Economy & Energy
Ano XII-No 71
Text for Discussion / Opinion:
Logistic Analysis of the Annual Occurrence of
Dengue Fever in Rio de Janeiro, Brazil (1985-2008)
José Israel Vargas* (firstname.lastname@example.org ),
Carlos Feu Alvim* (email@example.com),
Omar Campos Ferreira* e
Pedro Maciel Corgozinho** (firstname.lastname@example.org )
This study tries to estimate the possible onset of a dengue fever epidemic, mainly by examining the temporal variation of the disease measured by the report of its occurrence from 1986-2008 in the Brazilian state of Rio de Janeiro. We herein suggest criteria to permit an early detection and, therefore, measures to be adopted which could prevent and attenuate its spread.
Key words: Dengue fever epidemic, Rio de Janeiro, prevention.
* Economy and Energy
** CNPq-UFMG - scientific initiation
According to the World Health Organization (WHO) (1), dengue fever is the most important vector-borne viral disease. According to the WHO, 2.5 billion people live in areas at risk of contamination, and from 50 to 100 million cases of dengue fever are reported annually, while 500,000 are of the hemorrhagic type which accounts for a mortality rate of approximately 5%.
In September 2007, according to the WHO Regional Technical Assessment Group, (for Southeast Asia), the number of cases of dengue fever reported to the organization had doubled every ten years for the past fifty years. The report further warns of the possibility of an epidemic of new diseases such as Chikungunya fever, which has caused epidemics in India and Italy in recent years and which has the same vector mosquito (Aedes aegypti) as that of dengue and yellow fevers (2).
There is no vaccine or specific treatment for this disease; the only means of control or prevention is by combating the mosquito. Noteworthy is that without standing water, there are no mosquitoes and without mosquitoes, there is no dengue fever.
The disease is transmitted by the adult female, which feeds on human blood. In order to combat this disease, it is necessary to understand the behavior and temporal distribution of this mosquito. These factors depend on environmental parameters such as rainfall and temperature (cloud cover). This study has mainly examined seasonal variations of the disease with time, measured by official reports of the epidemic’s outbreaks from 1986 to 2008. The correlation between the disease and climatic conditions are briefly discussed.
As seen above, the transmission of dengue fever depends on the population of the mosquito vector, the number of persons infected, population density of the region and the meteorological factors (temperature and rainfall). When the process becomes critical, that is, such that for each person infected there is the possibility of other persons also being infected, an epidemic may occur. In this paper, progression of the disease is described by the non-linear differential equation of Volterra (3) and Lotka (4), as applied to a single species niche occupation:
dN(t)= a N(N*- N) dt (1),
dN/dt represents the number of infections in time dt; proportional to N, the number of individuals already infected; (N* -N) stands for the remaining population potentially at risk of contamination; N* represents the total population susceptible to the disease; a describes the rate of infection.
The solution of the equation (1), obtained by integration is the logistic, also called epidemiologic equation, which reads:
F/(1-F) = exp(-at+b),
where F is the fraction N/N* of the infected population, at time t.
The actual number of mosquitoes is a function of the occurrence of standing water which, in turn, depends on the intensity of rainfall and temperature, which create the appropriate conditions for the mosquito to develop. Thus, the curve describing the number of cases of dengue fever reported will, in general, accompany the seasonal rainfall.
Since propagation is cyclic, the entire susceptible population each year, N*, varies according to the conditions at the beginning of the cycle as well as on the vector concentration, the remaining number of infected persons and the variable dynamics of rainfall and temperature throughout the year. In this study, we emphasize the importance of the number of infected persons remaining during the months of lower incidence as an indicator of possible epidemics.
In this study, we have tried to analytically describe a future profile for epidemics in the state of Rio de Janeiro. The procedure adopted consists of an attempt to estimate at the very beginning of the rainy season, a possible onset of a dengue fever epidemic, in order to adopt measures of prevention and possible attenuation of its effects.
Examination of this subject is based on data available at the Rio de Janeiro Health Department, which covers not only the number of classic cases of dengue fever reported monthly, but also of the hemorrhagic form and the annual lethality of the disease. The tables obtained from this reference are found in Annex 1.
Information referring to average monthly temperatures as well as for precipitation, observed for Rio de Janeiro State as of 1985, are found in Annex II. We are grateful to Dr. Antonio Divino Moura, director of the Instituto Nacional de Meteorologia,for providing this data.
The occurrence of dengue fever has varied greatly over the years. The worst epidemics occurred in 2002 (228,000 cases reported until September of that year), and again in 2008 (260,000 reported until the same month of that year). Figure 1 shows the evolution of occurrences as of 1986. The monthly averages representative curve (Figure 2) reveals maximum intensity around the month of March. Months of lesser occurrence are September and October, the beginning of the rainy season (still under the influence, in epidemic terms, of the eventual remaining previous year’s infection).
Reports of dengue fever in the state of Rio de Janeiro
(1986 - 2008 monthly averages)
Figure 2 shows the corresponding curves of the epidemic’s greatest occurrence. Monthly variations may be observed, which seem to be related to the seasonal rainfall as well as to the average temperatures observed for that period (see figure 8).
This data allows a representative curve of the total average monthly variations, using the occurrences for each month, for an annual period to be drawn. The numbers thus obtained, as well as the results of the adjustment with the logistic equation, are shown in Figures 3a and 3b. Figure 3a represents equation 1 and, hence, the average disease intensity. A good agreement may be observed between the average numbers in the reference tables, and the adjusted values obtained with the logistic function (figure 3a)
This calculated function is obtainable via the Fischer and Pry (5) representation ( the logarithm of the logistic equation 2 ), shown in Figure 3b. The adjustment made allowed us to describe the infection’s occurrences for periods between October of the previous year to September of the current year under consideration. This time interval was used for the integration of the composite disease intensity curve (Figure 3a).
Estes dados permitem traçar curva representativa do total das variações mensais médias, ao somar as ocorrências observadas para cada mês, nos períodos anuais. Os dados correspondentes aos anos de 1986 a 2008 são, como se viu, disponíveis. Os valores assim obtidos, bem como os resultantes de ajuste logístico (solução da equação diferencial 1) são mostrados nas Figuras 3a e 3b. Pode-se observar boa concordância entre os valores médios observados a partir das tabelas de referência, com aqueles ajustados através da função logística.
The Fischer Pry representation (5) (adjustment of ln (f/(1-f)) where f is the fraction of events relative to the total in the period, that is, N/N*) - that permits the linearization of the logistic function - is shown in Figure 4. The adjustment of the line obtained, using the values represented above, permits to determine the coefficients of the original logistic curve l, Y= 1/(1+exp-(ax+b)), that well represents the adjusted equation (Figura 3b).
À vista do sucesso do ajuste realizado, adotaram-se para exame os períodos transcorridos entre outubro do ano anterior e setembro do ano em curso. Este período foi utilizado para a integração da função logística (Figura 3a).
Note: the logistic function, or epidemiology, is the solution of the Volterra-Lotka equation, frequently used by environmental biologists to describe the Darwinian competition among living organisms
The very nature of this phenomenon, for which the simultaneous presence of mosquitoes and persons infected is necessary, leads to the conclusion that the occurrence of reported cases for the dry months (September and October) should exercise a major influence over the progression of the epidemic in the following months, and serve to predict the disease’s outbreak. In the absence of occurrences of cases of this disease during the first months of the rainy season (September and October), the observation of a future outbreak might depend upon the arrival of a large number of contaminated persons from other geographical areas to the Rio de Janeiro region. This hypothesis is, however, very improbable and hardly would explain the (unexpected) outbreak of an epidemic.
It is noteworthy that Aedes aegypti is also the vector for yellow fever, but there have been no great occurrences of this disease, probably due to the absence of a significant number of infected persons, and thanks to programs of vaccination. It is now essential to maintain strict control over programs of vaccination as well as of the arrival of infected persons from regions affected by this disease.
Figure 4: Adjustment in the logistic curve using the Fischer Pry representation
Considering again the problematic of dengue fever, it becomes clear that the distribution of reports over the year, together with verification of the cases of contamination in the months during which it’s incidence is usually lowest (August and September), a prediction of the probability of occurrence of an outbreak over the months of the coming rainy season becomes possible. A correlation between confirmed reports of dengue fever during the months of least incidence (the last dry month and the first month of the rainy season) and the total number of annual reports of the disease was sought.
It thus becomes clear that ideally, evidence of a possible epidemic should be sought right at the beginning of the rainy season. The procedure we herein propose consists of taking the reports for September and October as our basis. In Figure 5, corresponding numbers are represented, as well as the reports for the total period of the two months (September and October), for which pertinent data is shown on the right side of Figure 5, on a scale 100-fold smaller for annual reports of this disease.
Figure 5: Reports for September and October and Totals
Examination of the results shows that the use of this criterion permits early detection of risks for major outbreaks of future epidemics, establishing, as a result, a threshold of 400 reported cases of the disease during the months of September and October for the previous year. This criterion would have allowed the prediction of the outbreaks for the years 1987, 1991, 2002, 2007 and 2008 (excepting the years 1992 and 2003).
It is noteworthy that 1992 and 2003 were followed by important outbreaks of the disease. Since the month of September frequently coincides with the beginning of the previous year’s rainy cycles, it seems natural that it might be contaminated by this occurrence. In this case, an assessment of the events of the following months may be useful in evaluating the anticipated behavior.
It must be noted, however, that even in years during which the number of cases of dengue fever are not significant (2003, 2004 and 2005), the number of reported cases for the two months of the dry season has been systematically rising. Thus, there is an increasing probability for outbreaks of this epidemic.
It therefore seems obvious that the authorities as well as the general population should take steps to reduce the number of cases by reducing the number of transmitters, combating them even during the dry season months, to break the epidemic cycle.
With relation to the behavior of the disease in 2009, it is essential to evaluate the dates referring to September and October of 2008. The total result, because of diagnostic difficulties or delays in data, varies significantly within the statistics published for the following months. This total for the two months, calculated until January 2009, was 1525, indicating a considerable probability of an emergent epidemic for the current year. It was noted that in December 2008, a mere 633 cases had been reported, which indicates the high probability of an outbreak.
Figure 6 shows the data representing the annual cycle (September and October) as a function of the data calculated for these two months (September and October for the previous year). A provisory projection for 2009 would be 200,000 reported cases of dengue fever. Obviously, there are factors, such as sudden drop in temperature or lack of rainfall, which might interrupt this process. The data collected so far, however, merits maximum attention.
The correlation between the recurrence of reported cases of dengue fever with the occurrence of greater periods of rainfall and higher temperatures is a well established phenomenon and influences the behavior of mosquito population and consequent infection of people. The average meteorological behavior is shown in Figures 7 and 8.
As for the increase in the number of reported cases of the disease during periods of minimum incidence, it is possible that this occurs either because of greater efficiency of sanitation officials for the period in question, or periods of rising temperatures, which afford greater emergence of the transmitter’s eggs. Figures 8 and 9 show the qualitative relationship between the occurrence of the disease and environmental variations.
Figure 6: Annual reports in function of those for September and October
Figure 7: Average variability of reported cases with rainfall and temperature
Figure 8: Average temperatures for the months of October through April for the following year; total number of reports of classic dengue fever and hemorrhagic dengue fever
Observation of figures 7, 8 and 9 suggests that the correlations examined agree qualitatively, while they do not allow us to establish evident relations for average data of reports of the disease with observed average temperature and rainfall. However, as observed in Figure 7, there is an annual cycle with discrepancy of a few months between average atmospheric temperature and rainfall.
In the hypothesis that a trigger mechanism, evidenced by the number of reported cases for the initial months of the rainy season, (September and October) is confirmed and, in the absence of efficient action both by public authorities and by the society in general for that period, it is likely that the state of Rio de Janeiro will be victim of a new dengue epidemic this year. For the future, the preventive measures suggested should begin right at the beginning of the dengue cycle in the months mentioned above. As recommended by the WHO (2), it will be paramount to be more efficient in informing the public to take the necessary measures to avoid an outbreak.
The analytical instruments used in this study should be applied in dealing with occurrences in other regions of Brazil.
Pedro Maciel is grateful to Universidade Federal de Minas Gerais (UFMG) and the CNPq for providing a scholarship and to Dr. José Israel Vargas for supervision.
The authors thank
Dr. M. Carvalho Dias, Director of CPETEC for the meteorological data provided.
ANNEX I - Table A1-
Distribution of reported cases of dengue in Rio de Janeiro State
Table A2 – Distribution of cases of hemorrhagic dengue – Rio de Janeiro State
Data from Secretaria Estadual de Saúde do Rio de Janeiro, January 2009,
Table A3 –
Deaths from hemorrhagic (HD) and other forms of dengue – Rio de Janeiro State
* 2007 and 2008: data from SINAN-RJ subject to revision and update 22/01/2009,
ANNEX II - Table A4 –
Average monthly rainfall (mm) – Rio de Janeiro State
Note: This table was obtained from data averages referring to 18 rainy seasons in Rio de Janeiro State – data bank, Delaware University, provided by the National Institute of Meterorology,
Table A5 – Average monthly temperature for Rio de Janeiro State (°C)
Note: This table was obtained from data averages referring to 18 rainy seasons in Rio de Janeiro State – data bank, Delaware University, provided by the National Institute of Meteorology
- Mariane Hopp, Development of a Dengue Alert System, Lecture at International Institute of Climate Research, University of Columbia, December 2001,
- From the University of Delaware date bank
 - References for the model used in this study are available in the article at:
Graphic Edition/Edição Gráfica:
Monday, 31 October 2011.