**ARTIGOS ORIGINAIS **ORIGINAL ARTICLES

**Imported and autochthonous cases in the dynamics of dengue epidemics in Brazil**

**Casos importados e autóctones na dinâmica da epidemia de dengue no Brasil**

**Casos importados y autóctonos en la dinámica de epidemias de dengue en Brasil**

**Nicolas Degallier; Charly Favier; Jean-Philippe Boulanger; Christophe Menkes**

Institut de Recherches pour le Développement. Paris, France

**ABSTRACT**

**OBJECTIVE:** To estimate the basic reproduction number (R_{0}) of dengue fever including both imported and autochthonous cases. **METHODS:** The study was conducted based on epidemiological data of the 2003 dengue epidemic in Brasília, Brazil. The basic reproduction number is estimated from the epidemic curve, fitting linearly the increase of initial cases. Aiming at simulating an epidemic with both autochthonous and imported cases, a "susceptible-infectious-resistant" compartmental model was designed, in which the imported cases were considered as an external forcing. The ratio between R_{0} of imported versus autochthonous cases was used as an estimator of real R_{0}. **RESULTS:** The comparison of both reproduction numbers (only autochthonous versus all cases) showed that considering all cases as autochthonous yielded a R_{0} above one, although the real R_{0} was below one. The same results were seen when the method was applied on simulated epidemics with fixed R_{0}. This method was also compared to some previous proposed methods by other authors and showed that the latter underestimated R_{0} values. **CONCLUSIONS:** It was shown that the inclusion of both imported and autochthonous cases is crucial for the modeling of the epidemic dynamics, and thus provides critical information for decision makers in charge of prevention and control of this disease.

**Descriptors:** Dengue, epidemiology. Flavivirus Infections, transmission. Risk. Communicable Period. Disease Outbreaks. Epidemiologic Surveillance.

**RESUMO**

**OBJETIVO:** Estimar o número básico de reprodução da dengue (R_{0}), com base nos casos importados, além dos casos autóctones. **MÉTODOS:** O estudo foi feito sobre dados epidemiológicos da epidemia de dengue em Brasília, 2003. O número básico de reprodução é determinado a partir da curva epidêmica, ajustando uma reta ao crescimento inicial do número de casos. Para simular uma epidemia com casos autóctones e importados, foi criado um modelo compartimentado do tipo "suscetíveis-infectados-resistentes". O R_{0} real foi estimado pela fração entre R_{0} dos casos autóctones e dos importados. **RESULTADOS:** A comparação de ambos valores de reprodução (apenas autóctones versus todos os casos) mostrou que considerando todos casos como autóctones, o valor de R_{0} foi superior a um, enquanto o R_{0} real era inferior a um. O mesmo resultado foi obtido com o conjunto de dados simulando uma epidemia com R_{0} fixo. O método foi também comparado a outros, observando-se que estes últimos subestimaram os valores do R_{0}. **CONCLUSÕES:** A inclusão de tanto casos autóctones como os importados é essencial para modelar a dinâmica da epidemia, possibilitando informação crítica aos tomadores de decisão, responsáveis pelo controle da doença.

**Descritores:** Dengue, epidemiologia. Infecções por Flavivirus, transmissão. Risco. Período de Transmissibilidade. Surtos de Doenças. Vigilância Epidemológica.

**RESUMEN**

**OBJETIVO:** Estimar el número de reproducción básica (R0) de la fiebre del dengue incluyendo casos importados y autóctonos. **MÉTODOS:** El estudio fue realizado basándose en datos epidemiológicos de la epidemia del dengue ocurrida en Brasilia, Districto Federal de Brasil, en el 2003. El número de reproducción básica es estimado de la curva epidémica, fijando el incremento lineal de los casos iniciales. Señalando casos importados y autóctonos en una simulación epidémica, fue diseñado un compartimiento "infeccioso-susceptible-resistente", en el cual los casos importados fueron considerados una fuerza externa. La tasa entre R0 de casos importados versus casos autóctonos fue usado como una estimación real de R0. **RESULTADOS:** La comparación de ambos números de reproducción (sólo autóctonos versus todos los casos) mostró que considerando todos los casos como autóctonos produjo un R0 por encima de uno, a pesar de que el valor real de R0 era menor que uno. Los mismos resultados fueron obtenidos cuando se aplicó el método a epidemias simuladas con valor fijo de R0. Este método fue también comparado con métodos propuestos anteriormente por otros autores y mostró los mismos valores subestimados de R0. **CONCLUSIONES:** Fue demostrado que la inclusión de casos importados y autóctonos es crucial para el modelaje de dinámicas epidémicas, y que provee información crítica para aquellos que participan en la toma de decisiones en la prevención y control de esta enfermedad.

**Descriptores:** Dengue, epidemiología. Infecciones por Flavivirus, transmisión. Riesgo. Periodo de Transmisión. Brotes de Enfermedades. Vigilancia Epidemiológica.

**INTRODUCTION**

Dengue fever has been endemic in Brazil since 1986.^{5} Its transmission involves only urban mosquitoes^{3} and humans, and many fatal cases occur every year due to epidemics where different serotypes are transmitted.^{8}

The quick expansion of vectors and viruses throughout nearly all Brazilian states is facilitated by ever increasing moving and traveling of people in the country. Until recent years in most localities the origin of dengue cases was neither reported nor investigated. To the authors' best knowledge, no study has been published investigating the impact of "imported" cases on the dynamics of the epidemics. The origin of each case in the 2002-2003 epidemics in Brasília was inferred after a detailed survey of the patient's background, taking into account the duration of the incubation phase of the disease. As these data about the most probable origin of these cases were available, we investigated its possible impact on the dynamics of dengue epidemics. We compared its dynamics with the outputs of a simple "susceptible-infectious-resistant" (SIR) model modified in order to include the input of daily numbers of imported cases when estimating the basic reproduction number (R_{0}).

Mathematical modeling is a powerful tool to simulate real past situations, to input data from the present and hopefully to forecast future outcomes.^{6} It has been widely used to understand the dynamics of infectious diseases.^{1} A commonly used family of epidemiological models are the compartmental or "box ones": individuals are grouped in classes according to their immunological status (e.g. SIR models consider the progress in the numbers of susceptible, infectious, and removed immune people).^{1} The basic reproduction number (R_{0}) is an essential parameter to describe the dynamics of an outbreak. It is the number of secondary infections generated by one case when the virus is introduced into a wholly susceptible population and when the probability of contact with the pathogen is homogeneous.^{7}

The objective of the present study was to estimate the basic reproduction number (R_{0}) of dengue fever including both imported and autochthonous cases.

**METHODS**

Data analyzed refer to the dengue epidemics occurred during 2002-2003 in Brasília, Brazilian Federal District (Figure 1). Brasília is a city situated in Central-West Brazil and has a rather recent history of contact with dengue.^{4}

When the number of imported cases is disregarded, the basic reproduction number is generally estimated from the epidemiological curve. At least four different methods for this estimation have been proposed. According to the classical formula,^{1,13,15 } where *m* is the relative density of mosquitoes to humans, *a* is the daily biting rate of the mosquitoes, *b* and *c* are the probabilities of viral transmission from an infected mosquito to a susceptible human, and from an infected human to a mosquito, respectively, *µ* is the daily mortality rate of the vector,τ* _{e}* the extrinsic incubation period, and γ the inverse of the duration of viremia. In the present study, we modified the method used by Massad et al

^{13}(2001) to estimate the basic reproduction number.

^{2}Considering that the initial growth of the number of cases is exponential with parameter , with τ

*the intrinsic incubation period and the other parameters as above. Then, λis estimated by a linear least squares fitting of the initial part of the curve of the cumulated number of cases against the daily number of new cases. The number of cases, which was taken into account for the regression of the increase at the beginning of each outbreak (slope λ in Fig. 2 A-D), was chosen to ensure a statistical significance of*

_{i}*p*<10

^{-4}.

A simple SIR model is used to characterize the progress of local (autochthonous) cases with continuous influx of exogenous cases (imported). We set S_{A}, I_{A} and N_{A} as susceptible, infectious and cumulative number of autochthonous cases, N as the total susceptible population and I_{I} and N_{I} infectious and cumulative numbers of imported cases. The progress of N_{I} is thus considered as an external forcing.

The parameter η is introduced to represent the possible differences between the durations of exposure times of autochthous and imported hosts, the latter arriving at various times of viremia. The basic reproductive rate of autochthonous cases is while for imported cases is is thus the ratio between the basic reproductive rate of imported and autochthonous cases. Since the final number of autochthonous cases is far lesser than the whole population, the total number of people can be assumed very large (here N = 100,000) and, as we only consider the initial model behavior, results are independent of N provided it is large enough. The other two parameters are computed by a fit of the ratio between the cumulative autochthonous cases over the cumulative imported cases in the Nelder-Mead simplex algorithm pre-implemented using MATLAB. Using this index (the ratio between autochthonous and imported cumulative cases) for the fit has yielded results that were more robust to the initial guess than the cumulative number of autochthonous cases. Bootstrap methods were used to estimate confidence intervals and R_{0}s values were compared to R_{0} estimated as if all cases were autochthonous.

**RESULTS**

In the case of the 2003 dengue epidemic in Brasília, the imported cases were included, and the modified model fitted when only autochthonous cases were considered (Figure 3B), as well when autochthonous cases were divided by the daily numbers of imported cases (Figure 3A). The comparison of R_{0} estimated using the method above (when the number of imported cases is unknown or disregarded) with R_{0} estimated when exogenous cases are explicitly included is striking. In the first case, R_{0} is far greater than one, indicating eco-epidemiological conditions favorable to local transmission of the virus. In the second case, the local R_{0} (estimated for autochthonous transmission) is clearly lower than one. The results of the comparison between the three methods are shown in Table 1, for different epidemics and for a SIR-simulated one with pre-defined R_{0} = 8.0. Besides great variability between the epidemics, the R_{0} values estimated by the method here described are always greater than those obtained using other methods. Furthermore, this method is the only one that provides a correct estimation of the R_{0} of the SIR-modeled epidemic. Figure 4 shows an almost linear relationship between R_{0aut} and R_{0imp}, when the former is set under one. Thus, under such assumptions, it becomes evident that in case of an epidemic, the reproduction number due to imported cases cannot be above 0.45 and that if the reproduction number due to autochthonous cases is near one, the latter becomes negligible.

**DISCUSSION**

The methods used by Marques et al^{12} (1994) and Massad et al^{13} (2001) systematically underestimated R_{0}, probably because they failed to consider the whole delay between two successive cases (viremia, and intrinsic and extrinsic incubation periods). The differences between the R_{0} values obtained for different epidemics (Table 1) reflect various eco-epidemiological conditions influencing hidden parameters (such as mosquito density, effective exposure rates between hosts and vectors^{10}).

Except for the viremic period in infected people, these parameters vary with climatic factors, mainly temperature and relative humidity, which affect vector survival and virus extrinsic cycle. Other factors may influence R_{0} but are generally not addressed due to unavailability of relevant data such as the degree of urbanization and social structure,^{11} transmission history and/or immune status of human population. The human population is no more fully susceptible in most Brazilian cities, and the case of Belém (Northern Brazil) was an exception since the 1997-1998 epidemic was the first in this city.^{16} In the cases of Brasília and Fortaleza (Northeastern Brazil) epidemics, where dengue transmission is known since 1997 and 1986, respectively,^{4,a} our R_{0} values are thus probably underestimated. As illustrated by Keeling & Grenfell^{9} (2000), assumptions about case distribution are rarely met in natural situations, and the assumption of homogeneity of contact is also generally not met, as it has been previously shown that cases are often aggregated inside and around the houses of the first cases. This may also contribute to underestimating R_{0} values. An unknown proportion of cases remain undetected as they are asymptomatic or are diagnosed as other fever-causing diseases. This fact makes the determination of the number of cases uncertain but not the estimation of R_{0} if we postulate that it is independent of the intensity of symptoms. A similar, or even more serious difficulty is the number of uncharacterized imported cases, which may negatively affect the estimation of R_{0} and thus invalidate any decision about prevention and control. This clearly implies that dengue epidemics may occur even when local ecological conditions are unfavorable, as was probably the case in Brasília in 2003. This is the first report emphasizing this aspect. Taking into account the imported *vs*. autochthonous nature of each diagnosed case will allow a better evaluation of the risk of dengue transmission or epidemic. Health authorities involved with dengue monitoring and prevention should develop and strengthen their methods to determine the imported or autochthonous origin of the cases. Future studies should incorporate these data in models, along with the yet missing entomological field data. As the populations of arthropod vectors are much influenced by climatic factors, it may thus be possible to link the dynamics of outbreaks to climatic factors through the underlying R_{0} and vectorial capacity.

**ACKNOWLEDGEMENTS**

Cristiane Oliveira of Diretoria de Vigilância Ambiental de Brasília, Brazil; José Rubens Costa Lima of Célula de Vigilância Epidemiológica de Fortaleza, Brazil; and Bernard Mondet of Institut de Recherche pour le Développement, Montpellier, France provided the data on epidemics of dengue from Brasília, Fortaleza and Belém, respectively.

**REFERENCES**

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13. Massad E, Coutinho FA, Burattini MN, Lopez LF. The risk of yellow fever in a dengue infested area. *Trans R Soc Trop Med Hyg.* 2001;95(4):370-4. DOI: 10.1016/S0035-9203(01)90184-1

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15. Nájera JA. A critical review of the field application of a mathematical model of malaria eradication. *Bull World Health Organ.* 1974;50(5):449-57.

16. Travassos da Rosa JFS, Mahagama AK, Pinto E, Magalhães MTF. Epidemia de dengue na grande Belém: aspectos conceituais e abordagem clínico-epidemiológica, no ano de 1997. *Rev Soc Bras Med Trop.* 1998;31(Supl):130.

** **** Correspondence:**

Nicolas Degallier

Institut de Recherches pour le Développement

LOCEAN-IPSL UPMC 4 pl. Jussieu case 100

T.45-55, 4eme étage

Faculté des Sciences, Paris 6

75252 Paris, France

E-mail: nicolas.degallier@ird.fr

Received: 3/11/2008

Revised: 7/21/2008

Accepted: 9/16/2008

Study financially supported by IRD-UMR182 (France), and National Health Foundation - Ministry of Health (Brazil). C. Favier was supported by the Project Modélisation des Arboviroses Tropicales Emergentes CLImato-Dépendantes (MATECLID; post graduate grant, Process APR GICC 2002).

Presented as poster at the Entomological Society of America Annual Meeting and Exhibition, held in Salt Lake City, Utah, USA, in November 17th, 2004.

a Degallier N, Hervé J-P, Travassos da Rosa APA, Travassos da Rosa ES, Vasconcelos PFC, Monteiro HAO, Sá Filho G, Travassos da Rosa JFS. Entomological studies on dengue fever vectors in Brazil: the epidemics of Boa Vista, Roraima, 1982, Niterói, Rio de Janeiro, 1986, and Ceará State, 1986, 1994. In: Travassos da Rosa APA, Vasconcelos PFC, Travassos da Rosa JFS. *An overview of arbovirology in Brazil and neighbouring countries.* Belem, Pará: Instituto Evandro Chagas; 1998. p.261-71.