First Principles Modeling of Nonlinear Incidence Rates in Seasonal Epidemics

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.Author Summary: 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.