Disentangling Extrinsic from Intrinsic Factors in Disease Dynamics: A Nonlinear Time Series Approach with an Application to Cholera

Abstract

Alternative explanations for disease and other population cycles typically include extrinsic environmental drivers, such as climate variability, and intrinsic nonlinear dynamics resulting from feedbacks within the system, such as species interactions and density dependence. Because these different factors can interact in nonlinear systems and can give rise to oscillations whose frequencies differ from those of extrinsic drivers, it is difficult to identify their respective contributions from temporal population patterns. In the case of disease, immunity is an important intrinsic factor. However, for many diseases, such as cholera, for which immunity is temporary, the duration and decay pattern of immunity is not well known. We present a nonlinear time series model with two related objectives: the reconstruction of immunity patterns from data on cases and population sizes and the identification of the respective roles of extrinsic and intrinsic factors in the dynamics. Extrinsic factors here include both seasonality and long-term changes or interannual variability in forcing. Results with simulated data show that this semiparametric method successfully recovers the decay of immunity and identifies the origin of interannual variability. An application to historical cholera data indicates that temporary immunity can be long-lasting and decays in approximately 9 yr. Extrinsic forcing of transmissibility is identified to have a strong seasonal component along with a long-term decrease. Furthermore, noise appears to sustain the multiple frequencies in the long-term dynamics. Similar semiparametric models should apply to population data other than for disease.

Shifting patterns: malaria dynamics and rainfall variability in an African highland

“…In the model, the transmission rate parameter (b t ) is allowed to vary in time in a non-specified fashion, with this variation determined by fitting the model to the data. This semi-mechanistic (semi-parametric) approach allows us to identify evidence for extrinsic forcing by considering both the variability in b t and in the residuals of the model, that is, in the error terms reflecting the variability in the transmission rate that is unaccounted for by the model itself (Koelle & Pascual 2004;Koelle et al 2005). Two key questions are addressed with this approach.…”

“…The first is a procedure to reconstruct the time series of susceptibles and the second is a transmission equation ( Finkenstadt & Grenfell 2000;Koelle & Pascual 2004). Our model here is a simplification of the TSIRS (Time Series Susceptible-Infectious-Recovered-Susceptible) model in Koelle & Pascual (2004), originally formulated for diseases with temporary immunity. Here, we consider that there is no loss of immunity and that the total population is constant in time with a constant turnover time T of individuals in the tea estate.…”

“…Because b t is not specified but determined from the model fit itself, the model is semi-parametric. Thus, we fit the model with the semi-parametric approach described in Koelle & Pascual (2004) and Koelle et al (2005), using log-transformed malaria cases. The approach essentially consists of two steps: (i) the fit of the transmission equation (2) using weighted linear regression after a log transformation and (ii) the smoothing of the residuals to obtain b t from the residuals of the linear regression.…”

Independent and Competing Disease Risks: Implications for Host Populations in Variable Environments

“…Understanding the relationship between disease impacts and environmental variability is particularly relevant in the context of global climate change (Colwell 1996;Anderson et al 2004;Harvell et al 2002). Some recent studies examine the influence of environmental variability on disease dynamics through its effects on host contact patterns or susceptibility and, therefore, disease transmission rates (Pascual et al 2000;Hay et al 2002;Koelle and Pascual 2004). Studies on ungulate population dynamics, too, have emphasized the importance of environmental variability (reviews in Saether 1997 andGaillard et al 2000;Ogutu and Owen-Smith 2003).…”

The transmission dynamics of syphilis and the CDC’s elimination plan

“…The annual cycle of cholera in Bangladesh can be explained by the immune status of the local population (an endogenous factor), and the El Nino southern oscillation and the Indian Ocean temperature (both exogenous factors) [14,26]. Koelle et al have recently developed a methodology that enables the effects of endogeneous and exogeneous factors to be independently isolated [26,27].…”