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Mathematical modeling of dengue epidemic: control methods and vaccination strategies

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Abstract

Dengue is, in terms of death and economic cost, one of the most important infectious diseases in the world. So, its mathematical modeling can be a valuable tool to help us to understand the dynamics of the disease and to infer about its spreading by the proposition of control methods. In this paper, control strategies, which aim to eliminate the Aedes aegypti mosquito, as well as proposals for the vaccination campaign are evaluated. In our mathematical model, the mechanical control is accomplished through the environmental support capacity affected by a discrete function that represents the removal of breedings. Chemical control is carried out using insecticide and larvicide. The efficiency of vaccination is studied through the transfer of a fraction of individuals, proportional to the vaccination rate, from the susceptible to the recovered compartments. Our major find is that the dengue fever epidemic is only eradicated with the use of an immunizing vaccine because control measures, directed against its vector, are not enough to halt the disease spreading. Even when the infected mosquitoes are eliminated from the system, the susceptible ones are still present, and infected humans cause dengue fever to reappear in the human population.

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Notes

  1. IBGE: Brazilian population aging in rhythm accelerated

References

  • Abboubakar H, Kamgang JC, Nkamba LN, Tieudjo D, Emini L (2015) Modeling the dynamics of arboviral diseases with vaccination perspective. Biomath 4:1507241. https://doi.org/10.11145/j.biomath.2015.07.241

    Article  Google Scholar 

  • Aldila D, Götz T, Soewono E (2013) An optimal control problem arising from a dengue disease transmission model. Math Biosci 242(1):9–16. https://doi.org/10.1016/j.mbs.2012.11.014

    Article  PubMed  Google Scholar 

  • Anderson KB, Gibbons RV, Edelman R, Eckels KH, Putnak RJ, Innis BL, Sun W (2011) Interference and facilitation between dengue serotypes in a tetravalent live dengue virus vaccine candidate. J Infect Dis 204(3):442–450

    CAS  PubMed  Google Scholar 

  • Andersson N, Nava-Aquilera E, Arostequi J, Morales-Perez A, Suazo-Laguna H, Legorreta-Soberanis J, Hernandez-Alvarez C, Fernandez-Salas I, Paredes-Solis S, Balmaseda A, Cortes-Guzman AJ, Serrano de los Santos R, Coloma J, Ledogar RJ, Harris E (2015) Evidence based community mobilization for dengue prevention in Nicaragua and Mexico (Camino Verde, the Green Way): cluster randomized controlled trial. BMJ 8(351):h3267

    Google Scholar 

  • Andraud M, Hens N, Marais C, Beutels P (2012) Dynamic epidemiological models for dengue transmission: a systematic review of structural approaches. PLoS ONE 7(11):E49085

    CAS  PubMed  PubMed Central  Google Scholar 

  • Antonio M, Yoneyama T (2001) Optimal and sub-optimal control in dengue epidemics. Optim Control Appl Methods 22(2):63–73. https://doi.org/10.1002/oca.683

    Article  Google Scholar 

  • Arduino MB (2014) Assessment of Aedes aegypti pupal productivity during the dengue vector control program in a costal urban centre of São Paulo state, Brazil. http://dx.doi.org/10.1155/2014/301083

  • Bartley LM, Donnely CA, Garnett GP (2002) The seasonal patterns of dengue in endemic areas: mathematical models of mechanism. Tran R Soc Trop Med Hyg 96:387–397

    CAS  Google Scholar 

  • BarZeev M (1958) The effect of temperature on the growth rate and survival of the immature stages of Aedes aegypti. Bull Entomol Res 49:157–163

    Google Scholar 

  • Beserra EB, Castro FP Jr, Santos JW, Santos TS, Fernandes CRM (2006) Biology and thermal exigency of Aedes aegypti (L.) (Diptera: Culicidae) from four bioclimatic localities of Paraiba. Neotrop Entomol 35(6):853–860

    PubMed  Google Scholar 

  • Blayneh KW, Gumel AB, Lenhart S, Clayton T (2010) Backward bifurcation and optimal control in transmission dynamics of West Nile virus. Bull Math Biol 72(4):1006–1028. https://doi.org/10.1007/s11538-009-9480-0

    Article  PubMed  Google Scholar 

  • Braga IA, Valle D (2007) Aedes aegypti: inseticidas, mecanismos de ação e resistência. Epidemiologia e Serviços da Saúde 16(4):279–293

    Google Scholar 

  • Bricks LF (2004) Vacinas para a dengue: perspectivas. Pediatria 26(4):268–281

    Google Scholar 

  • Burattini MN, Chen M, Coutinho FA, Goh KT, Lopez S, Ma E (2008) Modelling the control strategies against dengue in Singapore. Epidemiol Infect 136(3):309–319

    CAS  PubMed  Google Scholar 

  • Câmara FPC, Theophilo RLG, Santos GT, Pereira SRFG, Câmara DCP, Matos RRC (2007) Regional and dynamics characteristics of dengue in Brazil: a retrospective study. Rev Soc Bras Med Trop 40(2):192–196

    PubMed  Google Scholar 

  • Carvalho A, Roy RV, Andrus J (2016) Vaccine communication and advocacy: challenges and way forward. Exp Rev Vaccines 4(15):539–545. https://doi.org/10.1586/14760584.2016.1152187

    Article  CAS  Google Scholar 

  • Chan M, Johansson MA (2012) The incubation periods of dengue viruses. PLoS ONE 7(11):E30972

    Google Scholar 

  • Christofferson RC, Mores CN (2015) A role for vector control in dengue vaccine programs. Vaccine 33(50):69–74

    Google Scholar 

  • Cirino S, Silva JAL (2004) Modelo Epidemiológico SEIR de Transmissão da Dengue em Redes de Populações Acopladas. Tend Mat Apl Comput 5(1):55–64

    Google Scholar 

  • Coutinho FAB, Burattini MN, Lopez LF, Massad E (2006) Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue. Bull Math Biol 68(8):2263–2282

    CAS  PubMed  Google Scholar 

  • Cummings DA, Huang NE (2004) Travelling wave in the occurrence of dengue hemorrhagic fever in the Thailand. Nature 4(27):344–347

    Google Scholar 

  • De Jong MC, Diekmann O, Heesterbeek H (1995) How does transmission of infection depend on population size. Epidemic Models: Their Struct Relat Data 5(2):84–94

    Google Scholar 

  • Derouich M, Boutayeb A (2006) Dengue fever: mathematical modelling and computer simulation. Appl Math Comput 177(2):528–544

    Google Scholar 

  • Dias WO, Wanner EF, Cardoso RTN (2015) A multiobjective optimization approach for combating Aedes aegypti using chemical and biological alternated step-size control. Math Biosci 269:37–47

    PubMed  Google Scholar 

  • Dumont Y, Chiroleu F (2010) Vector control for the chikungunya disease. Math Biosci Eng 7:313–345

    PubMed  Google Scholar 

  • Esteva L, Vargas C (1998) Analysis of a dengue disease transmission model. Math Biosci 150:131–151

    CAS  PubMed  Google Scholar 

  • Esteva L, Yang HM (2005) Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique. Math Biosci 198:132–147

    PubMed  Google Scholar 

  • Garba SM, Gumel AB, Abu Bakar MR (2008) Backward bifurcations in dengue transmission dynamics. Math Biosci 215(1):11–25. https://doi.org/10.1016/j.mbs.2008.05.002

    Article  CAS  PubMed  Google Scholar 

  • Getis A, Morrison A, Gray K, Scott TW (2003) Characteristics of the spatial pattern of the dengue vector, Aedes aegypti, in Iquitos, Peru. Am J Trop Med Hyg 69(5):494–505

    PubMed  Google Scholar 

  • Gotelli NJ (2009) Ecologia, 4th edn. Editora Planta, Londrina

    Google Scholar 

  • Gubler DJ (1986) Dengue. The arboviruses: epidemiology and ecology, vol II. CeC, Boca Raton, p 213

    Google Scholar 

  • Gubler DJ (1997) Dengue and dengue hemorrhagic fever: its history and resurgence as a global health problem. In: Gubler DJ, Kuno G (eds) Dengue and dengue hemorrhagic fever. CAB International, New York

    Google Scholar 

  • Gubler DJ (2002) The global emergence/resurgence of arboviral disease as public health problems. Arc Med 33(4):330–342

    Google Scholar 

  • Gubler DJ, Suharyono W, Tan R, Albidini M, Sle (1981) Viremia in patients with naturally acquired dengue infection. Bull World Health Organ 59:623–630

    CAS  PubMed  PubMed Central  Google Scholar 

  • Hadinegoro SR, ArredondoGarcía JL, Capeding MR, Deseda C, Chotpitayasunondh T, Dietze R, Hj Muhammad Ismail HI, Reynales H, Limkittikul K, RiveraMedina DM, Tran HN, Bouckenooghe A, Chansinghakul D, Cortés M, Fanouillere K, Forrat R, Frago C, Gailhardou S, Jackson N, Noriega F, Plennevaux E, Wartel TA, Zambrano B, Saville M (2015) Efficacy and long-term safety of a dengue vaccine in regions of endemic disease. N Engl J Med 373(13):1195–1206. https://doi.org/10.1056/NEJMoa1506223

    Article  CAS  PubMed  Google Scholar 

  • Hendron RW, Bonsall MB (2016) The interplay of vaccination and vector control on small dengue network. J Theor Biol 407(21):349–361

    PubMed  PubMed Central  Google Scholar 

  • Horsfall WR (1955) Mosquitoes: their bionomics and relation to disease. Ronald, New York

    Google Scholar 

  • IBGE, Instituto Brasileiro de Geografia e Estatstica (2011) Dados sobre População do Brasil, PNAD (Pesquisa Nacional por Amostra de Domicílios), http://www.ibge.gov.br/home/estatistica/populacao/trabalhoerendimento/pnad2011/

  • Johansson MA, Hombach J, Sinha P, Cummings DA (2011) Models of the impact of dengue vaccines: a review of current research and potential approaches. Vaccine 29(35):5860–5868

    PubMed  PubMed Central  Google Scholar 

  • Johnson AJ, Roehring JT (1999) New mouse model for dengue virus vaccine testing. J Virol 73(1):783–786

    CAS  PubMed  PubMed Central  Google Scholar 

  • Julander JG, Perry ST, Shresta S (2011) Important advances in the field of anti-dengue virus research. Antivir Chem Chemother 21(3):105–116

    CAS  PubMed  Google Scholar 

  • Kim JE, Lee H, Lee CH, Lee S (2017) Assessment of optimal strategies in a two-patch dengue transmission model with seasonality. PLoS ONE 12(3):e0173673

    PubMed  PubMed Central  Google Scholar 

  • Knerer G, Currie CSM, Brailsford SC (2015) Impact of combined vector control and vaccination strategies on transmission dynamics of dengue fever: a model-based analysis. Health Care Manag Sci 18(2):205–217

    PubMed  Google Scholar 

  • Koopman JS, Prevots DR, Vaca MAM, Gomez HD, Zarate MLA, Longini IM Jr, Sepulveda JA (1999) Determinants and predictors of dengue infection in Mexico. Am J Epidemiol 133(11):1168–1178

    Google Scholar 

  • Koppen W (1948) Climatology con un studio de los climas de la tierra. México: Fondo da Cultura Económica

  • Korobeinicov A (2009) Global properties of SIR and SEIR epidemic models with multiple parallel interactions stages. Bull Math Biol 71:75–83

    Google Scholar 

  • Kurane I, Takasaki T (2001) Dengue fever and dengue hemorrhagic fever: challenges of controlling an enemy still at large. Rev Med Virol 11(5):301–311

    CAS  PubMed  Google Scholar 

  • Lana RM, Carneiro TGS, Honório NA, Codeço CT (2014) Seasonal and nonseasonal dynamics of Aedes aegypti in Rio de Janeiro, Brazil: fitting mathematical models to trap data. Acta Trop 129:25–32

    PubMed  Google Scholar 

  • Luz PM, Codeço CT, Medlock J, Struchiner CJ, Valle D, Galvani AP (2009) Profile of insecticide interventions on the abundance and resistance profile of Aedes aegypti. Epidemiol Infect 137(8):1203–1215

    CAS  PubMed  Google Scholar 

  • Maidana NA, Yang HM (2007) Describing the geographic spread of dengue disease by traveling waves. Math Biosci 215:64–77

    Google Scholar 

  • Maidana NA, Yang HM (2009) Spatial spreading of West Nile Virus described by traveling waves? J Theor Biol 258:403–417

    PubMed  Google Scholar 

  • MDS (2005) Ministério da Saúde. Guia de Vigilância Epidemiológica. 6nd ed. Brasília

  • MDS (2010) Ministério da Saúde. O uso racional de inseticidas no controle do Aedes aegypti e sua utilização oportuna em áreas com transmissão de dengue. Nota Técnica \(\text{N}^{o}\) 109/2010 CGPNCD/DEVEP/SVS/MS Brasília

  • Monica Das Gupta M, Engelman R, Levy J, Luchsinger G, Merrick T, Rosen JE (2014) Situação da População Mundial 2014. UNFPA, http://www.unfpa.org.br/swop2014/link/SWOP2014.pdf

  • Morrison AC, Getis A, Santiago M (1997) Exploratory space-time analysis of reported dengue cases during an outbreak in Florida, Puerto Rico, 1991–1992. Am J Trop Med Hyg 57:119–125

    Google Scholar 

  • Moulay D, Aziz-Alaoui MA, Cadivel M (2011) The chikungunya disease: modeling, vector and transmission global dynamics. Math Biosci 229(1):50–63

    CAS  PubMed  Google Scholar 

  • Moulay D, Aziz-Alaoui MA, Kwon HD (2012) Optimal control of chikungunya disease: larvae reduction, treatment and prevention. Math Biosci Eng 9(2):369–392. https://doi.org/10.3934/mbe.2012.9.369

    Article  PubMed  Google Scholar 

  • Nakhapakorn K, Tripathi NK (2005) An information value-based analysis of physical and climatic factors affecting dengue fever and dengue hemorrhagic fever incidence. Int J Health Geogr 4:1–13

    Google Scholar 

  • Newton EA, Reiter P (1992) A model of the transmission of dengue fever with an evolution of the impact of ultra-low volume (ULV) insecticide applications on dengue epidemics. Am J Trop Med Hyg 47(6):709–720

    CAS  PubMed  Google Scholar 

  • Otero M, Solari HG, Schweigmann N (2006) A stochastic population dynamics model for Aedes aegypti: formulation and application to a city with temperate climate. Bull Math Biol 68:1945–1974

    PubMed  Google Scholar 

  • Parks W, Lloyd L (2004) Planning social mobilization and communication for dengue fever prevention and control. World Health Organization, Geneva, pp 1–158

    Google Scholar 

  • Pio C, Yang HM, Esteva L (2008) Assessing the suitability of sterile insect technique applied to Aedes aegypti. J Biol Syst 16:565577

    Google Scholar 

  • Ramos MM, Mohammed H, Zielinski-Gutierrez E, Hayden MH, Lopez JL, Fournier M, Trujillo AR, Burton R, Brunkard JM, Anaya-Lopez L, Banicki AA, Morales PK, Smith B, Muñoz JL, Waterman SH (2008) Epidemic dengue and dengue hemorrhagic fever at the Texas–Mexico border: results of a household-based seroepidemiologic survey. Am J Trop Med Hyg 78(3):364–9

    PubMed  Google Scholar 

  • Rodrigues HS, Monteiro MT, Torres DF (2014) Vaccination models and optimal control strategies to dengue. Math Biosci 247:1–12

    PubMed  Google Scholar 

  • Rodrigues NCP, Lino VTS, Daumas RP, Andrade MKN, O’Dwyer G, Monteiro DLM, Gerardi A, Fernandes GHBV, Ramos JAS, Ferreira CEG, Leite IC (2016) Temporal and spatial evolution of dengue incidence in Brazil, 2001–2012. PLoS ONE 11(11):e0165945

    PubMed  PubMed Central  Google Scholar 

  • Rueda LM, Patel KJ, Axtell RC, Stinner RE (1990) Temperature-dependent development and survival rates of Culex quinquefasciatus and Aedes aegypti (diptera: Culicidae). J Med Entomol 27:892–898

    CAS  PubMed  Google Scholar 

  • Setzer J (1966) Climate and Ecological Atlas of the State of São Paulo. Comissão Interestadual da Bacia do Paraná-Uruguai em Colaboração com as Centrais Elétricas de SP (CESP), São Paulo, Brazil

  • Silva LJ, Richtmann R (2006) Vaccines under development: group B streptococcus, herpes-zoster, HIV, malaria and dengue. J Pediatr 82(3):115–124

    Google Scholar 

  • Stoddard ST, Morrison AC, Vazquez-Prokopec GM, Soldan VP, Kochel TJ, Kitron U, Elder JP, Scott TW (2009) The role of human movement in the transmission of vector-borne pathogens. PLoS Negl Trop Dis 3(7):e481

    PubMed  PubMed Central  Google Scholar 

  • Teixeira MG, Barreto ML, Costa MC, Ferreira LD, Vasconcelos PF, Cairncross S (2002) Dynamics of dengue virus circulation: a silent epidemic in a complex urban area. Trop Med Int Health 7(9):757–762

    Google Scholar 

  • Thomé RCA, Yang HM, Esteva L (2010) Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide. Math Biosci 223:12–23

    PubMed  Google Scholar 

  • Trips M (1972) Dry season survival of Aedes aegypti eggs in various breeding sites in the Dar Salaam area, Tanzania. Bull WHO 47:433–437

    Google Scholar 

  • Toro-Zapata HD, Restrepo LD, Vergaño-Salazar JG, Muñs-Loaiz A (2010) Classical dengue transmission dynamics involving mechanical control and prophylaxis. Rev Salud Publica (Bogota) 12(6):1020–1032

    Google Scholar 

  • Villar L, Dayan GH, Arredondo-García JL, Rivera DM, Cunha R, Deseda C, Reynales H, Costa MS, Morales-Ramírez JO, Carrasquilla G, Rey LC, Dietze R, Luz K, Rivas E, Montoya MCM, Supelano MC, Zambrano B, Langevin E, Boaz M, Tornieporth N, Saville M, Noriega F (2015) Efficacy of a tetravalent dengue vaccine in children in Latin America. N Engl J Med 372:113–23. https://doi.org/10.1056/NEJMoa1411037

    Article  CAS  PubMed  Google Scholar 

  • Watts DM, Burke DS, Harrison BA, Whitmire RE, Nisalak A (1987) Effect of temperature on the vector efficiency of Aedes aegypti for dengue 2 virus. Am J Trop Med Hyg 36:143–152

    CAS  PubMed  Google Scholar 

  • Westaway EG, Brinton MA, Gaidamovich S, Horzinek MC, Igarashi A, Kaariainen L et al (1985) Flaviviridae. Intervirology 24:183–192

    CAS  PubMed  Google Scholar 

  • WHO (1998) World Health Organization (WHO), Health of older persons in the western pacific region; Country Profiles. http://apps.who.int/iris/bitstream/10665/206868/1/Health_older_persons_WPR_eng.pdf?ua$=1$

  • WHO (2012) World Health Organization (WHO), Global strategy for dengue prevention and control. http://www.who.int/immunization/sage/meetings/2013/april/5_Dengue_SAGE_Apr2013_Global_Strategy.pdf

  • WHO (2016) World Health Organization (WHO), Immunization, Vaccines and Biologicals, Geneva. http://www.who.int/immunization/sage/meetings/2016/april/en/

  • Wilder-Smith A, Gubler DJ (2008) Geographic expansion of dengue: the impact of international travel. Med Clin North Am 92:1377–1390

    PubMed  Google Scholar 

  • Wilson EB, Woecester J (1945) The law of mass action in epidemiology. Proc Natl Acad Sci USA 31:24–34 (part I) and 109–116 (part II)

  • Yang HM, Ferreira CP (2008) Assessing the effects of the vector control on dengue transmission. Appl Math Comput 198:401–413

    Google Scholar 

  • Yang HM, Marcoris MLG, Galvani KC, Andrighetti MTM, Wanderley DMV (2009) Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue. Epidemiol Infect 137:1188–1202

    CAS  PubMed  Google Scholar 

  • Yang HM, Boldrini JL, Fassoni AC, Freitas LFS, Gomez MC, LIMA KKB, Andrade VR, Freitas ARR (2016) Fitting the Incidence data from the city of Campinas, Brazil, based on dengue transmission modellings considering time-dependent entomological parameters. PLoS ONE 11:e0152186

    PubMed  PubMed Central  Google Scholar 

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Acknowledgements

This work was partially supported by the Brazilian agencies CAPES, CNPq and FAPEMIG. We thank Dr. Marcelo Lobato Martins of the Physics Department—Federal University of Viçosa—for the kindness of your priceless suggestions. We also thank the comments and suggestions provided by anonymous referees, which contributed to improving this paper.

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  1. Departamento de Física, Universidade Federal de Viçosa, Viçosa, Minas Gerais, 36570-900, Brazil

    Sylvestre Aureliano Carvalho

  2. Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras, Minas Gerais, 37200-000, Brazil

    Stella Olivia da Silva & Iraziet da Cunha Charret

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Carvalho, S.A., da Silva, S.O. & Charret, I.C. Mathematical modeling of dengue epidemic: control methods and vaccination strategies. Theory Biosci. 138, 223–239 (2019). https://doi.org/10.1007/s12064-019-00273-7

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