Extending the susceptible-exposed-infected-removed (SEIR) model to handle the false negative rate and symptom-based administration of COVID-19 diagnostic tests: SEIR-fansy

Abstract

False negative rates of severe acute respiratory coronavirus 2 diagnostic tests, together with selection bias due to prioritized testing can result in inaccurate modeling of COVID-19 transmission dynamics based on reported "case" counts. We propose an extension of the widely used Susceptible-Exposed-Infected-Removed (SEIR) model that accounts for misclassification error and selection bias, and derive an analytic expression for the basic reproduction number $R_0$ as a function of false negative rates of the diagnostic tests and selection probabilities for getting tested. Analyzing data from the first two waves of the pandemic in India, we show that correcting for misclassification and selection leads to more accurate prediction in a test sample. We provide estimates of undetected infections and deaths between April 1, 2020 and August 31, 2021. At the end of the first wave in India, the estimated under-reporting factor for cases was at 11.1 (95% CI: 10.7,11.5) and for deaths at 3.58 (95% CI: 3.5,3.66) as of February 1, 2021, while they change to 19.2 (95% CI: 17.9, 19.9) and 4.55 (95% CI: 4.32, 4.68) as of July 1, 2021. Equivalently, 9.0% (95% CI: 8.7%, 9.3%) and 5.2% (95% CI: 5.0%, 5.6%) of total estimated infections were reported on these two dates, while 27.9% (95% CI: 27.3%, 28.6%) and 22% (95% CI: 21.4%, 23.1%) of estimated total deaths were reported. Extensive simulation studies demonstrate the effect of misclassification and selection on estimation of $R_0$ and prediction of future infections. A R-package SEIRfansy is developed for broader dissemination.

Keywords: R package SEIRfansy; compartmental models; infection fatality rate; reproduction number; selection bias; sensitivity; undetected infections.

FIGURE 1

Compartmental model incorporating false negative test results. There are total 10 compartments in this model. The S and E compartments corresponds to Susceptible (who have not been infected till now) and Exposed (who are infected with the virus but are still not infectious). There are three infectious compartments, namely, U (Untested infectious), P (Tested Positive), and F (Tested False Negative). Finally the four compartments RU, RR, DU, and DR denote the Recovered Unreported, Recovered Reported, Deceased Unreported, and Deceased Reported, respectively. $\beta$ corresponds to the rate of transmission by false negative (F) individuals, while for U and P, it is multiplied by scaling factors ${\alpha }_u$ and ${\alpha }_p$. Other parameters include $D_e$ which corresponds to the incubation period, f which is the false negative rate (= 1‐sensitivity) and r which denotes the rate of ascertainment. $D_r$, ${\beta }_1.D_r$, and $D_r/{\beta }_2$ correspond to recovery period for P, $U,$ and F, respectively, while ${\mu }_c$, ${\delta }_1$, ${\mu }_c$, and ${\mu }_c/{\delta }_2$ denote the death rates for P, U, and F, respectively

FIGURE 2

Flowchart showing the estimation process of misclassification model using a Metropolis‐Hastings Markov chain Monte Carlo algorithm

FIGURE 3

Estimates of $R_0$ in India across phases for (A) wave 1 and (B) wave 2. The mean and 95% credible intervals (in parentheses) are provided under the multinomial‐2‐parameter model. The reproduction numbers are estimated for the training periods corresponding to each of the two waves: April 1, 2020 to January 31, 2021 for the first wave and February 1, 2021 to June 30, 2021 for the second wave

FIGURE 4

COVID‐19 cases in India for wave 1 with estimated number of reported, false negative, and untested cases. We have taken April 1, 2020 to January 31, 2021 as the first wave training period and in the first wave, we have not taken any testing period. (A) Total active COVID cases in India from April 1, 2020 to January 31, 2021 including reported active cases, false negatives active, and untested active cases. (B) Proportion of reported active cases among Active COVID cases in India (C) Total cumulative cases in India from April 1, 2020 to January 31, 2021 including reported cumulative cases, cumulative false negatives and untested cumulative cases. (D) Proportion of reported cases among total cumulative COVID cases in India . (E) Total deaths in India from April 1, 2020 to January 31, 2021 including reported and unreported deaths. (F) Proportion of reported deaths among total deaths in India. The dotted curves in subfigures A, C, and E represent the observed data

FIGURE 5

COVID‐19 cases in India for wave 2 with estimated number of reported, false negative, and untested cases. In the wave 2, we have taken February 1, 2021 to June 30, 2021 as the training period, while, July 1, 2021 to August 31, 2021 was taken to be the test period. (A) Total active COVID cases in India from February 1, 2021 to August 31, 2021 including reported active cases, false negatives active, and untested active cases. (B) Proportion of reported active cases among active COVID cases in India (C) Total cumulative cases in India from February 1, 2021 to August 31, 2021 including reported cumulative cases, cumulative false negatives, and untested cumulative cases. (D) Proportion of reported cases among total cumulative COVID cases in India . (E) Total deaths in India from February 1, 2021 to August 31, 2021 including reported and unreported deaths. (F) Proportion of reported deaths among total deaths in India. The dotted curves in subfigures A, C, and E represent the observed data

FIGURE 6

Effect of misclassification on estimates for India (A) Estimates of Total Active Cases for wave 1 with f = 0, 0.15 and 0.3 (B) Estimates of Reported Active Cases for wave 1 with f = 0, 0.15 and 0.3 with the observed data (C) Estimates of Total Active Cases for wave 2 with f = 0, 0.15 and 0.3 (D) Estimates of Reported Active Cases for wave 2 with f = 0, 0.15, and 0.3 with the observed data. April 1, 2020 to January 31, 2021 was taken as the first wave training period and there was no test period for the first wave. In the wave 2, February 1, 2021 to June 30, 2021 was taken as the training period, while July 1, 2021 to August 31, 2021 was taken to be the test period for the second wave

FIGURE 7

(A) Effect of misclassification on Total Active Cases; (B) Effect of selection on Total Active Cases; and (C) Effect of selection on Reported Active Cases: simulations were carried out in order to assess the effect of misclassification, selection on Reported Active and Total Active Cases. For misclassification, the data was generated for a period of 101 days which was divided into five time periods: days $1-10$, $11-31$, $32-50$, $51-64$, $65-101$. The values of ${\mathit{\beta }}_t$ across the five periods were set at 0.8, 0.65, 0.4, 0.3, 0.3 and the corresponding values of ${\mathit{r}}_t$ were set at 0.1, 0.2, 0.15, 0.15, 0.2. For selection model, additional parameters were set as $p_0=(10^{-6},10^{-5},1-10^{-6}-10^{-5})$ and $p_1=(0.02,0.18,0.8)$. As before, the data are generated for a period of 101 days with five periods $1-10$, $11-31$, $32-50$, $51-64$ and $65-101$ while the values of ${\mathit{\beta }}_t$ used to generate the data were $0.6,0.4,0.3,0.25,$ and 0.2 for the five periods, respectively. Predictions are based on the multinomial‐2‐parameter model, where the probability of being tested is assumed to be independent of symptoms with $f=0.3$ (the simulation truth)