Stochastic eco-epidemiological model of dengue disease transmission by Aedes aegypti mosquito

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Abstract

We present a stochastic dynamical model for the transmission of dengue that takes into account seasonal and spatial dynamics of the vector Aedes aegypti. It describes disease dynamics triggered by the arrival of infected people in a city. We show that the probability of an epidemic outbreak depends on seasonal variation in temperature and on the availability of breeding sites. We also show that the arrival date of an infected human in a susceptible population dramatically affects the distribution of the final size of epidemics and that early outbreaks have a low probability. However, early outbreaks are likely to produce large epidemics because they have a longer time to evolve before the winter extinction of vectors. Our model could be used to estimate the risk and final size of epidemic outbreaks in regions with seasonal climatic variations.

Introduction

Arboviruses is a shortened name given to arthropod-borne viruses from various families which are transmitted by arthropods. Some Arboviruses are able to cause re-emergent diseases such as St. Louis Encephalitis, Chikungunya, Dengue, Ross River disease, West Nile, Yellow Fever, Equine Encephalitis, etc. [1]. Arthropods are able to transmit the virus upon biting the host, allowing the virus to enter the host’s bloodstream. The virus replicates in the vector but usually does not harm it. In the mosquito-borne diseases, the virus establishes a persistent infection in the mosquito salivary glands and there is sufficient virus in the saliva to infect another host during feeding. Each arbovirus usually grows only in a limited number of mosquito species. The work presented in this article is focused on mosquito-borne diseases (mainly dengue fever) transmitted by Aedes aegypti. This is one of the most efficient mosquito vectors for arboviruses, because it is highly anthropophilic, thrives in close proximity to humans and often lives indoors.

Dengue is spread only by adult females, which require blood to complete oogenesis. During the blood meal the female ingests dengue viruses from an infectious human. The viruses develop within the mosquito and are re-injected in later blood meals into the blood stream of susceptible humans. Dengue is an acute febrile viral disease (with four serotypes of flaviviruses DEN1,DEN2,DEN3 and DEN4) which presents headaches, bone, joint and muscular pains, rash and leukopenia as symptoms. Dengue epidemics were reported throughout the 19th and 20th centuries in the Americas, southern Europe, northern Africa, the eastern Mediterranean, Asia, Australia and on various islands in the Indian Ocean, Central Pacific and Caribbean [2].

The history of dengue in Argentina began as early as in 1916 when an epidemic affected the cities of Concordia and Paraná. In 1947 the Pan American Health Organization (PHO) led a continental mosquito eradication program and by 1967 the mosquito was considered to be eradicated in Argentina. The mosquito was detected again in 1986 and since 1997 several epidemic outbreaks took place in the northwestern and northeastern regions of the country. A brief history of dengue epidemics in Argentina is found in Appendix A.

Nowadays A. aegypti is a permanent inhabitant of the city of Buenos Aires [3], [4], [5]. Every summer there is a potential risk of dengue virus transmission because of the arrival of viremic people from Bolivia, Paraguay, Brazil and other endemic countries. However, no autochthonous cases of the disease have been detected until present [5], but in the last years some clinical studies confirmed dengue infection in people arriving from neighboring endemic countries [6]. Therefore, the development of mathematical models which permit the estimation of the probability of an epidemic outbreak and its final size has become a matter of sanitary necessity.

The first model of dengue was performed by Newton and Reiter in 1992 [7]. They developed a deterministic model in which the populations of hosts and vectors were divided into subpopulations representing disease status and the flow between subpopulations was described by differential equations. Several deterministic models have been developed taking into account different possible aspects of the disease: constant human population and variable vector population [8], variable human population size [9], vertical and mechanical transmission in the vector population [10], seasonally varying parameters and presence of two simultaneous dengue serotypes [11], age structure in the human population [12] and presence of two serotypes of dengue at separated intervals of time [13]. In addition, in 2006 Tran and Raffy proposed a spatial and temporal dynamical model based on diffusion equations using remote-sensing data [14].

There are also other approaches. Focks et al. developed a stochastic model that describes the daily dynamics of dengue virus transmission in an urban environment based on the simulation of a human population growing in response to country- and age-specific birth and death rates [15]. Santos et al. developed a periodically forced two-dimensional cellular automata model that describes complex features of the disease taking into account external seasonality (rainfall) and human and mosquito mobility [16].

Our proposal in this article is the third in a series of minimalist stochastic models. The first describes the seasonal dynamics of A. aegypti populations in a homogeneous environment [17]. The second one describes the A. aegypti dispersal driven by the availability of oviposition sites in an urban environment [18]. This new model takes into account the seasonal and spatial dynamics of the vectors and describes the disease dynamics triggered by the arrival of viremic people in a city.

Our main goal is the development of a mathematical tool that allows the study of different epidemic scenarios in an urban environment, the estimation of the epidemic risk and the study of the growth and final size of an epidemic outbreak due to the spatial dynamics of the vector. A particular aim of the work is the estimation of dengue epidemic risk in the city of Buenos Aires, Argentina.

Populations of hosts (Humans) and vectors (A. aegypti) were divided into subpopulations representing disease status: susceptible (S), exposed (E) and infectious (I) for adult female vectors, and susceptible (S), exposed (E), infectious (I) and removed (R) for the human population. The population of adult male mosquitoes is not taken into account explicitly and we consider that, on average, one half of the emerging adults are females [19]. Three kinds of females were considered: adult females in their first gonotrophic cycle (A1 females), females in subsequent gonotrophic cycles (A2 females) and flyers (F), which are the adult females that have already finished their gonotrophic cycles and fly in order to deposit their eggs.

The following sections will describe the populations and events of the stochastic transmission model (Section 2), the mathematical description of the stochastic model (Section 3), the parameters, initial values and boundary conditions (Section 4), results and discussion (Section 5), the transcription of the dengue model into a yellow fever model, the choice of dengue parameters as well as some minimal computations in the validation direction (Section 6), and finally, summary and conclusions (Section 7).

Section snippets

Populations of the stochastic process

We consider a two-dimensional space as a mesh of squared patches where the dynamics of the immature stages of the mosquito and the evolution of the disease take place, and where only Flyers can fly from one patch to the next according to a diffusion-like process. We take into account that during the gonotrophic cycles the mosquito dispersal is negligible, and once the gonotrophic cycle ends the female begins to fly, becoming a Flyer in search of oviposition sites. A detailed explanation of the

Mathematical description of the stochastic model

The evolution of the subpopulations is modelled by a state-dependent Poisson process [25], [26] where the probability of the state:(E,L,P,A1,A2s,A2e,A2i,Fs,Fe,Fi,Hs,He,Hi.Hr)(i,j)evolves in time following a Kolmogorov forward equation that can be constructed directly from the information collected in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and in Eq. (4). Fig. 1 shows the subpopulations of vector populations and the events which affect these populations collected in Table 1, Table 2

Parameters related to mosquito biology

The description of the development of the transition rates and the choice of the model parameters related to mosquito biology and dispersal such as mortality of eggs (me), hatching rate (elr), mortality of larvae (ml), density-dependent mortality of larvae (α), pupation rate (lpr), mortality of pupae (mp), pupae into adults development coefficient (par), emergence factor (ef), mortality of adults (ma), gonotrophic cycle coefficients (cycle1, cycle2) for adult females in stages A1 and A2,

Effect of the date of arrival of one exposed human in the final size of the epidemics

Fig. 3 shows the frequency of the final size of epidemics as a function of the date of arrival of one exposed human in the susceptible human population. By final size of epidemics we understand the total number of susceptible humans who were infected during the epidemic outbreak. We performed the simulations using a grid with 13 × 13 patches and with a density of breeding sites of 200 BS/patch. We started with 100 susceptible humans and 10,000 mosquito eggs in every patch July 1st and we ran the

Dengue and yellow fever

Dengue and yellow fever are two kinds of encephalitis that produce hemorrhagic fever. At the level of description explored in the present work they are not distinguishable, except perhaps for different characteristic times of the clinical phase and the extrinsic cycle of the virus.

From a clinical point of view, the main difference between dengue and yellow fever is the mortality of the toxic period. In both diseases, fever takes a saddle back pattern, with fever dropping or disappearing during

Summary and conclusions

We have developed a stochastic dynamical model for the transmission of dengue that takes into account the seasonal and spatial dynamics of the vectors and describes the disease dynamics triggered by the arrival of infected people in the city and modulated by the seasonal and the spatial vector dynamics.

The model takes into account the populations of both hosts (Humans) and vectors (A. aegypti), which are divided into subpopulations representing the disease status: susceptible (S), exposed (E)

Acknowledgements

The authors acknowledge CONICET and the support given by the University of Buenos Aires under Grant X308 (2004–2007), X210 (2008–2010) and by the Agencia Nacional de Promoción Cientf´ica y Tecnológica (Argentina) under Grants PICTR 87/2002 and PICT 00932/2006.

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