In this paper we used a general stochastic processes framework to derive from first principles the incidence rate function that characterizes epidemic models. We investigate a particular case, the Liu-Hethcote-van den Driessche's (LHD) incidence rate function, which results from modeling the number of successful transmission encounters as a pure birth process. This derivation also takes into account heterogeneity in the population with regard to the per individual transmission probability. We adjusted a deterministic SIRS model with both the classical and the LHD incidence rate functions to time series of the number of children infected with syncytial respiratory virus in Banjul, Gambia and Turku, Finland. We also adjusted a deterministic SEIR model with both incidence rate functions to the famous measles data sets from the UK cities of London and Birmingham. Two lines of evidence supported our conclusion that the model with the LHD incidence rate may very well be a better description of the seasonal epidemic processes studied here. First, our model was repeatedly selected as best according to two different information criteria and two different likelihood formulations. The second line of evidence is qualitative in nature: contrary to what the SIRS model with classical incidence rate predicts, the solution of the deterministic SIRS model with LHD incidence rate will reach either the disease free equilibrium or the endemic equilibrium depending on the initial conditions. These findings along with computer intensive simulations of the models' Poincaré map with environmental stochasticity contributed to attain a clear separation of the roles of the environmental forcing and the mechanics of the disease transmission in shaping seasonal epidemics dynamics. Nonlinearity in the infection incidence is one of the main components that shape seasonal epidemics. Here, we revisit classical incidence and propose a first principles derivation of the infection incidence rate. A qualitative analysis of the SIRS model with both the classical and the proposed incidence rate showed that the new model is physically more meaningful. We conducted a statistical analysis confronting the SIRS and SEIR models formulated using both incidence rate functions with four data sets of seasonal childhood epidemics. Two data sets were hospital records of cases of syncytial respiratory virus (RSV). The other two data sets were taken from the well-known UK measles epidemics database. We found that seasonal epidemics is better explained using our incidence rate model embedded in a Poisson sampling process. The results presented here are not by any means an exhaustive exploration of the interplay between nonlinear dynamics and stochasticity. Our results may be viewed as the starting point of multiple research avenues. Three such research topics could be: the first-principles derivation of non-linear incidence rate functions, the role of bistability and demographic stochasticity for disease persistence and the simulation of environmental and demographic stochasticity in the Poincaré map.

We present a forecasting model aim to predict hospital occupancy in metropolitan areas during the current COVID-19 pandemic. Our SEIRD type model features asymptomatic and symptomatic infections with detailed hospital dynamics. We model explicitly branching probabilities and non-exponential residence times in each latent and infected compartments. Using both hospital admittance confirmed cases and deaths, we infer the contact rate and the initial conditions of the dynamical system, considering breakpoints to model lockdown interventions and the increase in effective population size due to lockdown relaxation. The latter features let us model lockdown-induced 2nd waves. Our Bayesian approach allows us to produce timely probabilistic forecasts of hospital demand. We have applied the model to analyze more than 70 metropolitan areas and 32 states in Mexico.

We introduce a Bayesian sequential data assimilation and forecasting method for non-autonomous dynamical systems. We applied this method to the current COVID-19 pandemic. It is assumed that suitable transmission, epidemic and observation models are available and previously validated. The transmission and epidemic models are coded into a dynamical system. The observation model depends on the dynamical system state variables and parameters, and is cast as a likelihood function. The forecast is sequentially updated over a sliding window of epidemic records as new data becomes available. Prior distributions for the state variables at the new forecasting time are assembled using the dynamical system, calibrated for the previous forecast. Epidemic outbreaks are non-autonomous dynamical systems depending on human behavior, viral evolution and climate, among other factors, rendering it impossible to make reliable long-term epidemic forecasts. We show our forecasting method’s performance using a SEIR type model and COVID-19 data from several Mexican localities. Moreover, we derive further insights into the COVID-19 pandemic from our model predictions. The rationale of our approach is that sequential data assimilation is an adequate compromise between data fitting and dynamical system prediction.

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.

In most cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior derived from the error in the numerical solver. To compare an approximate posterior distribution with the theoretical one, we propose using Bayes factors (BFs), considering both of them as models for the data at hand. From a theoretical point of view, we prove that the theoretical vs. numerical posterior BF tends to 1, in the same order as the numerical solver used. In practice, we illustrate the fact that for higher order solvers (e.g., Runge--Kutta) the BF is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error-free posteriors. Two examples are presented where nearly 90% CPU time is saved, with all inference results being identical to those obtained using a solver with a much finer time step.