pandemic: How Epidemiological Dynamics Work

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

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

The mathematical calculation

The mathematical interpretation is exponentially growing. If we consider an R0 = 5, knowing that each secondary case occurs after an average of 5 days (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 in force of containment measures. The objective is to reduce the transmissibility to below 1. What few realize is that, like the exponential force for growth with Rt above 1, with values ​​below 1 we have the force of the exponential decrease.

What we have 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 case growth for an R0 = 5

Figure 1b: Example of exponential drop with Rt = 0.5

What are the factors that influence Rt?

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

If we can get everyone to reduce the number of interactions by 20% and 80% of the population to use masks with 70% effectiveness, the Rt will drop to 0.9 and Voilá! We went into an exponential drop.

In this last example, we assume that we do not know who the transmitters are and we resort to population measures to reduce social interactions, and thus indirectly reduce the interactions of transmitters. But you see, we know that the average viral prevalence is less than 1% (it can reach 3-4% in the peaks).

So, to reduce interactions by 1% of the population, because I don’t know who they are, I have to intervene in the 100%, including the 99% who are not transmitters. It is effective, at the expense of intervention in a large part of people who do not interfere in the account. Population actions should have the least possible side effect (ideally none), as they interfere with many people who do not affect the epidemiological dynamics.

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