-
-
-
-The current version of serofoi supports
-three different models for estimating the Force-of-Infection
-(FoI), including constant and time-varying trajectories. For
-fitting the model to the sero-prevalence data we use a suit of bayesian
-models that include prior and upper prior distributions
-
-
What is the Force-of-Infection
-
-
The force of infection, also known as the hazard rate or the
-infection pressure, is a key concept in mathematical modelling of
-infectious diseases. It represents the rate at which susceptible
-individuals become infected, given their exposure to a pathogen. In
-simple terms, the force of infection quantifies the risk of a
-susceptible individual becoming infected over a period of time. It is
-usually expressed as a rate per unit of time (e.g., per day or per
-year).
-
-
Constant vs Time-varying FoI
-
-
The FoI is one of the most important parameters in
-epidemiology, but it is often incorrectly assumed to be constant over
-time. Identifying whether the FoI follows a constant or a
-time-varying trend can be important in the identification and
-characterization of the spread of disease. In Table 1 there is a summary
-of the models currently supported by serofoi.
-
-
-
-
-
-
-
-
-
-constant |
-Constant FoI |
-
-
-tv_normal |
-Time-varying normal FoI: slow change in FoI |
-
-
-tv_normal_log |
-Time-varying normal-log FoI: fast epidemic change in
-FoI |
-
-
-
-
-
Table 1. Model options and descriptions.
-
-
Model 1. Constant Force-of-Infection (endemic model)
-
-
The endemic constant model is a simple mathematical model
-used in epidemiology to describe the seroprevalence of an infectious
-disease within a population, as a product of a long-term
-transmission.
-
For a constant FoI endemic model, the rate of infection acquisition
-\(\lambda\) is constant over time for
-each trajectory, and the seroprevalence \(P\) behaves as a cumulative process
-increasing monotonically with age. For the seroprevalence at age \(a\) and time \(t\), we have: \[
-P(a,t) = 1-\exp\left(-\lambda a\right)
-\] The number of positive cases follows a binomial distribution,
-where \(n\) is the number of trials
-(size of the age group) and \(P\) is
-the probability of successes (seroprevalence) for a certain age group:
-\[
-p(a,t) \sim binom(n(a,t), P(a,t))
-\] In serofoi, for the constant model,
-the FoI (\(\lambda\)) is
-modelled within a Bayesian framework using a uniform prior distribution
-\(\sim U(0,2)\). Future versions of the
-package may allow to choose different default distributions. This model
-can be implemented for the previously prepared dataset
-data_test by means of the run_seromodel
-function specifying run_seromodel="constant".
-
The object simdata_constant contains a minimal simulated
-dataset that emulates an hypothetical endemic situation where the
-FoI is constant with value 0.2 and includes data for 250
-samples of individuals between 2 and 47 years old with a number of
-trials \(n=5\). The following code
-shows how to implement the constant model to this simulated
-serosurvey:
-
-
-Figure 1. Constant serofoi model plot. Simulated (red) vs modelled
-(blue) FoI.
-
In this case, 800 iterations are enough to ensure convergence. The
-plot_seromodel method provides a visualisation of the
-results, including a summary where the expected log pointwise predictive
-density (elpd) and its standard error (se) are
-shown. We say that a model converges if all the R-hat estimates are
-below 1.1.
-
-
Time-varying FoI models
-
-
For the time-varying FoI models, the probability for a case
-to be positive at age a at time \(t\)
-also follows a binomial distribution, as described above. However, the
-seroprevalence is obtained from a cumulative of the yearly-varying
-values of the FoI over time: \[
-P(a,t) = 1 - \exp\left(-\sum_{i=t-a+1}^{t}\lambda_i\right)
-\] The corresponding serosurvey completed at time \(t_{sur}\) is informative for the interval
-\([t_{sur}-a_{max}, t_{sur}]\).
-
-
Model 2. Time-varying FoI - Slow Time-Varying FoI
-
-
The time-varying slow normal model relies on the following
-prior distributions for the FoI to describe the spread of a
-given infectious disease within a population over time: \[
-\lambda(t)\sim normal(\lambda(t-1), \sigma) \\
-\lambda(t=1) \sim normal(0, 1)
-\] The object simdata_sw_dec contains a minimal
-simulated dataset that emulates a situation where the FoI
-follows a stepwise decreasing tendency (FoI panel in Fig. 2).
-The simulated dataset contains information about 250 samples of
-individuals between 2 and 47 years old with a number of trials \(n=5\). The following code shows how to
-implement the slow time-varying normal model to this simulated
-serosurvey:
-
-
-Figure 2. Slow time-varying serofoi model plot. Simulated (red) vs
-modelled (blue) FoI.
-
The number of iterations required may depend on the number of years,
-reflected by the difference between the year of the serosurvey and the
-maximum age-class sampled.
-
-
Model 3. Time-varying FoI - Fast Epidemic Model
-
-
The time-varying fast epidemic model, relies on normal prior
-distributions for the FoI in the logarithmic scale, i.e: \[
-\lambda(t)\sim normal(\log(\lambda(t-1)), \sigma) \\
-\lambda(t=1) \sim normal(-6, 4)
-\] This is done in order to capture fast changes in the
-FoI trend. Importantly, the standard deviation parameter of
-this normal distribution of the FoI \(\lambda(t)\) is set using an upper prior
-that follows a Cauchy distribution.
-
In order to test this model we use the minimal simulated dataset
-contained in the simdata_large_epi object. This dataset
-emulates a hypothetical situation where a three-year epidemic occurs
-between 2032 and 2035. The simulated serosurvey tests 250 individuals
-from 0 to 50 years of age in the year 2050. The implementation of the
-fast epidemic model can be obtained running the following lines of
-code:
-
-data("simdata_large_epi")
-serodata_large_epi <- prepare_serodata(simdata_large_epi)
-model_3 <- run_seromodel(serodata = serodata_large_epi,
- foi_model = "tv_normal_log",
- n_iters = 1500)
-model_3_plot <- plot_seromodel(model_3, size_text = 6)
-plot(model_3_plot)
-
-Figure 3. Time-varying fast epidemic serofoi model plot.
-Simulated (red) vs modelled (blue) FoI.
-
In Fig 3 we can see that the fast epidemic serofoi model is
-able to identify the large epidemic simulated on the
-simdata_large_epi dataset.
-
-
Models Comparison
-
-
The statistical details of the three models are described in Table
-2.
-
-
-
-
-
-
-
-
-
-
-
-constant |
-\(\sim binom(n(a,t),
-P(a,t))\) |
-\(\lambda\sim
-uniform(0,2)\) |
- |
-
-
-tv_normal |
-\(\sim binom(n(a,t),
-P(a,t))\) |
-\(\lambda\sim
-normal(\lambda(t-1),\sigma)\\ \lambda(t=1)\sim
-normal(0,1)\) |
-\(\sigma\sim
-Cauchy(0,1)\) |
-
-
-tv_normal_log |
-\(\sim binom(n(a,t),
-P(a,t))\) |
-\(\lambda\sim
-normal(log(\lambda(t-1)),\sigma)\\ \lambda(t=1)\sim
-normal(-6,4)\) |
-\(\sigma\sim
-Cauchy(0,1)\) |
-
-
-
-
-
Table 2. Statistical characteristics of
-serofoi’s currently supported models for the
-FoI (\(\lambda\)). Here \(n\) is the size of an age group \(a\) at time-step \(t\) and \(P\) is its corresponding
-seroprevalence.
-
Above we showed that the fast epidemic model
-(tv_normal_log) is able to identify the large epidemic
-outbreak described by the simdata_large_epi dataset, which
-was simulated according to a step-wise decreasing FoI (red line
-in Fig 3).
-
Now, we would like to know whether this model actually fits this
-dataset better than the other available models in
-serofoi. For this, we also implement both the
-endemic model (constant) and the slow time-varying normal
-model (tv_normal):
-
Using the function cowplot::plot_grid we can visualise
-the results of the three models simultaneously:
-
-cowplot::plot_grid(model_1_plot, model_2_plot, model_3_plot,
- nrow = 1, ncol = 3, labels = "AUTO")
-
-Figure 4. Model comparison between the three serofoi models for a
-large-epidemic simulated dataset.
-
A common criterion to decide what model fits the data the best is to
-choose the one with the larger elpd. According to this
-criterion, in this case the best model is the fast epidemic model, which
-is the only one that manages to identify the large epidemic (see the
-second row of panel C in Figure 4).
-
NOTE: Running the serofoi models for the
-first time on your local computer may take a few minutes for the rstan
-code to compile locally. However, once the initial compilation is
-complete, there is no further need for local compilation.
-
-For details on the implementation of time-varying models and model
-comparison see 
-Figure 1. Serofoi models for FoI estimates of Chikungunya virus
-transmission in an urban remote area of Brazil.
-Figure 2. serofoi models for FoI
-estimates of Venezuelan Equine Encephalitis Virus (VEEV)
-transmission in a rural remote area of Panama.
-Figure 3. serofoi endemic models for
-FoI estimates of Trypanosoma cruzi in a rural area of
-Colombia. ## References