Model description

PRIME implements an epidemiological model to characterize and forecast the rate at which people turn symptomatic from disease over time. For the purpose of this model, we assume that once people develop symptoms, they have ready access to medical services and can be diagnosed readily. From this perspective, these forecasts represent a lower bound on the actual number of people that are infected with COVID-19 as the people currently infected, but still incubating, are not accounted for. A fraction of the population infected might also exhibit minor or no symptoms at all and might not seek medical advice. Therefore, these cases will not be part of patient counts released by health officials. The epidemiological model consists of two canonical elements: an infection rate model and an incubation rate model. One or more infection rate models are then combined through a convolution with the incubation rate model to yield the number of cases that turn symptomatic daily.

Infection Rate Model

The infection rate component is modeled as a Gamma distribution

\begin{equation} f_\Gamma(t;k_j,\theta_j)=\theta_j^{-k_j}t^{k_j-1}\exp(-t/\theta_j)\big/\Gamma(k_j) \end{equation}

with shape \(k_j\) and scale \(\theta_j\). The choice of values for the pair (\(k_j\),|thj|) can accomodate both sharp increases in the number of infections, which would correspond to strained medical resources, as well as weaker gradients corresponding to a smaller pressure on the available medical resources. Fig. 1 show example infection rate curves for several shape and scale parameter values.

Fig. 1 Infection rate models with fixed scale parameters \(\theta=10\) (left frame) and fixed shape parameter \(k=3\) (right frame).

Incubation Rate Model

PRIME employs a lognormal incubation distribution for COVID-19 [Lauer2020]. The probability density function (PDF), \(f_{LN}\), and cumulative distribution function (CDF), \(F_{LN}\), of the lognormal distribution are given by

\begin{equation} f_{LN}(t;\mu,\sigma)=\frac{1}{t\sigma\sqrt{2}}\exp\left(-\frac{(\log t-\mu)^2}{2\sigma^2}\right),\, F_{LN}(t;\mu,\sigma)=\frac{1}{2}\mathrm{erfc}\left(-\frac{\log t-\mu}{\sigma\sqrt{2}}\right) \label{eq:inceq} \end{equation}

In this toolkit we model the mean \(\mu\) as Student’s t distribution with \(n=36\) degrees of freedom which provided the closest agreement for the 95% confidence interval with the data in [Lauer2020]. Similarly, the standard deviation \(\sigma\) is assumed to have a chi-square distribution. The resulting 95% CIs are \([1.48, 1.76]\) and \([0.320, 0.515]\) for \(\mu\) and \(\sigma\), respectively. The left frame in Fig. 2 shows the family of PDFs with \(\mu\) and \(\sigma\) drawn from Student’s t and \(\chi^2\) distributions, respectively. The nominal incubation PDF is shown in black in this frame. The impact of the uncertainty in the incubation model parameters is displayed in the right frame of this figure. For example, 7 days after infection, there is a large variability (60%-90%) in the fraction of infected people that completed the incubation phase and started displaying symptoms. This variability decreases at later times, e.g. after 10 days more then 85% of case completed the incubation process.

Fig. 2 Probability density functions for the incubation model (left frame) and fraction of people for which incubation ran its course after 7, 10, and 14 days respectively (right frame)

Single Wave Model

With these assumptions the number of people infected and with completed incubation period in the time range \([t_{i-1},t_i]\) can be written as a convolution between the infection rate and the incubation rate [Safta2020]

\begin{equation} n_i\approx N(t_i-t_{i-1})\int_{t_0}^{t_i} f_\Gamma(\tau-t_0;k,\theta) f_{LN}(t_i-\tau;\mu,\sigma)d\tau \label{eq:symptApprox} \end{equation}

Here \(N\) represents the total number of cases over the course of the epidemic, \((t_i-t_{i-1})\) is typically equal to 1 day, and the time parameter \(t_0\) represents the start of the epidemic. In the expression above the sub-script j was neglected since the model contains one wave only.

Multi-Wave Model

The multiple wave model is an extension of the single wave model presented above. In the multi-wave model, a set of infection curves are superimposed to approximate the evolution of the epidemic that exhibits multiple peaks across certain regions. The resulting model is written as

\begin{equation} n_i\approx (t_i-t_{i-1}) \int_{t_0}^{t_i} \left( \sum_{j=1}^K N_j f_\Gamma(\tau-t_0-\Delta t_j;k_j,\theta_j) \right) f_{LN}(t_i-\tau;\mu,\sigma)d\tau \label{eq:symptApproxMultiWave} \end{equation}

Here \(N_j\) represents the total number of cases over the course of the j-th wave, and \(\Delta t_j\) represents the time shift for the j-th infection curve with respect to the start of the epidemic \(t_0\).

Lauer2020(1,2)

Lauer S.A., Grantz K.H., Bi Q., Jones F.K., Zheng Q., Meredith H.R., Azman A.S., Reich N.G., Lessler J., The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application, Annals of Internal Medicine (2020),

Safta2020

Safta C., Ray J., and Sargsyan K., Characterization of partially observed epidemics through Bayesian inference: application to COVID-19, Computational Mechanics (2020),