Learning the seasonality of disease incidences from empirical data

Abstract

Investigating the seasonality of disease incidences is very important in disease surveillance in regions with periodical climatic patterns. In lieu of the paradigm about disease incidences varying seasonally in line with meteorology, this work seeks to determine how well simple epidemic models can capture such seasonality for better forecasts and optimal futuristic interventions. Once incidence data are assimilated by a periodic model, asymptotic analysis in relation to the long-term behavior of the disease occurrences can be performed using the classical Floquet theory, which explains the stability of the existing periodic solutions. For an illustrative case, we employed infected-recovered models with an infection rate of a single period and that of commensurate periods to assimilate weekly dengue incidence data from the city of Jakarta, Indonesia, which we present in their raw and moving-average-filtered version. To estimate the infection rate of a single period, eight optimization schemes were assigned returning magnitudes of the rate that vary insignificantly across schemes. Three schemes were assigned to estimate the infection rate of commensurate periods based on three different sets of periods used. Each scheme involving commensurate periods gives better fitting than that involving only a single period. The computation results combined with the analytical results indicate that if the disease surveillance in the city does not improve, then the incidence will raise to a certain positive orbit and remain cyclical.

Introduction

Investigating the seasonality of disease incidences plays a fundamental role in the detection of future outbreaks, and thus in the provision of economically viable control interventions. In the case of vector-borne diseases, previous studies suggested that the incidence is sensitive to the behavior of meteorological factors (Babin, 2003, Bartley, Donnelly, Garnett, 2002, Mutheneni, Morse, Caminade, Upadhyayula, 2017, Stratton, Ehrlich, Mor, Naumova, 2017). The rationale behind this is that the aforementioned factors, including water precipitation, temperature, and air humidity, have a major influence on the growth of virus-carrying vectors. Let us look, for example, at the case of dengue. A high temperature, on the one hand, prolongs the life of mosquitoes Aedes aegypti and shortens the extrinsic incubation period of the dengue virus, which then increases the number of living infected mosquitoes (Lambrechts, Paaijmans, Fansiri, Carrington, Kramer, Thomas, Scott, 2011, Mutheneni, Morse, Caminade, Upadhyayula, 2017). On the other hand, high rainfall induces the development of breeding sites for mosquitoes (Bicout, Vautrin, Vignolles, Sabatier, 2015, Paaijmans, Wandago, Githeko, Takken, 2007), while high wind speeds disrupt the development of the aquatic phases on leaf blades, light plastics, or other unstable sites that further induce negative correlation with the dengue incidence (Cheong et al., 2013). The message here is that one can achieve a better understanding of the seasonality of disease incidences in some regions where the corresponding meteorological factors fluctuate periodically.

One idea to approach fluctuating data and generate some forecasts is to develop epidemic models in which several biological properties can be adjusted. A promising feature of epidemic models is their accessibility to further development, i.e., by including more biological properties, to further increase realism. As far as assimilating almost-periodic data is concerned, one can designate some parameters in the model to be periodic by an argument that they can be correlated with some periodical meteorological factors (Cushing, 1998, He, Earn, 2007, Henson, Cushing, 1997, Ireland, Mestel, Norman, 2007, Sauvage, Langlais, Pontier, 135., 2007, Sauvage, Langlais, Yuccoz, Pontier, 2003). Decision making comes into play when prior knowledge about the long-term behavior of the model solution that assimilates the data is gained. This asymptotic analysis is the part where the basic reproductive number plays a significant role. The basic reproductive number is defined as the number of secondary infections that happen when a single infective individual comes into a completely susceptible population during the infection period (Diekmann, Heesterbeek, 2000, Diekmann, Heesterbeek, Metz, 1990, Diekmann, Heesterbeek, Roberts, 2010). In the case of epidemic models where all the involved parameters are constant, determination of the basic reproductive number is straightforward. For models where the healthy and infective subpopulations are separated, the so-called next-generation method has widely been used to generate the basic reproductive number by associating it with the local stability of the disease-free equilibrium and the endemic equilibrium (Diekmann, Heesterbeek, Roberts, 2010, van den Driessche, Watmough, 2002). For models with periodic parameters, equilibria are no longer present, but periodic solutions are. Here, the determination of the true basic reproductive number becomes more challenging, since it should be associated with the local stability of the trivial and nontrivial periodic solutions (Tian and Wang, 2014). In lieu of the asymptotic analysis of nonautonomous periodic models, the local stability of periodic solutions can be understood with the aid of the Floquet theory. It addresses some conditions under which a periodic solution is locally asymptotically stable, which can bear a relation to the basic reproductive number from the autonomous counterpart or bear a definition of the true basic reproductive number.

Some epidemic models have previously been introduced together with periodic parameters, with Tian and Wang (2014), Rocha et al. (2015) or without the analysis of periodic solutions (Aguiar, Ballesteros, Kooi, Stollenwerk, 2011, Aguiar, Kooi, Rocha, Ghaffari, Stollenwerk, 2013, Andraud, Hens, Beutels, 2013). One apparent issue from those with the analysis is that their parameters are never calibrated using real field data. Some other models brought forward time-varying parameters to pronounce a perfect match between their solutions and field data, however requiring further effort to determine the prediction of the value of the optimal parameters at each time in the next time window (Götz, Altmeier, Bock, Rockenfeller, Sutimin, Wijaya, 2017, Wijaya, Magdalena, Naiborhu, 2012). In the same way, our data assimilation here is based on estimating periodic parameters in a model governed by a nonautonomous system of differential equations. A relatively small number of unobservable entities in the parameters give a much fewer degrees of freedom rather than letting them completely time-varying. Moreover, the cyclicality of the modelled phenomenon can be learned as this framework still takes the seasonality of both data and model solution into account. Unlike the usual least square method applied to several functional forms, this framework preserves the explicitness of the physical processes behind the phenomenon owing to the mathematical model, leading it to deeper understanding than just relating one to other data. Our further contribution lies heavily in combining the results from this data assimilation and asymptotic analysis results from Floquet theory to perform a simple decision-making process. We then use the final result to not only detect future outbreaks of a disease but also predict the unforeseen behavior of the incidence trajectory in the long run.