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2013, 10(5&6): 1455-1474. doi: 10.3934/mbe.2013.10.1455

The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile

1. 

Mathematical, Computational & Modeling Sciences Center, School of Human Evolution and Social Change, Arizona State University, Box 872402, Tempe, AZ 85287

2. 

Department of Epidemiology, Ministerio de Salud, Santiago, Chile

3. 

Universidad del Desarrollo, Santiago, Chile, Chile

4. 

School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, United States

5. 

Tulane University, New Orleans, LA, 70118, United States

Received  September 2012 Revised  February 2013 Published  August 2013

We use a stochastic simulation model to explore the effect of reactive intervention strategies during the 2002 dengue outbreak in the small population of Easter Island, Chile. We quantified the effect of interventions on the transmission dynamics and epidemic size as a function of the simulated control intensity levels and the timing of initiation of control interventions. Because no dengue outbreaks had been reported prior to 2002 in Easter Island, the 2002 epidemic provided a unique opportunity to estimate the basic reproduction number $R_0$ during the initial epidemic phase, prior to the start of control interventions. We estimated $R_0$ at $27.2$ ($95 \%$CI: $14.8$, $49.3$). We found that the final epidemic size is highly sensitive to the timing of start of interventions. However, even when the control interventions start several weeks after the epidemic onset, reactive intervention efforts can have a significant impact on the final epidemic size. Our results indicate that the rapid implementation of control interventions can have a significant effect in reducing the epidemic size of dengue epidemics.
Citation: Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
References:
[1]

X. Aguilera, A. Olea, J. Mora, K. Abarca and J. Segovia, Brote de dengue en Isla de Pascua,, El Vigia (Boletín de Vigilancia en Salud Püblica), 16 (2002), 37.

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford University Press, (1991). doi: 10.1016/0046-8177(90)90224-S.

[3]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer-Verlag, (2000). doi: 10.1007/978-1-4614-1686-9.

[4]

G. Chowell, B. Cazelles, H. Broutin and C. V. Munayco, The influence of climatic and geographic factors on the timing of dengue epidemics in Peru, 1994-2008,, BMC Infectious Diseases, 11 (2011). doi: 10.1186/1471-2334-11-164.

[5]

G. Chowell, P. Diaz-Duenas, J. C. Miller, P. W. Fenimore, J. M. Hyman and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data,, Mathematical Biosciences, 208 (2007), 571. doi: 10.1016/j.mbs.2006.11.011.

[6]

G. Chowell, C. A. Torre, C. Munyaco-Escate, L. Suarez -Ognio, R. Lopez-Cruz, J. M. Hyman and C. Castillo-Chavez, Spatial and temporal dynamics of dengue fever in Peru: 1994-2006,, Epidemiology and Infection, 8 (2008), 1. doi: 10.1017/S0950268808000290.

[7]

G. Chowell and F. Sanchez, Climate-based descriptive models of dengue fever: The 2002 epidemic in Colima, Mexico,, Journal of Environmental Health, 68 (2006), 40.

[8]

O. Diekmann and J. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,", Wiley Series in Mathematical and Computational Biology, (2000).

[9]

, Dirección Meteorológica de Chile., Available from: , ().

[10]

C. Favier, N. Degallier, M. G. Rosa-Freitas, J. P. Boulanger, J. R. Costa Lima, J. F. Luitgards-Moura, C. E. Menkes, B. Mondet, C. Oliveira, E. T. Weimann and P. Tsouris, Early determination of the reproduction number for vector-borne diseases: The case of dengue in Brazil,, Tropical Medicine and International Health, 11 (2006), 332. doi: 10.1111/j.1365-3156.2006.01560.x.

[11]

N. M. Ferguson, C. A. Donnelly and R. M. Anderson, Transmission dynamics and epidemiology of dengue: Insights from age-stratified sero-prevalence surveys,, Phil. Trans. Roy. Soc. Lond. B, 354 (1999), 757. doi: 10.1098/rstb.1999.0428.

[12]

D. A. Focks, E. Daniels, D. G. Haile and J. E. Keesling, A simulation model of the epidemiology of urban dengue fever: Literature analysis, model development, preliminary validation, and samples of simulation results,, American Journal of Tropical Medicine and Hygiene, 53 (1995), 489.

[13]

D. J. Gubler, Dengue and dengue hemorrhagic fever,, Clinical Microbiology Reviews, 11 (1998), 480.

[14]

L. Gustafsson and M. Sternad, Bringing consistency to simulation of population models-Poisson simulation as a bridge between micro and macro simulation,, Mathematical Biosciences, 209 (2007), 361. doi: 10.1016/j.mbs.2007.02.004.

[15]

S. B. Halstead, Dengue virus-mosquito interactions,, Annu Rev Entomol., 53 (2007), 273. doi: 10.1146/annurev.ento.53.103106.093326.

[16]

Y. H. Hsieh and C. W. Chen, Turning points, reproduction number, and impact of climatological events for multi-wave dengue outbreaks,, Trop. Med. Int. Health., 14 (2009), 628. doi: 10.1111/j.1365-3156.2009.02277.x.

[17]

Y. H. Hsieh and S. Ma, Intervention measures, turning point, and reproduction number for dengue, Singapore, 2005,, American Journal of Tropical Medicine and Hygiene, 80 (2009), 66.

[18]

J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Mathematical Biosciences, 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3.

[19]

Instituto Nacional de Estadísticas (INE), Estadísticas Demográficas y Vitales, 2009., Available from: , ().

[20]

J. A. Jacquez, "Compartmental Analysis in Biology and Medicine,", Michigan Thompson-Shore, (1996).

[21]

T. H. Jetten and D. A. Focks, Potential changes in the distribution of dengue transmission under climate warming,, American Journal Tropical Medicine and Hygiene, 57 (1997), 285.

[22]

J. S. Koopman, D. R. Prevots, M. A. Vaca Marin, H. Gomez Dantes, M. L. Zarate Aquino, I. M. Longini, Jr and J. Sepulveda Amor, Determinants and predictors of dengue infection in Mexico., American Journal of Epidemiology, 133 (1991), 1168.

[23]

C. F. Li, T. W. Lim, L. L. Han and R. Fang, Rainfall, abundance of Aedes aegypti and dengue infection in Selangor, Malaysia,, Southeast Asian Journal of Tropical Medicine and Public Health, 16 (1985), 560.

[24]

P. M. Luz, C. T. Codeco, E. Massad and C. J. Struchiner, Uncertainties regarding dengue modeling in Río de Janeiro, Brazil,, Memórias do Instituto Oswaldo Cruz, 98 (2003), 871. doi: 10.1590/S0074-02762003000700002.

[25]

G. MacDonald, "The Epidemiology and Control of Malaria,", Chapter Epidemics, (1957), 45.

[26]

C. A. Marques, O. P. Forattini and E. Massad, The basic reproduction number for dengue fever in São Paulo state, Brazil: 1990-1991 epidemic,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 88 (1994), 58.

[27]

E. Massad, F. A. Coutinho, M. N. Burattini and L. F. Lopez, The risk of yellow fever in a dengue-infested area,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 95 (2001), 370. doi: 10.1016/S0035-9203(01)90184-1.

[28]

D. T. Mourya, P. Yadav and A. C. Mishra, Effect of temperature stress on immature stages and susceptibility of Aedes aegypti mosquitoes to chikungunya virus,, American Journal Tropical Medicine and Hygiene, 70 (2004), 346.

[29]

L. E. Muir and B. H. Kay, Aedes aegypti survival and dispersal estimated by mark-release-recapture in northern Australia,, American Journal of Tropical Medicine and Hygiene, 58 (1998), 277.

[30]

M. Otero, H. G. Solari and N. Schweigmann, A stochastic population dynamics model for Aedes aegypti: Formulation and application to a city with temperate climate,, Bulletin of Mathematical Biology, 68 (2006), 1945. doi: 10.1007/s11538-006-9067-y.

[31]

C. Perret, K. Abarca, J. Ovalle, P. Ferrer, P. Godoy, A. Olea, X. Aguilera and M. Ferrés, Dengue-1 virus isolation during first dengue fever outbreak on Easter Island, Chile,, Emerg Infect Dis., 9 (2003), 1465. doi: 10.3201/eid0911.020788.

[32]

R. Ross, "The Prevention of Malaria,", John Murray, (1910).

[33]

G. W. Schultz, Seasonal abundance of dengue vectors in Manila, Republic of the Philippines,, Southeast Asian Journal of Tropical Medicine and Public Health, 24 (1993), 369.

[34]

D. L. Smith, J. Dushoff and F. E. Mckenzie, The risk of a mosquito-borne infection in a heterogeneous environment,, PLoS Biology, 2 (2004), 1957.

[35]

S. T. Stoddard, A. C. Morrison, G. M. Vazquez-Prokopec, V. Paz Soldan, T. J. Kochel, U. Kitron, J. P. Elder and T. W. Scott, The role of human movement in the transmission of vector-borne pathogens,, PLoS Negl. Trop. Dis., 3 (2009). doi: 10.1371/journal.pntd.0000481.

[36]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[37]

D. W. Vaughn, S. Green, S. Kalayanarooj, B. L. Innis, S. Nimmannitya, S. Suntayakorn, T. P. Endy, B. Raengsakulrach, A. L. Rothman, F. A. Ennis and A. Nisalak, Dengue viremia titer, antibody response pattern, and virus serotype correlate with disease severity,, Journal of Infectious Diseases, 181 (2000), 2. doi: 10.1086/315215.

[38]

H. J. Wearing, P. Rohani and M. J. Keeling, Appropriate models for the management of infectious diseases,, PLoS Medicine, 2 (2005). doi: 10.1371/journal.pmed.0020174.

[39]

World Health Organization, Dengue and dengue hemorrhagic fever, accessed March 04, 2006., Available from: , ().

[40]

World Health Organization, Clinical diagnosis of dengue, accessed April 04, 2006., Available from: , (): 012.

show all references

References:
[1]

X. Aguilera, A. Olea, J. Mora, K. Abarca and J. Segovia, Brote de dengue en Isla de Pascua,, El Vigia (Boletín de Vigilancia en Salud Püblica), 16 (2002), 37.

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans,", Oxford University Press, (1991). doi: 10.1016/0046-8177(90)90224-S.

[3]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer-Verlag, (2000). doi: 10.1007/978-1-4614-1686-9.

[4]

G. Chowell, B. Cazelles, H. Broutin and C. V. Munayco, The influence of climatic and geographic factors on the timing of dengue epidemics in Peru, 1994-2008,, BMC Infectious Diseases, 11 (2011). doi: 10.1186/1471-2334-11-164.

[5]

G. Chowell, P. Diaz-Duenas, J. C. Miller, P. W. Fenimore, J. M. Hyman and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data,, Mathematical Biosciences, 208 (2007), 571. doi: 10.1016/j.mbs.2006.11.011.

[6]

G. Chowell, C. A. Torre, C. Munyaco-Escate, L. Suarez -Ognio, R. Lopez-Cruz, J. M. Hyman and C. Castillo-Chavez, Spatial and temporal dynamics of dengue fever in Peru: 1994-2006,, Epidemiology and Infection, 8 (2008), 1. doi: 10.1017/S0950268808000290.

[7]

G. Chowell and F. Sanchez, Climate-based descriptive models of dengue fever: The 2002 epidemic in Colima, Mexico,, Journal of Environmental Health, 68 (2006), 40.

[8]

O. Diekmann and J. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,", Wiley Series in Mathematical and Computational Biology, (2000).

[9]

, Dirección Meteorológica de Chile., Available from: , ().

[10]

C. Favier, N. Degallier, M. G. Rosa-Freitas, J. P. Boulanger, J. R. Costa Lima, J. F. Luitgards-Moura, C. E. Menkes, B. Mondet, C. Oliveira, E. T. Weimann and P. Tsouris, Early determination of the reproduction number for vector-borne diseases: The case of dengue in Brazil,, Tropical Medicine and International Health, 11 (2006), 332. doi: 10.1111/j.1365-3156.2006.01560.x.

[11]

N. M. Ferguson, C. A. Donnelly and R. M. Anderson, Transmission dynamics and epidemiology of dengue: Insights from age-stratified sero-prevalence surveys,, Phil. Trans. Roy. Soc. Lond. B, 354 (1999), 757. doi: 10.1098/rstb.1999.0428.

[12]

D. A. Focks, E. Daniels, D. G. Haile and J. E. Keesling, A simulation model of the epidemiology of urban dengue fever: Literature analysis, model development, preliminary validation, and samples of simulation results,, American Journal of Tropical Medicine and Hygiene, 53 (1995), 489.

[13]

D. J. Gubler, Dengue and dengue hemorrhagic fever,, Clinical Microbiology Reviews, 11 (1998), 480.

[14]

L. Gustafsson and M. Sternad, Bringing consistency to simulation of population models-Poisson simulation as a bridge between micro and macro simulation,, Mathematical Biosciences, 209 (2007), 361. doi: 10.1016/j.mbs.2007.02.004.

[15]

S. B. Halstead, Dengue virus-mosquito interactions,, Annu Rev Entomol., 53 (2007), 273. doi: 10.1146/annurev.ento.53.103106.093326.

[16]

Y. H. Hsieh and C. W. Chen, Turning points, reproduction number, and impact of climatological events for multi-wave dengue outbreaks,, Trop. Med. Int. Health., 14 (2009), 628. doi: 10.1111/j.1365-3156.2009.02277.x.

[17]

Y. H. Hsieh and S. Ma, Intervention measures, turning point, and reproduction number for dengue, Singapore, 2005,, American Journal of Tropical Medicine and Hygiene, 80 (2009), 66.

[18]

J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Mathematical Biosciences, 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3.

[19]

Instituto Nacional de Estadísticas (INE), Estadísticas Demográficas y Vitales, 2009., Available from: , ().

[20]

J. A. Jacquez, "Compartmental Analysis in Biology and Medicine,", Michigan Thompson-Shore, (1996).

[21]

T. H. Jetten and D. A. Focks, Potential changes in the distribution of dengue transmission under climate warming,, American Journal Tropical Medicine and Hygiene, 57 (1997), 285.

[22]

J. S. Koopman, D. R. Prevots, M. A. Vaca Marin, H. Gomez Dantes, M. L. Zarate Aquino, I. M. Longini, Jr and J. Sepulveda Amor, Determinants and predictors of dengue infection in Mexico., American Journal of Epidemiology, 133 (1991), 1168.

[23]

C. F. Li, T. W. Lim, L. L. Han and R. Fang, Rainfall, abundance of Aedes aegypti and dengue infection in Selangor, Malaysia,, Southeast Asian Journal of Tropical Medicine and Public Health, 16 (1985), 560.

[24]

P. M. Luz, C. T. Codeco, E. Massad and C. J. Struchiner, Uncertainties regarding dengue modeling in Río de Janeiro, Brazil,, Memórias do Instituto Oswaldo Cruz, 98 (2003), 871. doi: 10.1590/S0074-02762003000700002.

[25]

G. MacDonald, "The Epidemiology and Control of Malaria,", Chapter Epidemics, (1957), 45.

[26]

C. A. Marques, O. P. Forattini and E. Massad, The basic reproduction number for dengue fever in São Paulo state, Brazil: 1990-1991 epidemic,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 88 (1994), 58.

[27]

E. Massad, F. A. Coutinho, M. N. Burattini and L. F. Lopez, The risk of yellow fever in a dengue-infested area,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 95 (2001), 370. doi: 10.1016/S0035-9203(01)90184-1.

[28]

D. T. Mourya, P. Yadav and A. C. Mishra, Effect of temperature stress on immature stages and susceptibility of Aedes aegypti mosquitoes to chikungunya virus,, American Journal Tropical Medicine and Hygiene, 70 (2004), 346.

[29]

L. E. Muir and B. H. Kay, Aedes aegypti survival and dispersal estimated by mark-release-recapture in northern Australia,, American Journal of Tropical Medicine and Hygiene, 58 (1998), 277.

[30]

M. Otero, H. G. Solari and N. Schweigmann, A stochastic population dynamics model for Aedes aegypti: Formulation and application to a city with temperate climate,, Bulletin of Mathematical Biology, 68 (2006), 1945. doi: 10.1007/s11538-006-9067-y.

[31]

C. Perret, K. Abarca, J. Ovalle, P. Ferrer, P. Godoy, A. Olea, X. Aguilera and M. Ferrés, Dengue-1 virus isolation during first dengue fever outbreak on Easter Island, Chile,, Emerg Infect Dis., 9 (2003), 1465. doi: 10.3201/eid0911.020788.

[32]

R. Ross, "The Prevention of Malaria,", John Murray, (1910).

[33]

G. W. Schultz, Seasonal abundance of dengue vectors in Manila, Republic of the Philippines,, Southeast Asian Journal of Tropical Medicine and Public Health, 24 (1993), 369.

[34]

D. L. Smith, J. Dushoff and F. E. Mckenzie, The risk of a mosquito-borne infection in a heterogeneous environment,, PLoS Biology, 2 (2004), 1957.

[35]

S. T. Stoddard, A. C. Morrison, G. M. Vazquez-Prokopec, V. Paz Soldan, T. J. Kochel, U. Kitron, J. P. Elder and T. W. Scott, The role of human movement in the transmission of vector-borne pathogens,, PLoS Negl. Trop. Dis., 3 (2009). doi: 10.1371/journal.pntd.0000481.

[36]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[37]

D. W. Vaughn, S. Green, S. Kalayanarooj, B. L. Innis, S. Nimmannitya, S. Suntayakorn, T. P. Endy, B. Raengsakulrach, A. L. Rothman, F. A. Ennis and A. Nisalak, Dengue viremia titer, antibody response pattern, and virus serotype correlate with disease severity,, Journal of Infectious Diseases, 181 (2000), 2. doi: 10.1086/315215.

[38]

H. J. Wearing, P. Rohani and M. J. Keeling, Appropriate models for the management of infectious diseases,, PLoS Medicine, 2 (2005). doi: 10.1371/journal.pmed.0020174.

[39]

World Health Organization, Dengue and dengue hemorrhagic fever, accessed March 04, 2006., Available from: , ().

[40]

World Health Organization, Clinical diagnosis of dengue, accessed April 04, 2006., Available from: , (): 012.

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