First principles modeling of nonlinear incidence rates in seasonal epidemics

Abstract

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.

Figure 1. Observed time series of infected individuals in Gambia and Finland.

Plotted are the monthly number of reported syncytial virus cases in two cities: Banjul in Gambia (from October 1991 to September 1994) and Turku in Finland (from October 1981 to March 1990). Plotted also is the mean monthly temperature range for both localities, for the same time spans.

Figure 2. Predicted vs. observed time series of infected individuals.

Using the ML estimates in Table 1, the predicted infected dynamics of the classical SIRS model was compared against number of infected individuals reported in Gambia and in Finland. Panels A and B show the predictions for the Classical SIRS model and panels C and D show the predictions for the LHD SIRS model.

Figure 3. Predicted vs. observed time series of infected individuals.

Using the ML estimates in Table 2, the predicted infected dynamics of the SEIR model was plotted against the number of infected individuals reported in London and Birmingham. Panels A and B show the predictions for the classical SEIR model and panels C and D show the predictions for the LHD SEIR model.

Figure 4. Predicted model dynamics by the classical SIRS model.

Using the ML estimates in Table 1, the predicted dynamics of the classical SIRS model was plotted without seasonal forcing for both localities, Gambia and Finland (subplots A and B respectively). When seasonal forcing is added (subplots C and D), a limit cycle arises and the endemic equilibrium becomes a function of time (see “Qualitative analysis of the fitted SIRS models”). If the strength of seasonality is large enough as it is the case in Finland, the limit cycle undergoes a period doubling bifurcation creating a small loop in the phase plane. This loop corresponds to the alternating small epidemic outbreaks observed in the predicted and recorded time series of infected individuals for Finland.

Figure 5. Predicted model dynamics by the nonlinear LHD SIRS model.

Using the ML estimates in Table 1, the predicted dynamics of the nonlinear SIRS model with the LHD incidence rate function was plotted without seasonal forcing for both localities, Gambia and Finland (subplots A and B respectively). When seasonal forcing is added (subplots C and D), a limit cycle arises and the endemic equilibrium becomes a function of time (see “Qualitative analysis of the fitted SIRS models”). If the strength of seasonality is large enough as it is the case in Finland, the limit cycle undergoes a period doubling bifurcation creating a small loop in the phase plane. This loop corresponds to the small alternating epidemic outbreaks observed in the predicted and recorded time series of infected individuals for Finland.

Figure 6. Numerical evaluation of the Poincaré map for the SIRS model with environmental stochasticity.

The stationary state was initially perturbed with environmental stochasticity (green diamond). Subsequently, the map was iterated and plotted 3000 times, and at each time, environmental stochasticity was incorporated in the mapping function. The final location of the trajectory was plotted as a red diamond. In panels A and B the environmental stochasticity noise was and both the initial and final states of the trajectory are in a neighborhood of the attractor. In panels C and D was . Note that in the classical model increasing the environmental noise results in transient visits to the disease free equilibrium stable submanifold (panel C), whereas in the LHD model, with a large enough perturbation the trajectory visits the disease free equilibrium basin of attraction and remains there.