models: Models used in the PandemicLP package

In PandemicLP: Long Term Prediction for Epidemic and Pandemic Data

Models used in the PandemicLP package

Description

This document explains the models used in the PandemicLP package in some detail.

Growth curve for the single wave mean cases

The count data for number of cases or deaths is modeled according to an epidemiological model of growth. In particular, the average counts are μ(t) modeled with a generalized logistic curve:

μ(t) = a c f \frac{e^{-c t}}{(b+e^{-c t})^{f+1}}.

All parameters, that is a, b, c and f are positive.

Parameter c is interpreted as the infection rate. Parameter f controls the asymmetry, so if it is equal to 1, then the curve is symmetric. If it is lesser than 1, then the cases grow slower before the peak than they decrease after. The behavior is inverted when f is greater than 1.

The counts for the Covid-19 pandemic typically had a behavior with positive asymmetry, and so the default for the package functions is to use a greater than 1 truncation for f.

It was common in the early stages of the Covid-19 pandemic that the predictions would result in very high and absurd values for the total number of cases (TNC). It is straightforward to show that

TNC = \frac{a}{b^f}.

Since all locations displayed a total number of cases that never exceeded 5% of that location's population, another truncation is applied, so that a≤ b^f 0.08 Pop, where Pop is the location's population. This is the reason why the model requires the region's population in order to run the model estimation.

Probabilistic model

The simplest probabilistic model for the counts is the Poisson model. If y_t is the count at time t, then

y_t | θ \sim Poisson(μ(t)),

where θ represents the model parameters.

Advanced modeling

Here we present some other forms for the growth curve in the mean. The respective parameters can be adjusted in the pandemic_model function.

Seasonal effects

A weekly seasonal effect can be added. This is done by multiplying μ(t) by a positive effect d when t is the desired weekday. If d < 1 then that weekday represents under-reporting. It is over-reporting if d > 1. Currently, only weekdays are accepted as seasonal effects.

Multiple curves

Additionally, two or more curves can be fitted, as happened in the Covid-19 pandemic in many locations. In this case the model is slightly different. In this case,

μ(t) = μ_1(t)+...+μ_K(t)

μ_j(t) = a_j c_j \frac{e^{-c_j t}}{(b_j+e^{-c_j t})^2}Φ(α_j (t-δ_j)), j = 1, ..., K,

where Φ(.) is the probit function. The probit function induces asymmetry in the curve, similarly to parameter f, which is thus excluded in this case.

Negative Binomial

In addition to the Poisson family, it is possible to fit a Negative Binomial model. The model is parameterized so that the overdispersion does not depend on the mean. This particular parameterization has shown best results when combined with the multiple waves and seasonal effects described above. The model is

y_t | λ_t \sim Poisson(λ_t)

λ_t | θ \sim Gamma(φ μ(t), φ)

.

Prior distribution

Apart from the truncation mentioned above, the prior is defined as independent priors, detailed below. The format is as follows. p\sim D(h1, h2): def1, def2, where p is the parameter, D is the distribution family, h1 and h2 are the hyperparameter encoding such that they can be changed in the prior_parameters argument of the pandemic_model function. Finally, def1 and def2 are the default values if they are not changed by the user. Note that every available model used in the pandemic_model function uses only a subset of these parameters. The parameterization of the distributions is such that the values are passed directly to the stan code.

a_j\sim Gamma(a_alpha, a_beta), j = 1, ..., K: 0.1, 0.1

b_j\sim LogNormal(mu_{b_1}, sigma2_{b_1}), j = 1, ..., K: 0, 20

c_j\sim Gamma(c_alpha, c_beta), j = 1, ..., K: 2, 9

f\sim Gamma(f_alpha, f_beta): 0.01, 0.01

d_j\sim Gamma(d_{j_alpha}, d_{j_beta}), j = 1, 2, 3: 2, 1

δ_j\sim Normal(mu_delta, sigma2_delta), j = 1, ..., K: 0, 100

α_j\sim Gamma(alpha_alpha, alpha_beta), j = 1, ..., K: 0.01, 0.01

φ\sim Gamma(phi_alpha, phi_beta): 0.1, 0.1

Note that the prior for waves parameters are the same for all waves. However, it is possible to use a specific prior for each seasonal effect. For example, if the user wants to change the mu_b_1 and d_2_beta for a model with at least two seasonal effects, they would include the argument prior_parameters = list(mu_b_1 = 1, d_2_beta = 0.001) in the pandemic_model function.

Options for the pandemic_model function

Four arguments in the function change the fitted model, as described below:

• 'seasonal_effect': By leaving this argument NULL, the standard model is fitted. By supplying it with a vector of up to three weekdays, the desired seasonal effects are added to the model.

• 'n_waves': By leaving this argument equal to 1, the standard model is fitted. By changing it to 2 or more implies a multiple waves model.

• 'family': The standard model is fitted with the default value of "poison". When changed to "negbin", the negative binomialm model is used.

• 'prior_parameters': If left as NULL, the default prior values are used. By setting a list with any objects as described above, the provided values will be used.

References

Dani Gamerman, Marcos O. Prates, Thais Paiva and Vinicius D. Mayrink (2021). Building a Platform for Data-Driven Pandemic Prediction: From Data Modelling to Visualisation - The CovidLP Project. CRC Press

URL: http://est.ufmg.br/covidlp/home/en/

See Also

pandemic_model and posterior_predict.pandemicEstimated.

PandemicLP documentation built on March 18, 2022, 6:22 p.m.