Modeling dengue outbreaks

Abstract

We introduce a dengue model (SEIR) where the human individuals are treated on an individual basis (IBM) while the mosquito population, produced by an independent model, is treated by compartments (SEI). We study the spread of epidemics by the sole action of the mosquito. Exponential, deterministic and experimental distributions for the (human) exposed period are considered in two weather scenarios, one corresponding to temperate climate and the other to tropical climate. Virus circulation, final epidemic size and duration of outbreaks are considered showing that the results present little sensitivity to the statistics followed by the exposed period provided the median of the distributions are in coincidence. Only the time between an introduced (imported) case and the appearance of the first symptomatic secondary case is sensitive to this distribution. We finally show that the IBM model introduced is precisely a realization of a compartmental model, and that at least in this case, the choice between compartmental models or IBM is only a matter of convenience.

Introduction

Dengue fever is a vector-born disease produced by a flavivirus of the family flaviviridae [1]. The main vectors of dengue are Aedes aegypti and Aedes albopictus.

The research aimed at producing dengue models for public policy use began with Newton and Reiter [2] who introduced the minimal model for dengue in the form of a set of Ordinary Differential Equations (ODE) for the human population disaggregated in susceptible, exposed, infected and recovered compartments. The mosquito population was not modeled in this early work. A different starting point was taken by Focks et al. [3], [4] that began by describing mosquito populations in a computer framework named Dynamic Table Model where later the human population (as well as the disease) was introduced [5].

Newton and Reiter’s model (NR) favours economy of resources and mathematical accessibility, in contrast, Focks’ model emphasize realism. These models represent in dengue two contrasting compromises in the standard trade-off in modeling. A third starting point has been recently added. Otero et al. (OSS) developed a dengue model [6] which includes the evolution of the mosquito population [7], [8] and is spatially explicit. This last model is somewhat in between Focks’ and NR as it is formulated as a state-dependent Poisson model with exponentially distributed times (a detailed description of our model is given in Section 2).

All modeling approaches have been further developed [9], [10], [11], [12], [13], [14], [15]. ODE models have received most of the attention. Some of the works explore: variability of vector population [9], human population [10], the effects of hypothetical vertical transmission of dengue in vectors [11], seasonality [12], age structure [13] as well as incomplete gamma distributions for the incubation and infectious times [16]. Contrasting modeling outcomes with those of real epidemics has shown the need to consider spatial heterogeneity as well [17].

The development of computing technology has made possible to produce Individual Based Models (IBM) for epidemics [18], [19]. IBM have been advocated as the most realistic models [19] since their great flexibility allows the modeler to describe disease evolution and human mobility at the individual level. When the results are only to be analysed numerically, IBM are probably the best choice. However, they are frequently presented in a most unfriendly way for mathematicians as they usually lack a formulation (expression in closed formulae) and are – at best – presented as algorithms if not just in words [18]. In contrast, working on the ODE side, it has been possible, for example, to achieve an understanding of the influence of distribution of the infectious period in epidemic modeling [20], [21], [22]. IBM have been used to study the time interval between primary and secondary cases [23] which is influenced, in the case of dengue, mainly by the extrinsic (mosquito) and intrinsic (human) incubation period.

In this work an IBM model for human population in a dengue epidemic is presented. The model is driven by mosquito populations modeled with spatial heterogeneity with the method introduced in [8] (see Section 2). The IBM model is then used to examine the actual influence of the distribution of the incubation period comparing the most relevant information produced by dengue models: dependence of the probability of dengue circulation with respect to the mosquito population and the total epidemic size. Exponential, delta (fixed times, deterministic) and experimental [24] distributions are contrasted (Section 3). The infectious period and the extrinsic incubation period is modeled using experimental data and measured transmission rates (human to mosquito) [24].

The IBM model produced is critically discussed. We show that it can be mapped exactly into a stochastic compartmental model of a novel form (see Section 4) thus crossing for the first time the valley separating IBM from compartmental models. This result opens new perspectives which we also discuss in Section 4. Finally, Section 5 presents the conclusions of this work.