ESTIMATING CUMULATIVE COVID-19 INFECTIONS BY A NOVEL "PANDEMIC RATE EQUATION" (preprint)

ABSTRACT

A fundamental problem dealing with the Covid-19 pandemic has been to estimate the rate of infection, since so many cases are asymptomatic and contagious just for a few weeks. For example, in the US, estimate the proportion P(t) = N/330 where N is the US total who have ever been infected (in millions)at time t (months, t =0 being March 20). This is important for decisions on social restrictions, and allocation of medical resources, etc. However, the demand for extensive testing has not produced good estimates. In the US, the CDC has used the blood supply to sample for anti-bodies. Anti-bodies do not tell the whole picture, according to the Karolinska Instituet , many post infection cases show T-cell immunity, but no anti-bodies. We introduce a method based on a difference-differential equation (dde) for P(t). We emphasize that this is just for the present, with no prediction on how the pandemic will evolve. The dde uses only x=x(s), which is the number/million testing positive, and y=y(s), the number/million who have been tested for all time 0 < s < t (months), with no assumptions on the dynamics of the pandemic. However, we need two parameters. First, R , the ratio of asymptomatic to symptomatic infected cases. Second, T , the period of active infection when the virus can be detected. Both are random variables with distribution which can be estimated. For fixed R, we prove uniform bounds (1+ R) x/(y +1) < P(t) < (1+ R) x(t) , are best possible, with range depending on T . One advantage of our theory is being able to estimate P for many regions and countries where x and y is the only information available.