2012, 9(2): 369-392. doi: 10.3934/mbe.2012.9.369

Optimal control of chikungunya disease: Larvae reduction, treatment and prevention

1. 

LMAH, Université du Havre, 25 rue Philippe Lebon, BP540, 76058 Le Havre Cedex, France, France

2. 

Department of Mathematics, Inha University, Incheon, 402-751

Received  April 2011 Revised  December 2011 Published  March 2012

Since the 1980s, there has been a worldwide re-emergence of vector-borne diseases including Malaria, Dengue, Yellow fever or, more recently, chikungunya. These viruses are arthropod-borne viruses (arboviruses) transmitted by arthropods like mosquitoes of Aedes genus. The nature of these arboviruses is complex since it conjugates human, environmental, biological and geographical factors. Recent researchs have suggested, in particular during the Réunion Island epidemic in 2006, that the transmission by Aedes albopictus (an Aedes genus specie) has been facilitated by genetic mutations of the virus and the vector capacity to adapt to non tropical regions. In this paper we formulate an optimal control problem, based on biological observations. Three main efforts are considered in order to limit the virus transmission. Indeed, there is no vaccine nor specific treatment against chikungunya, that is why the main measures to limit the impact of such epidemic have to be considered. Therefore, we look at time dependent breeding sites destruction, prevention and treatment efforts, for which optimal control theory is applied. Using analytical and numerical techniques, it is shown that there exist cost effective control efforts.
Citation: Djamila Moulay, M. A. Aziz-Alaoui, Hee-Dae Kwon. Optimal control of chikungunya disease: Larvae reduction, treatment and prevention. Mathematical Biosciences & Engineering, 2012, 9 (2) : 369-392. doi: 10.3934/mbe.2012.9.369
References:
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[40]

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[41]

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B. V. Tandale, P. S. Sathe, V. A. Arankalle, R. Wadia, R. Kulkarni, S. V. Shah, S. K. Shah, J. K. Sheth, A. Sudeep, A. S. Tripathy and A. C. Mishra, Systemic involvements and fatalities during chikungunya epidemic in India, 2006,, Journal of Clinical Virology, 46 (2009), 145.   Google Scholar

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R. C. Thomé, H. M. Yang and L. Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide,, Mathematical Biosciences, 223 (2010), 12.  doi: 10.1016/j.mbs.2009.08.009.  Google Scholar

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M. Vazeille, C. Jeannin, E. Martin, F. Schaffner and A. B. Failloux, Chikungunya: A risk for Mediterranean countries,, Acta Tropica, 105 (2008), 200.  doi: 10.1016/j.actatropica.2007.09.009.  Google Scholar

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H. M. Yang and C. P. Ferreira, Assessing the effects of vector control on dengue transmission,, Applied Mathematics and Computation, 198 (2008), 401.  doi: 10.1016/j.amc.2007.08.046.  Google Scholar

show all references

References:
[1]

B. M. Adams, H. T. Banks, M. Davidian, H.-D. Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols,, J. Comput. Appl. Math., 184 (2005), 10.  doi: 10.1016/j.cam.2005.02.004.  Google Scholar

[2]

B. M. Adams, H. T. Banks, H.-D. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches,, Mathematical Biosciences and Engineering, 1 (2004), 223.  doi: 10.3934/mbe.2004.1.223.  Google Scholar

[3]

J. Adhami and P. Reiter, Introduction and establishment of Aedes (Stegomyia) albopictus skuse (diptera : Culicidae) in Albania,, American Mosquito Control Association, 14 (1998), 340.   Google Scholar

[4]

N. Alphey, M. B. Bonsall and L. Alphey, Modeling resistance to genetic control of insects,, Journal of Theoretical Biology, 270 (2011), 42.  doi: 10.1016/j.jtbi.2010.11.016.  Google Scholar

[5]

, Be dry with mosquitoes, 2011., Available from: \url{http://www.albopictus.eid-med.org/}., ().   Google Scholar

[6]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bulletin of Mathematical Biology, 69 (2007), 1067.  doi: 10.1007/s11538-006-9166-9.  Google Scholar

[7]

M. Q. Benedict, R. S. Levine, W. A. Hawley and L. P. Lounibos, Spread of the tiger: Global risk of invasion by the mosquito Aedes albopictus,, Vector Borne and Zoonotic Diseases, 7 (2007), 76.  doi: 10.1089/vbz.2006.0562.  Google Scholar

[8]

K. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne disease: Treatment and prevention,, Discrete and Continuous Dynamical Systems Series B, 11 (2009), 587.  doi: 10.3934/dcdsb.2009.11.587.  Google Scholar

[9]

C. Cosner, J. Beier, R. Cantrell, D. Impoinvil, L. Kapitanski, M. Potts, A. Troyo and S. Ruan, The effects of human movement on the persistence of vector-borne diseases,, Journal of Theoretical Biology, 258 (2009), 550.  doi: 10.1016/j.jtbi.2009.02.016.  Google Scholar

[10]

N. Curcó, N. Gimènez, M. Serra, A. Ripoll, M. García and P. Vives, Asian tiger mosquito bites: Perception of the affected population after Aedes albopictus became established in Spain,, Actas Dermo-Sifiliogràficas (English Edition), 99 (2008), 708.   Google Scholar

[11]

T. Das, M. C. Jaffar-Bandjee, J. J. Hoarau, P. K. Trotot, M. Denizot, G. Lee-Pat-Yuen, R. Sahoo, P. Guiraud, D. Ramful, S. Robin, J. L. Alessandri, B. A. Gauzere and P. Gasque, Chikungunya fever: CNS infection and pathologies of a re-emerging arbovirus,, Progress in Neurobiology, 91 (2010), 121.  doi: 10.1016/j.pneurobio.2009.12.006.  Google Scholar

[12]

H. Delatte, G. Gimonneau, A. Triboire and D.Fontenille, Influence of temperature on immature development, survival, longevity, fecundity, and gonotrophic cycles of Aedes albopictus, vector of chikungunya and dengue in the Indian Ocean,, Journal of Medical Entomology, 46 (2009), 33.  doi: 10.1603/033.046.0105.  Google Scholar

[13]

H. Delatte, C. Paupy, J. S. Dehecq, J. Thiria, A. B. Failloux and D. Fontenille, Aedes albopictus, vector of chikungunya and dengue viruses in reunion island: Biology and control,, Parasite, 15 (2008), 3.   Google Scholar

[14]

E. Depoortere, S. Salmaso, M. Pompa, P. Guglielmetti and D. Coulombier, Chikungunya in Europe,, The Lancet, 371 (2008), 723.   Google Scholar

[15]

M. Diallo, J. Thonnon, M. Traore-Lamizana and D. Fontenille, Vectors of chikungunya virus in Senegal: Current data and transmission cycles,, The American Journal of Tropical Medicine and Hygiene, 60 (1999), 281.   Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

M. Dubrulle, L. Mousson, S. Moutailler, M. Vazeille and A. B. Failloux, Chikungunya virus and Aedes mosquitoes: Saliva is infectious as soon as two days after oral infection,, PLoS ONE, 4 (2009).  doi: 10.1371/journal.pone.0005895.  Google Scholar

[19]

Y. Dumont and F. Chiroleu, Vector control for the chikungunya disease,, Mathematical Biosciences and Engineering, 7 (2010), 313.  doi: 10.3934/mbe.2010.7.313.  Google Scholar

[20]

Y. Dumont, F. Chiroleu and C. Domerg, On a temporal model for the chikungunya disease: Modeling, theory and numerics,, Mathematical Biosciences, 213 (2008), 80.  doi: 10.1016/j.mbs.2008.02.008.  Google Scholar

[21]

M. Enserink, Epidemiology: Tropical disease follows mosquitoes to Europe,, Science, 317 (2007).  doi: 10.1126/science.317.5844.1485a.  Google Scholar

[22]

L. Esteva and C. Vargas, A model for dengue disease with variable human population,, Journal of Mathematical Biology, 38 (1999), 220.  doi: 10.1007/s002850050147.  Google Scholar

[23]

W. H. Fleming and R. W. Rishel, "Deterministic and Stochastic Optimal Control,", Applications of Mathematics, (1975).   Google Scholar

[24]

P. J. Gullan and P. Cranston, "The Insects: An Outline of Entomology," 4th edition,, Wiley-Blackwell, (2010).   Google Scholar

[25]

M. G. Guzman and G. Kouri, Dengue and dengue hemorrhagic fever in the Americas: Lessons and challenges,, Journal of Clinical Virology, 27 (2003), 1.  doi: 10.1016/S1386-6532(03)00010-6.  Google Scholar

[26]

W. A. Hawley, The biology of Aedes albopictus,, J. Am. Mosq. Control Assoc. Suppl., 1 (1988), 1.   Google Scholar

[27]

H. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[28]

E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model,, Discrete and Continuous Dynamical Systems Series B, 2 (2002), 473.  doi: 10.3934/dcdsb.2002.2.473.  Google Scholar

[29]

K. Laras, N. C. Sukri, R. P. Larasati, M. J. Bangs, R. Kosim, Djauzi, T. Wandra, J. Master, H. Kosasih, S. Hartati, C. Beckett, E. R. Sedyaningsih, H. J. Beecham III and A. L. Corwin, Tracking the re-emergence of epidemic chikungunya virus in Indonesia,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 99 (2005), 128.  doi: 10.1016/j.trstmh.2004.03.013.  Google Scholar

[30]

D. L. Lukes, "Differential Equations. Classical to Controlled,", Mathematics in Science and Engineering, 162 (1982).   Google Scholar

[31]

W. H. R. Lumdsen, An epidemic of virus disease in Southern Province, Tanganyika territory, in 1952-1953. II. General description and epidemiology,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 49 (1955), 23.   Google Scholar

[32]

C. J. Mitchell, Geographic spread of Aedes albopictus and potential for involvement in arbovirus cycles in the Mediterranean basin,, Journal of Vector Ecology, 20 (1995), 44.   Google Scholar

[33]

D. Moulay, M. A. Aziz-Alaoui and M. Cadivel, The chikungunya disease: Modeling, vector and transmission global dynamics,, Mathematical Biosciences, 229 (2011), 50.  doi: 10.1016/j.mbs.2010.10.008.  Google Scholar

[34]

C. Paupy, H. Delatte, L. Bagny, V. Corbel and D. Fontenille, Aedes albopictus, an arbovirus vector: From the darkness to the light,, Microbes and Infection, 11 (2009), 1177.  doi: 10.1016/j.micinf.2009.05.005.  Google Scholar

[35]

G. Pialoux, B. A. Gaüzère and M. Strobel, Infection à virus chikungunya: Revue générale par temps d'épidémie,, Médecine et Maladies Infectieuses, 36 (2006), 253.  doi: 10.1016/j.medmal.2006.04.002.  Google Scholar

[36]

L. Pontryagin, V. Boltyanskii, R. Gamkrelidze and E. Mishchenko, "The Mathematical Theory of Optimal Processes,", A Pergamon Press Book, (1964).   Google Scholar

[37]

S. Rajapakse, C. Rodrigo and A. Rajapakse, Atypical manifestations of chikungunya infection,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 104 (2010), 89.  doi: 10.1016/j.trstmh.2009.07.031.  Google Scholar

[38]

G. Rezza, L. Nicoletti, R. Angelini, R. Romi, A. Finarelli, M. Panning, P. Cordioli, C. Fortuna, S. Boros, F. Magurano, G. Silvi, P. Angelini, M. Dottori, M. Ciufolini, G. Majori and A. Cassone, Infection with chikungunya virus in Italy: An outbreak in a temperate region,, The Lancet, 370 (2007), 1840.  doi: 10.1016/S0140-6736(07)61779-6.  Google Scholar

[39]

M. C. Robinson, An epidemic of virus disease in Southern Province, Tanganyika territory, in 1952-53. I. Clinical features,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 49 (1955), 28.  doi: 10.1016/0035-9203(55)90080-8.  Google Scholar

[40]

R. W. Ross, The newala epidemic. III. The virus: Isolation, pathogenic properties and relationship to the epidemic,, Epidemiology and Infection, 54 (1956), 177.   Google Scholar

[41]

T. Seyler, Y. Hutin, V. Ramanchandran, R. Ramakrishnan, P. Manickam and M. Murhekar, Estimating the burden of disease and the economic cost attributable to chikungunya, Andhra Pradesh, India, 2005-2006,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 104 (2010), 133.  doi: 10.1016/j.trstmh.2009.07.014.  Google Scholar

[42]

D. Sissoko, D. Malvy, C. Giry, G. Delmas, C. Paquet, P. Gabrie, F. Pettinelli, M. A. Sanquer and V. Pierre, Outbreak of chikungunya fever in Mayotte, Comoros Archipelago, 2005-2006,, Transactions of the Royal Society of Tropical Medicine and Hygiene, 102 (2008), 780.  doi: 10.1016/j.trstmh.2008.02.018.  Google Scholar

[43]

A. B. Sudeep and D. Parashar, Chikungunya: An overview,, Journal of Biosciences, 33 (2008), 443.   Google Scholar

[44]

B. V. Tandale, P. S. Sathe, V. A. Arankalle, R. Wadia, R. Kulkarni, S. V. Shah, S. K. Shah, J. K. Sheth, A. Sudeep, A. S. Tripathy and A. C. Mishra, Systemic involvements and fatalities during chikungunya epidemic in India, 2006,, Journal of Clinical Virology, 46 (2009), 145.   Google Scholar

[45]

R. C. Thomé, H. M. Yang and L. Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide,, Mathematical Biosciences, 223 (2010), 12.  doi: 10.1016/j.mbs.2009.08.009.  Google Scholar

[46]

M. Vazeille, C. Jeannin, E. Martin, F. Schaffner and A. B. Failloux, Chikungunya: A risk for Mediterranean countries,, Acta Tropica, 105 (2008), 200.  doi: 10.1016/j.actatropica.2007.09.009.  Google Scholar

[47]

World-Health-Organization, Dengue and severe dengue,, factsheet no. 117, (2008).   Google Scholar

[48]

H. M. Yang and C. P. Ferreira, Assessing the effects of vector control on dengue transmission,, Applied Mathematics and Computation, 198 (2008), 401.  doi: 10.1016/j.amc.2007.08.046.  Google Scholar

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