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December 2006 - January 2007 
ISSN 1518-2932

 

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Logistic Analysis of the Annual Occurrence of Dengue Fever in Rio de Janeiro, Brazil (1985-2008)

Economical Zoning of Hydrographic Basins Territories - Ecological Importance

 

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Text for Discussion / Opinion:

Logistic Analysis of the Annual Occurrence of

 Dengue Fever in Rio de Janeiro, Brazil (1985-2008)

José Israel Vargas* (jivargas@globo.com ),

Carlos Feu Alvim* (feu@ecen.com),

 Omar Campos Ferreira* e

 Pedro Maciel Corgozinho** (maciel.c@hotmail.com )

Abstract

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.

 Institutions

* Economy and Energy

** CNPq-UFMG - scientific initiation

 1.  Introduction:

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.[1]

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.

 2.  Discussion

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[2],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 1

  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).

 

Figure 2

 

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)[3].

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).

 

Figure 3a

Figure 3b

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.

 Caixa de texto: Note added in August 2009: It seems that the wrong prevision in this article of an epidemic outbreak in 2008/2009 can also be due to this type of contamination from the previous season.

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

 Figure 9 

 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.  

3. Conclusion

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.  

Acknowledgments

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.

References

1 – World Health Organization (2009) accessed 19/01/2009.

http://www.who.int/mediacentre/factsheets/fs117/en/

2 - World Health Organization (2008) First Meeting of Regional Technical Advisory Group (RTAG) on Dengue SEA-DEN-6 accessed 19/01/2009
http://203.90.70.117/PDS_DOCS/B3132.pdf

3 – Volterra, V. (1931). Leçon sur la Theorie Mathematique de la lute pour la Vie. Paris: Gauthier - Vilars

4 - Lotka, A. J. (1925). Elements of physical Biology. Baltimore M. D.: Williams & Wilkins Co.

5 - Fisher, J. C., & Pry, R. (1971). Simple substitution model for technology change. Technological Forecasting and Social Change, 3, №1, 75-88.

 ANNEX I -  Table A1-

  Distribution of reported cases of dengue in Rio de Janeiro State

Year/Month

Jan

Feb

Mar

Apr

Mai

Jun

1986

-

-

-

1,603

13,258

8,594

1987

14,074

16,422

15,217

8,182

3,696

1,129

1988

302

205

297

143

116

86

1989

123

99

93

213

160

88

1990

55

44

104

1,140

2,423

2,383

1991

34,636

20,036

14,783

9,903

3,565

1,290

1992

437

357

244

91

90

80

1993

62

50

300

43

18

8

1994

28

23

77

14

20

23

1995

586

7,817

16,442

7,743

2,235

124

1996

499

744

3,673

6,873

2,866

738

1997

308

370

289

158

100

75

1998

1,554

1,917

8,272

11,359

2,950

528

1999

667

835

1,770

2,437

1,459

319

2000

289

318

612

923

1,003

392

2001

3,055

7,435

11,086

19,287

20,596

8,530

2002

49,280

93,016

99,861

31,642

7,208

1,672

2003

1,797

2,390

1,755

1,000

545

263

2004

508

341

443

211

132

76

2005

358

256

218

167

126

157

2006

1,920

4,515

8,036

7,685

4,153

1,355

2007*

3,229

6,261

14,225

15,019

11,050

4,452

2008*

17,647

25,416

86,502

94,046

25,157

4,609

 

 

Jul

Aug

Sept

Oct

Nov

Dec

Total

1986

3,611

793

246

181

426

3,795

32,507

1987

351

58

22

21

74

109

59,355

1988

31

77

48

20

67

58

1,450

1989

48

19

13

23

83

182

1,144

1990

2,341

779

406

399

2,798

6,813

19,685

1991

403

217

249

191

242

376

85,891

1992

47

38

39

66

91

78

1,658

1993

9

17

24

20

34

38

623

1994

34

20

16

23

6

3

287

1995

85

65

21

20

53

49

35,240

1996

203

82

75

53

127

292

16,225

1997

43

37

47

28

67

782

2,304

1998

154

85

55

57

122

5,329

32,382

1999

137

113

96

83

63

1,104

9,083

2000

199

150

102

95

96

102

4,281

2001

3,386

1,303

657

931

926

3,023

80,215

2002

985

502

272

522

1,522

1,763

288,245

2003

140

100

104

159

415

574

9,242

2004

84

112

128

159

192

308

2,694

2005

115

191

146

189

267

390

2,580

2006

626

841

326

337

423

837

31,054

2007*

2,293

1,085

847

1,262

2,777

4,053

66,553

2008*

1,514

890

641

884

1251

865

259,422

SOURCE: SESDEC-RJ/SAS/SVS/CVE/DTI/SDTVZ

Table A2 – Distribution of cases of hemorrhagic dengue – Rio de Janeiro State

 

Number of cases

Year/ Month

Jan

Fev

Mar

Abr

Mai

Jun

Jul

Ago

Set

Out

Nov

Dez

Total

1990

-

-

-

2

2

10

7

5

1

3

29

161

220

1991

565

261

145

85

26

1

1

-

1

-

1

-

1086

1992

-

-

-

-

-

-

-

-

-

-

-

-

-

1993

-

-

-

-

-

-

-

-

-

-

-

-

-

1994

-

-

-

-

-

-

-

-

-

-

-

-

-

1995

3

57

51

9

11

2

-

-

1

3

1

-

138

1996

1

2

12

26

20

1

-

1

-

-

-

-

63

1997

1

1

2

2

-

-

-

-

-

-

-

1

7

1998

2

-

5

13

-

-

1

-

-

-

-

-

21

1999

-

3

3

3

2

-

-

-

-

-

-

-

11

2000

-

-

-

2

1

-

2

-

-

-

-

-

5

2001

15

22

49

80

140

71

14

5

5

12

15

39

467

2002

412

624

538

186

21

10

9

-

1

4

16

10

1831

2003

9

14

8

8

3

1

1

-

-

2

-

1

47

2004

1

1

-

1

-

1

-

-

-

-

-

-

4

2005

-

-

3

1

3

-

-

2

-

2

1

1

13

2006

5

8

10

17

16

10

3

2

1

3

2

2

79

2007*

8

6

47

40

46

16

14

4

6

5

9

17

218

2008*

264

528

645

264

57

14

2

1

-

-

-

-

1776

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

Year

DH/SCH

Other

Year

DH/SCH

Other

1990

15

-

 

 

 

1991

24

-

2000

-

-

1992

-

-

2001

13

-

1993

-

-

2002

91

-

1994

-

-

2003

-

-

1995

-

-

2004

-

-

1996

-

-

2005

2

1

1997

1

-

2006

12

-

1998

3

-

2007*

22

15

1999

3

-

2008*

100

140

Source: SESDEC-RJ/SAS/SVS/CVE/DTI/SDTVZ

* 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

Year/Month

Jan

Feb

Mar

Apr

Mai

Jun

Jul

Aug

Sept

Oct

Nov

Dec

TOTAL

1985

425

203

225

105

49

44

5

37

128

83

162

249

1715

1986

104

184

179

110

31

5

84

56

49

50

126

313

1290

1987

207

122

127

166

82

95

10

19

79

117

125

239

1386

1988

135

409

146

126

98

77

42

10

51

160

198

193

1645

1989

175

186

110

86

58

81

96

35

109

80

129

154

1300

1990

118

119

118

120

50

26

64

82

105

105

112

139

1160

1991

275

176

231

140

42

34

34

54

88

85

104

156

1419

1992

144

108

136

76

36

12

51

40

113

97

333

166

1312

1993

171

175

98

102

45

61

12

13

69

90

107

229

1172

1994

188

55

227

158

61

52

24

22

20

71

163

131

1170

1995

122

184

142

50

103

20

24

24

107

216

149

217

1359

1996

155

252

207

95

64

61

36

43

122

122

242

209

1609

1997

143

65

86

46

63

36

30

32

43

87

188

110

929

1998

97

71

49

31

30

29

29

30

42

80

95

98

679

1999

230

133

230

97

55

57

49

30

72

89

165

231

1438

2000

252

129

172

47

30

30

43

71

106

86

152

251

1369

2001

96

78

132

62

105

40

55

30

82

167

242

235

1324

2002

183

184

155

52

74

47

29

51

80

48

125

162

1188

2003

271

63

170

38

30

29

28

29

37

45

76

82

897

2004

174

218

90

51

30

28

30

29

33

58

138

150

1028

2005

110

69

75

38

33

29

35

33

98

98

222

186

1026

2006

139

167

153

70

37

38

35

61

123

97

168

263

1352

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)

Year/Month

Jan Feb Mar Apr Mai Jun Jul Aug Sept Oct Nov Dec

1985

22.5

24.5

23.9

22.6

19.5

17.7

17.4

19.6

19.1

21.0

21.9

22.6

1986

24.5

25.2

24.7

23.7

21.7

19.0

17.5

20.0

20.1

21.3

23.6

23.3

1987

24.6

24.5

23.3

23.1

20.4

18.0

19.1

18.5

19.2

21.3

23.1

23.6

1988

26.2

24.0

24.0

22.3

20.2

17.2

15.8

18.2

20.4

20.2

21.1

23.2

1989

23.8

23.9

24.1

22.8

19.1

18.7

16.7

18.7

19.6

19.8

21.5

22.9

1990

25.5

24.4

24.8

24.4

19.9

19.0

18.0

18.1

19.8

22.4

24.3

24.1

1991

23.6

24.3

23.7

22.8

20.3

19.6

17.7

18.9

19.2

21.4

22.2

24.7

1992

24.1

22.4

23.2

22.4

21.1

20.0

18.1

18.6

20.0

21.7

21.9

22.9

1993

25.1

24.8

24.5

23.2

20.3

18.4

18.9

18.4

20.6

22.3

23.6

24.1

1994

23.6

26.0

23.8

22.0

21.3

18.7

18.8

18.7

20.6

22.4

23.2

25.0

1995

25.9

24.9

24.2

22.4

20.2

19.0

19.9

20.7

20.8

20.9

22.5

23.8

1996

25.6

25.5

24.4

22.8

19.4

18.4

17.2

18.7

20.0

21.5

22.2

24.7

1997

24.4

25.2

23.2

22.2

19.9

18.9

18.8

19.2

21.4

21.6

22.7

25.0

1998

25.0

25.6

25.1

23.5

20.3

17.4

18.3

20.2

20.5

21.5

20.0

23.1

1999

24.7

25.1

24.4

21.7

19.4

18.6

18.6

18.2

21.0

19.6

20.5

23.0

2000

24.0

24.9

24.1

23.2

20.4

18.9

17.5

19.1

20.1

23.0

22.2

24.1

2001

25.2

25.8

25.0

23.6

19.9

19.8

18.0

19.7

20.3

21.0

22.3

23.5

2002

23.9

22.8

24.6

23.2

20.4

19.8

17.9

31.1

19.4

23.0

22.9

23.7

2003

23.9

25.6

23.9

22.8

19.8

19.9

18.4

18.0

19.6

21.2

22.4

23.6

2004

23.0

22.9

22.8

22.3

19.3

18.2

17.2

18.2

21.4

20.9

22.3

22.8

2005

23.7

23.2

23.8

23.1

20.7

19.2

17.7

20.1

19.9

22.8

21.6

22.5

2006

24.5

24.8

24.0

22.0

18.9

18.2

18.1

19.5

19.8

21.0

21.6

23.5

 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


[1]- Mariane Hopp, Development of a Dengue Alert System, Lecture at International Institute of Climate Research, University of Columbia, December 2001,

[2]- From the University of Delaware date bank

[3] - References for the model used in this study are available in the article at:

http://ecen.com/eee45/eee45e/thecnologicalperspective.htm

 

 

Graphic Edition/Edição Gráfica:
MAK
Editoração Eletrônic
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Revised/Revisado:
Monday, 31 October 2011
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