2017, 22(3): 687-715. doi: 10.3934/dcdsb.2017034

Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus

Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence KS 66045, USA

Email: fbagusto@gmail.com

Received  August 2015 Revised  December 2015 Published  December 2016

Chikungunya is an RNA viral disease, transmitted to humans by infected Aedes aegypti or Aedes albopictus mosquitoes. In this paper, an age-structured deterministic model for the transmission dynamics of Chikungunya virus is presented. The model is locally and globally asymptotically stable when the reproduction number is less than unity. A global sensitivity analysis using the reproduction number indicates that the mosquito biting rate, the transmission probability per contact of mosquitoes and of humans, mosquito recruitment rate and the death rate of the mosquitoes are the parameters with the most influence on Chikungunya transmission dynamics. Optimal control theory was then applied, using the results from the sensitivity analysis, to minimize the number infected humans, with time dependent control variables (impacting mosquito biting rate, transmission probability, death rate and recovery rates in humans).

The numerical simulations indicate that Chikungunya can be reduced by the application of these controls. The benefits associated with these health interventions are evaluated using cost-effectiveness analysis and these shows that using mono-control strategy involving treatment of infected individuals is the most cost-effective strategy of this category. With pairs of control, the pairs involving treatment of infected individuals and mosquitoes adulticiding, is the most cost-effective strategy of this category and is more cost-effective than using the triple control strategy involving personal protection, treatment of infected humans and mosquitoes adulticiding.

Citation: Folashade B. Agusto. Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 687-715. doi: 10.3934/dcdsb.2017034
References:
[1]

F. B. Agusto, S. Easley, K. Freeman and M. Thomas, Mathematical model of a three age-structured transmission dynamics of Chikungunya Virus Comput. Math. Methods Med. , (2016), Art. ID 4320514, 31 pp.

[2]

F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, BioSystems, 113 (2013), 155-164. doi: 10.1016/j.biosystems.2013.06.004.

[3]

F. B. AgustoJ. M. Lenhart and S. Cushing, Optimal Control of the spread of malaria super-infectivity, Journal of Biological Systems Special issue on Infectious Disease Modeling, 21 (2013), 1340002, 26pp. doi: 10.1142/S0218339013400020.

[4]

F. B. AgustoN. Marcus and K. O. Okosun, Application of optimal control to the epidemiology of malaria disease, Electronic Journal of Differential Equations, 2012 (2012), 1-22.

[5]

R. M. Anderson and R. May, Infectious Diseases of Humans Oxford University Press, New York, 1991.

[6]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example, Int. Stat. Rev., 62 (1994), 229-243. doi: 10.2307/1403510.

[7]

S. B. Cantor and T. G. Ganiats, Incremental cost-effectiveness analysis: The optimal strategy depends on the strategy set, Journal of Clinical Epidemiology, 52 (1999), 517-522. doi: 10.1016/S0895-4356(99)00021-9.

[8]

Centers for Disease Control and Prevention. Chikungunya virus, Available from: http://www.cdc.gov/Chikungunya/symptoms/index.html.

[9]

L. J. ChangK. A. DowdF. H. MendozaJ. G. Saunders and S. Sitar, Safety and tolerability of Chikungunya virus-like particle vaccine in healthy adults: A phase 1 dose-escalation trial, Lancet, 384 (2014), 2046-2052. doi: 10.1016/S0140-6736(14)61185-5.

[10]

N. ChitnisJ. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0.

[11]

A. CosteroK. Mormann and S. A. Juliano, Asymmetrical competition and patterns of abundance of Aedes albopictus and Culex pipiens (Diptera: Culicidae), Journal of Medical Entomology, 42 (2005), 559-570.

[12]

H. DelatteG. GimonneauA. 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), 33-41.

[13]

O. DiekmannJ. A. P. Heesterbeek and J. A. P. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.

[14]

M. Doucleff, Trouble In Paradise: Chikungunya outbreak grows in caribbean, Health News, 2013. Available from: http://www.npr.org/sections/health-shots/2013/12/18/255216192/trouble-in-paradise-Chikungunya-outbreak-grows-in-caribbean.

[15]

M. DubrulleL. MoussonS. MoutaillerM. 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), e5895. doi: 10.1371/journal.pone.0005895.

[16]

Y. DumontF. Chiroleu and C. Domerg, On a temporal model for the Chikungunya disease: Modeling, theory and numerics, Mathematical Biosciences, 213 (2008), 80-91. doi: 10.1016/j.mbs.2008.02.008.

[17]

Y. Dumont and F. Chiroleu, Vector control for the Chikungunya disease, Mathematical Biosciences and Engineering, 7 (2010), 313-345. doi: 10.3934/mbe.2010.7.313.

[18]

M. Enserink, Epidemiology: Tropical disease follows mosquitoes to Europe, Science, 317 (2007), 1485a.

[19]

K. Fiscella and P. Franks, Cost-effectiveness of the transdermal nicotine patch as an adjunct to physicians smoking cessation counseling, JAMA, 276 (1996), 1247-1251.

[20]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control Springer Verlag, New York, 1975.

[21]

K. A. FreedbergJ. A. ScharfsteinG. R. SeageE. LosinaM. C. Weinstein and D. E. Craven, The cost-effectiveness of preventing AIDS-related opportunistic infections, JAMA, 279 (1998), 130-136. doi: 10.1001/jama.279.2.130.

[22]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.

[23]

H. R. Joshi, Optimal control of an HIV immunology model, Optim. Control Appl. Math, 23 (2002), 199-213. doi: 10.1002/oca.710.

[24]

E. JungS. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.

[25]

D. KernS. LenhartR. Miller and J. Yong, Optimal control applied to native-invasive population dynamics, J Biol Dyn., 1 (2007), 413-426. doi: 10.1080/17513750701605556.

[26]

D. KirschnerS. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792. doi: 10.1007/s002850050076.

[27]

C. Lahariya and S. K. Pradhan, Emergence of Chikungunya virus in Indian subcontinent after 32 years: A review, Journal of Vector Borne Diseases, 43 (2006), 151-160.

[28]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007.

[29]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011.

[30]

C. ManoreJ. HickmannS. XuH. Wearing and J. Hyman, Comparing Dengue and Chikungunya emergence and endemic Transmission in A. aegypti and A. albopictus, Journal of Theoretical Biology, 356 (2014), 174-191. doi: 10.1016/j.jtbi.2014.04.033.

[31]

E. MassadS. MaM. N. BurattiniY. TunF. A. B. Coutinho and W. Lang, The risk of Chikungunya fever in a dengue-endemic area, Journal of Travel Medicine, 15 (2008), 147-155. doi: 10.1111/j.1708-8305.2008.00186.x.

[32]

R. G. McLeodJ. F. BrewsterA. B. Gumel and D. A. Slonowsky, Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs, Math. Biosci. Eng., 3 (2006), 527-544. doi: 10.3934/mbe.2006.3.527.

[33]

O. P. Misra and D. K. Mishra, Simultaneous effects of control Measures on the transmission dynamics of Chikungunya disease, Applied Mathematics, 2 (2012), 124-130. doi: 10.5923/j.am.20120204.05.

[34]

D. MoulayM. A. Aziz-Alaoui and M. Cadivel, The Chikungunya disease: Modeling, vector and transmission global dynamics, Mathematical Biosciences, 229 (2011), 50-63. doi: 10.1016/j.mbs.2010.10.008.

[35]

H. Nur AidaA. Abu HassanA. T. NuritaM. R. Che Salmah and B. Norasmah, Population analysis of Aedes albopictus (Skuse)(Diptera: Culicidae) under uncontrolled laboratory conditions, Tropical Biomedicine, 25 (2008), 117-125.

[36]

K. O. OkosunO. Rachid and N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems, 111 (2013), 83-101. doi: 10.1016/j.biosystems.2012.09.008.

[37]

K. PeskoC. J. WestbrookC. N. MoresL. P. Lounibos and M. H. Reiskind, Effects of infectious virus dose and bloodmeal delivery method on susceptibility of Aedes aegypti and Aedes albopictus to Chikungunya virus, Journal of Medical Entomology, 46 (2009), 395-399.

[38]

G. PialouxB. A. GaüzéreS. Jauréguiberry and M. Strobel, Chikungunya, an epidemic arbovirosis, The Lancet Infectious Diseases, 7 (2007), 319-327. doi: 10.1016/S1473-3099(07)70107-X.

[39]

S. D. PinkertonD. R. HoltgraveW. J. DiFranceiscoL. Y. Stevenson and J. A. Kelly, Cost-effectiveness of a community-level HIV risk reduction intervention, Am J Public Health, 88 (1998), 1239-1242. doi: 10.2105/AJPH.88.8.1239.

[40]

P. PolettiG. MesseriM. AjelliR. ValloraniC. Rizzo and S. Merler, Transmission potential of Chikungunya virus and control measures: The case of Italy, PLoS ONE, 6 (2011), e18860. doi: 10.1371/journal.pone.0018860.

[41]

P. Pongsumpun and S. Sangsawangl, Local stability analysis for age structural model of Chikungunya disease, Journal of Basic and Applied Scientific Research, 3 (2013), 302-312.

[42]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Wiley, New York, 1962.

[43]

D. Ruiz-MorenoI. S. VargasK. E. Olson and L. C. Harrington, Modeling dynamic introduction of Chikungunya virus in the United States, PLOS Neglected Tropical Disease, 6 (2012), e1918. doi: 10.1371/journal.pntd.0001918.

[44]

I. SchuffeneckerI. ItemanA. MichaultS. Murri and L. Frangeul, Genome microevolution of Chikungunya viruses causing the Indian Ocean outbreak, PLoS Med, 3 (2006), e263. doi: 10.1371/journal.pmed.0030263.

[45]

O. Schwartz and M. L. Albert, Biology and pathogenesis of Chikungunya virus, Nature Reviews Microbiology, 8 (2010), 491-500. doi: 10.1038/nrmicro2368.

[46]

M. R. SebastianR. Lodha and S. K. Kabra, Chikungunya infection in children, Indian Journal of Pediatrics, 76 (2009), 185-189. doi: 10.1007/s12098-009-0049-6.

[47]

R. K. SinghS. TiwariV. K. MishraR. Tiwari and T. N. Dhole, Molecular epidemiology of Chikungunya virus: Mutation in E1 gene region, Journal of Virological Methods, 185 (2012), 213-220. doi: 10.1016/j.jviromet.2012.07.001.

[48]

J. E. Staples, S. L. Hills and A. M. Powers, Chikungunya. Centers for Disease Control and Prevention (CDC), 2014, http://wwwnc.cdc.gov/travel/yellowbook/2014/chapter-3-infectious-diseases-related-to-travel/Chikungunya. Accessed June 8,2015.

[49]

K. A. TsetsarkinD. L. VanlandinghamC. E. McGee and S. Higgs, A Single mutation in Chikungunya virus affects vector specificity and epidemic potential, PLoS Pathogens, 3 (2007), e201. doi: 10.1371/journal.ppat.0030201.

[50]

M. J. TurellJ. R. Beaman and R. F. Tammariello, Susceptibility of selected strains of Aedes aegypti and Aedes albopictus (Diptera: Culicidae) to Chikungunya virus, Journal of Medical Entomology, 29 (1992), 49-53. doi: 10.1093/jmedent/29.1.49.

[51]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[52]

M. VazeilleS. MoutaillerD. CoudrierC. Rousseaux and H. Khun, Two Chikungunya isolates from the outbreak of La Reunion (Indian Ocean) exhibit different patterns of infection in the mosquito, Aedes albopictus, PLoS ONE, 2 (2007), e1168. doi: 10.1371/journal.pone.0001168.

[53]

World Health Organization, Chikungunya, 2014, Available from: http://www.who.int/denguecontrol/arbo-viral/other_arboviral_Chikungunya/en/.

[54]

World Health Organization, Chikungunya, 2014, Available from: http://www.who.int/mediacentre/factsheets/fs327/en/.

[55]

X. YanY. Zou and J. Li, Optimal quarantine and isolation strategies in epidemics control, World Journal of Modelling and Simulation, 3 (2007), 202-211.

[56]

X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Mathematical and Computer Modelling, 47 (2008), 235-245. doi: 10.1016/j.mcm.2007.04.003.

[57]

L. Yakob and A. C. A. Clements, A Mathematical model of Chikungunya dynamics and control: The major epidemic on Réunion Island, PLoS ONE, 8 (2013), e57448.

show all references

References:
[1]

F. B. Agusto, S. Easley, K. Freeman and M. Thomas, Mathematical model of a three age-structured transmission dynamics of Chikungunya Virus Comput. Math. Methods Med. , (2016), Art. ID 4320514, 31 pp.

[2]

F. B. Agusto, Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model, BioSystems, 113 (2013), 155-164. doi: 10.1016/j.biosystems.2013.06.004.

[3]

F. B. AgustoJ. M. Lenhart and S. Cushing, Optimal Control of the spread of malaria super-infectivity, Journal of Biological Systems Special issue on Infectious Disease Modeling, 21 (2013), 1340002, 26pp. doi: 10.1142/S0218339013400020.

[4]

F. B. AgustoN. Marcus and K. O. Okosun, Application of optimal control to the epidemiology of malaria disease, Electronic Journal of Differential Equations, 2012 (2012), 1-22.

[5]

R. M. Anderson and R. May, Infectious Diseases of Humans Oxford University Press, New York, 1991.

[6]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example, Int. Stat. Rev., 62 (1994), 229-243. doi: 10.2307/1403510.

[7]

S. B. Cantor and T. G. Ganiats, Incremental cost-effectiveness analysis: The optimal strategy depends on the strategy set, Journal of Clinical Epidemiology, 52 (1999), 517-522. doi: 10.1016/S0895-4356(99)00021-9.

[8]

Centers for Disease Control and Prevention. Chikungunya virus, Available from: http://www.cdc.gov/Chikungunya/symptoms/index.html.

[9]

L. J. ChangK. A. DowdF. H. MendozaJ. G. Saunders and S. Sitar, Safety and tolerability of Chikungunya virus-like particle vaccine in healthy adults: A phase 1 dose-escalation trial, Lancet, 384 (2014), 2046-2052. doi: 10.1016/S0140-6736(14)61185-5.

[10]

N. ChitnisJ. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0.

[11]

A. CosteroK. Mormann and S. A. Juliano, Asymmetrical competition and patterns of abundance of Aedes albopictus and Culex pipiens (Diptera: Culicidae), Journal of Medical Entomology, 42 (2005), 559-570.

[12]

H. DelatteG. GimonneauA. 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), 33-41.

[13]

O. DiekmannJ. A. P. Heesterbeek and J. A. P. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.

[14]

M. Doucleff, Trouble In Paradise: Chikungunya outbreak grows in caribbean, Health News, 2013. Available from: http://www.npr.org/sections/health-shots/2013/12/18/255216192/trouble-in-paradise-Chikungunya-outbreak-grows-in-caribbean.

[15]

M. DubrulleL. MoussonS. MoutaillerM. 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), e5895. doi: 10.1371/journal.pone.0005895.

[16]

Y. DumontF. Chiroleu and C. Domerg, On a temporal model for the Chikungunya disease: Modeling, theory and numerics, Mathematical Biosciences, 213 (2008), 80-91. doi: 10.1016/j.mbs.2008.02.008.

[17]

Y. Dumont and F. Chiroleu, Vector control for the Chikungunya disease, Mathematical Biosciences and Engineering, 7 (2010), 313-345. doi: 10.3934/mbe.2010.7.313.

[18]

M. Enserink, Epidemiology: Tropical disease follows mosquitoes to Europe, Science, 317 (2007), 1485a.

[19]

K. Fiscella and P. Franks, Cost-effectiveness of the transdermal nicotine patch as an adjunct to physicians smoking cessation counseling, JAMA, 276 (1996), 1247-1251.

[20]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control Springer Verlag, New York, 1975.

[21]

K. A. FreedbergJ. A. ScharfsteinG. R. SeageE. LosinaM. C. Weinstein and D. E. Craven, The cost-effectiveness of preventing AIDS-related opportunistic infections, JAMA, 279 (1998), 130-136. doi: 10.1001/jama.279.2.130.

[22]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.

[23]

H. R. Joshi, Optimal control of an HIV immunology model, Optim. Control Appl. Math, 23 (2002), 199-213. doi: 10.1002/oca.710.

[24]

E. JungS. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems-Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473.

[25]

D. KernS. LenhartR. Miller and J. Yong, Optimal control applied to native-invasive population dynamics, J Biol Dyn., 1 (2007), 413-426. doi: 10.1080/17513750701605556.

[26]

D. KirschnerS. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792. doi: 10.1007/s002850050076.

[27]

C. Lahariya and S. K. Pradhan, Emergence of Chikungunya virus in Indian subcontinent after 32 years: A review, Journal of Vector Borne Diseases, 43 (2006), 151-160.

[28]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models Chapman and Hall, 2007.

[29]

S. MarinoI. B. HogueC. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011.

[30]

C. ManoreJ. HickmannS. XuH. Wearing and J. Hyman, Comparing Dengue and Chikungunya emergence and endemic Transmission in A. aegypti and A. albopictus, Journal of Theoretical Biology, 356 (2014), 174-191. doi: 10.1016/j.jtbi.2014.04.033.

[31]

E. MassadS. MaM. N. BurattiniY. TunF. A. B. Coutinho and W. Lang, The risk of Chikungunya fever in a dengue-endemic area, Journal of Travel Medicine, 15 (2008), 147-155. doi: 10.1111/j.1708-8305.2008.00186.x.

[32]

R. G. McLeodJ. F. BrewsterA. B. Gumel and D. A. Slonowsky, Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs, Math. Biosci. Eng., 3 (2006), 527-544. doi: 10.3934/mbe.2006.3.527.

[33]

O. P. Misra and D. K. Mishra, Simultaneous effects of control Measures on the transmission dynamics of Chikungunya disease, Applied Mathematics, 2 (2012), 124-130. doi: 10.5923/j.am.20120204.05.

[34]

D. MoulayM. A. Aziz-Alaoui and M. Cadivel, The Chikungunya disease: Modeling, vector and transmission global dynamics, Mathematical Biosciences, 229 (2011), 50-63. doi: 10.1016/j.mbs.2010.10.008.

[35]

H. Nur AidaA. Abu HassanA. T. NuritaM. R. Che Salmah and B. Norasmah, Population analysis of Aedes albopictus (Skuse)(Diptera: Culicidae) under uncontrolled laboratory conditions, Tropical Biomedicine, 25 (2008), 117-125.

[36]

K. O. OkosunO. Rachid and N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems, 111 (2013), 83-101. doi: 10.1016/j.biosystems.2012.09.008.

[37]

K. PeskoC. J. WestbrookC. N. MoresL. P. Lounibos and M. H. Reiskind, Effects of infectious virus dose and bloodmeal delivery method on susceptibility of Aedes aegypti and Aedes albopictus to Chikungunya virus, Journal of Medical Entomology, 46 (2009), 395-399.

[38]

G. PialouxB. A. GaüzéreS. Jauréguiberry and M. Strobel, Chikungunya, an epidemic arbovirosis, The Lancet Infectious Diseases, 7 (2007), 319-327. doi: 10.1016/S1473-3099(07)70107-X.

[39]

S. D. PinkertonD. R. HoltgraveW. J. DiFranceiscoL. Y. Stevenson and J. A. Kelly, Cost-effectiveness of a community-level HIV risk reduction intervention, Am J Public Health, 88 (1998), 1239-1242. doi: 10.2105/AJPH.88.8.1239.

[40]

P. PolettiG. MesseriM. AjelliR. ValloraniC. Rizzo and S. Merler, Transmission potential of Chikungunya virus and control measures: The case of Italy, PLoS ONE, 6 (2011), e18860. doi: 10.1371/journal.pone.0018860.

[41]

P. Pongsumpun and S. Sangsawangl, Local stability analysis for age structural model of Chikungunya disease, Journal of Basic and Applied Scientific Research, 3 (2013), 302-312.

[42]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes Wiley, New York, 1962.

[43]

D. Ruiz-MorenoI. S. VargasK. E. Olson and L. C. Harrington, Modeling dynamic introduction of Chikungunya virus in the United States, PLOS Neglected Tropical Disease, 6 (2012), e1918. doi: 10.1371/journal.pntd.0001918.

[44]

I. SchuffeneckerI. ItemanA. MichaultS. Murri and L. Frangeul, Genome microevolution of Chikungunya viruses causing the Indian Ocean outbreak, PLoS Med, 3 (2006), e263. doi: 10.1371/journal.pmed.0030263.

[45]

O. Schwartz and M. L. Albert, Biology and pathogenesis of Chikungunya virus, Nature Reviews Microbiology, 8 (2010), 491-500. doi: 10.1038/nrmicro2368.

[46]

M. R. SebastianR. Lodha and S. K. Kabra, Chikungunya infection in children, Indian Journal of Pediatrics, 76 (2009), 185-189. doi: 10.1007/s12098-009-0049-6.

[47]

R. K. SinghS. TiwariV. K. MishraR. Tiwari and T. N. Dhole, Molecular epidemiology of Chikungunya virus: Mutation in E1 gene region, Journal of Virological Methods, 185 (2012), 213-220. doi: 10.1016/j.jviromet.2012.07.001.

[48]

J. E. Staples, S. L. Hills and A. M. Powers, Chikungunya. Centers for Disease Control and Prevention (CDC), 2014, http://wwwnc.cdc.gov/travel/yellowbook/2014/chapter-3-infectious-diseases-related-to-travel/Chikungunya. Accessed June 8,2015.

[49]

K. A. TsetsarkinD. L. VanlandinghamC. E. McGee and S. Higgs, A Single mutation in Chikungunya virus affects vector specificity and epidemic potential, PLoS Pathogens, 3 (2007), e201. doi: 10.1371/journal.ppat.0030201.

[50]

M. J. TurellJ. R. Beaman and R. F. Tammariello, Susceptibility of selected strains of Aedes aegypti and Aedes albopictus (Diptera: Culicidae) to Chikungunya virus, Journal of Medical Entomology, 29 (1992), 49-53. doi: 10.1093/jmedent/29.1.49.

[51]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[52]

M. VazeilleS. MoutaillerD. CoudrierC. Rousseaux and H. Khun, Two Chikungunya isolates from the outbreak of La Reunion (Indian Ocean) exhibit different patterns of infection in the mosquito, Aedes albopictus, PLoS ONE, 2 (2007), e1168. doi: 10.1371/journal.pone.0001168.

[53]

World Health Organization, Chikungunya, 2014, Available from: http://www.who.int/denguecontrol/arbo-viral/other_arboviral_Chikungunya/en/.

[54]

World Health Organization, Chikungunya, 2014, Available from: http://www.who.int/mediacentre/factsheets/fs327/en/.

[55]

X. YanY. Zou and J. Li, Optimal quarantine and isolation strategies in epidemics control, World Journal of Modelling and Simulation, 3 (2007), 202-211.

[56]

X. Yan and Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Mathematical and Computer Modelling, 47 (2008), 235-245. doi: 10.1016/j.mcm.2007.04.003.

[57]

L. Yakob and A. C. A. Clements, A Mathematical model of Chikungunya dynamics and control: The major epidemic on Réunion Island, PLoS ONE, 8 (2013), e57448.

Figure 1.  Systematic flow diagram of age-structured Chikungunya Model (1)-(2)
Figure 2.  Simulation of the age-structured Chikungunya model (1)-(2) as a function of time when $\mathcal{R}_0 < 1$. (a) Total number of infectious (asymptomatic and symptomatic) juveniles (b) Total number of infectious (asymptomatic and symptomatic) adults (c) Total number of infectious (asymptomatic and symptomatic) seniors (d) Total number of infectious mosquitoes. Parameter values used are as given in Table 2
Figure 3.  Simulation of the age-structured Chikungunya model (1)-(2) as a function of time when $\mathcal{R}_0 > 1$. With parameter values used are as given in Table 2. (a) Total number of infectious (asymptomatic and symptomatic) juveniles (b) Total number of infectious (asymptomatic and symptomatic) adults (c) Total number of infectious (asymptomatic and symptomatic) seniors (d) Total number of infectious mosquitoes
Figure 4.  PRCC values for the age-structured Chikungunya model (1)-(2), using the basic reproduction number ($\mathcal{R}_{0}$) as the response function. Parameter values (baseline) and ranges used are as given in Table 2
Figure 5.  Simulation of the age-structured Chikungunya model (3)-(4) as a function of time without control and with optimal control for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
Figure 6.  The optimal controls of the age-structured Chikungunya model (3)-(4) for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment; (d). Personal protection control; (e). Mosquitoes adulticiding control
Figure 7.  Simulation of the age-structured Chikungunya model (3)-(4) as a function of time using strategies A, B and C for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
Figure 8.  The optimal controls of the age-structured Chikungunya model (3)-(4) using strategies A, B and C for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment; (d). Personal protection control; (e). Mosquitoes adulticiding control
Figure 9.  Simulation of the age-structured Chikungunya model (3)-(4) as a function of time using strategies D, E and F for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
Figure 10.  The optimal controls of the age-structured Chikungunya model (3)-(4) using strategies D, E and F for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment; (d). Personal protection control; (e). Mosquitoes adulticiding control
Figure 11.  Simulation of the age-structured Chikungunya model (3)-(4) as a function of time using strategies B, E and G for: (a). Total number of infected juvenile; (b). Total number of infected adult; (c). Total number of infected seniors; (d). Total number of infected mosquitoes
Figure 12.  The optimal controls of the age-structured Chikungunya model (3)-(4) using strategies B, E and G for: (a). Juvenile treatment; (b). Adult treatment; (c). Seniors treatment
Table 1.  Description of the variables and parameters of the agestructured Chikungunya model (1)-(2)
Variable Description
$S_J, S_A, S_S$ Population of susceptible juvenile and adult humans
$E_J, E_A, E_S$ Population of exposed juvenile and adult humans
$I_{AJ}, I_{SJ}$ Population of asymptomatic and symptomatic Juvenile humans
$I_{AA}, I_{SA}$ Population of asymptomatic and symptomatic adult humans
$I_{AS}, I_{SS}$ Population of asymptomatic and symptomatic seniors humans
$R_J, R_A, R_S$ Population of recovered juvenile and adult humans
$S_M$ Population of susceptible mosquitoes
$E_M$ Population of exposed mosquitoes
$I_{M}$ Population of infected mosquitoes
Parameter Description
$\pi_J$ Recruitment rate of juvenile humans
$\pi_M$ Recruitment rate of mosquitoes
$\alpha, \xi$ Juvenile and adult maturation rates
$\beta_J, \beta_A, \beta_S$ Transmission probability per contact for susceptible humans
$\mu_J, \mu_A, \mu_S$ Natural death rate of juvenile, adult and senior humans
$\varepsilon_J, \varepsilon_A, \varepsilon_S$ Fraction of exposed humans becoming asymptomatic
$\sigma_J, \sigma_A, \sigma_S$ Progression rate of exposed juvenile, adult and senior humans
$\gamma_{AJ},\gamma_{SJ}$ Recovery rate of asymptomatic and symptomatic juvenile humans
$\gamma_{AA},\gamma_{SA}$ Recovery rate of asymptomatic and symptomatic adult humans
$\gamma_{AS},\gamma_{SS}$ Recovery rate of asymptomatic and symptomatic senior humans
$\beta_M$ Transmission probability per contact for susceptible mosquitoes
$b_M $ Mosquito biting rate
$\sigma_M$ Progression rate of exposed mosquitoes
$\mu_M$ Natural death rate of mosquitoes
Variable Description
$S_J, S_A, S_S$ Population of susceptible juvenile and adult humans
$E_J, E_A, E_S$ Population of exposed juvenile and adult humans
$I_{AJ}, I_{SJ}$ Population of asymptomatic and symptomatic Juvenile humans
$I_{AA}, I_{SA}$ Population of asymptomatic and symptomatic adult humans
$I_{AS}, I_{SS}$ Population of asymptomatic and symptomatic seniors humans
$R_J, R_A, R_S$ Population of recovered juvenile and adult humans
$S_M$ Population of susceptible mosquitoes
$E_M$ Population of exposed mosquitoes
$I_{M}$ Population of infected mosquitoes
Parameter Description
$\pi_J$ Recruitment rate of juvenile humans
$\pi_M$ Recruitment rate of mosquitoes
$\alpha, \xi$ Juvenile and adult maturation rates
$\beta_J, \beta_A, \beta_S$ Transmission probability per contact for susceptible humans
$\mu_J, \mu_A, \mu_S$ Natural death rate of juvenile, adult and senior humans
$\varepsilon_J, \varepsilon_A, \varepsilon_S$ Fraction of exposed humans becoming asymptomatic
$\sigma_J, \sigma_A, \sigma_S$ Progression rate of exposed juvenile, adult and senior humans
$\gamma_{AJ},\gamma_{SJ}$ Recovery rate of asymptomatic and symptomatic juvenile humans
$\gamma_{AA},\gamma_{SA}$ Recovery rate of asymptomatic and symptomatic adult humans
$\gamma_{AS},\gamma_{SS}$ Recovery rate of asymptomatic and symptomatic senior humans
$\beta_M$ Transmission probability per contact for susceptible mosquitoes
$b_M $ Mosquito biting rate
$\sigma_M$ Progression rate of exposed mosquitoes
$\mu_M$ Natural death rate of mosquitoes
Table 2.  Parameters values of the age-structured Chikungunya model (1)-(2)
Parameter Values Range References
$\pi_J$ $\frac{1}{15\times365}$ $\frac{1}{15\times365}$ -$\frac{1}{12\times365}$ [10,30]
$\alpha$ $\frac{1}{16\times365}$ $\frac{1}{18\times365}$ -$\frac{1}{15\times365}$ Assumed
$\xi$ $\frac{1}{60\times365}$ $\frac{1}{65\times365}$ -$\frac{1}{55\times365}$ Assumed
$\beta_J, \beta_A, \beta_S$ 0.24 0.001 -0.54 [16,17,30,40,50]
$b_M$ 0.25 0.19 -0.39 [12,30]
$\mu_J$ $\frac{1}{3\times365}$ $\frac{1}{5\times365}$ -$\frac{1}{1\times365}$ Assumed
$\mu_A$ $\frac{1}{40\times365}$ $\frac{1}{60\times365}$ -$\frac{1}{18\times365}$ Assumed
$\mu_S$ $\frac{1}{70\times365}$ $\frac{1}{80\times365}$ -$\frac{1}{60\times365}$ Assumed
$\varepsilon_J, \varepsilon_A, \varepsilon_S$ 0.155 0.03-0.28 [48]
$\sigma_J, \sigma_S$ $\frac{1}{2\times3}$ $\frac{1}{2\times4}$ -$\frac{1}{2\times2}$ Assumed
$\sigma_A$ $\frac{1}{3}$ $\frac{1}{4}$ -$\frac{1}{2}$ [17,27,30,38,46,45]
$\gamma_{AJ},\gamma_{SJ}$ $\frac{1}{1.5\times4.5}$ $\frac{1}{1.5\times8}$ -$\frac{1}{1.5\times3}$ Assumed
$\gamma_{AA},\gamma_{SA}$ $\frac{1}{4.5}$ $\frac{1}{7}$ -$\frac{1}{3}$ [30,34,46,45]
$\gamma_{AS},\gamma_{SS}$ $\frac{1}{2.5\times4.5}$ $\frac{1}{2.5\times8}$ -$\frac{1}{2.5\times3}$ Assumed
$\pi_M$ 0.24 0.015 -0.32 [10,11,30,35]
$\beta_M$ 0.24 0.005 -0.35 [16,31,37,40,50]
$\sigma_M$ $\frac{1}{3.5}$ $\frac{1}{6}$ -$\frac{1}{2}$ [15,17,34,46]
$\mu_M$ $\frac{1}{14}$ $\frac{1}{42}$ -$\frac{1}{14}$ [17,27,34,46,45]
Parameter Values Range References
$\pi_J$ $\frac{1}{15\times365}$ $\frac{1}{15\times365}$ -$\frac{1}{12\times365}$ [10,30]
$\alpha$ $\frac{1}{16\times365}$ $\frac{1}{18\times365}$ -$\frac{1}{15\times365}$ Assumed
$\xi$ $\frac{1}{60\times365}$ $\frac{1}{65\times365}$ -$\frac{1}{55\times365}$ Assumed
$\beta_J, \beta_A, \beta_S$ 0.24 0.001 -0.54 [16,17,30,40,50]
$b_M$ 0.25 0.19 -0.39 [12,30]
$\mu_J$ $\frac{1}{3\times365}$ $\frac{1}{5\times365}$ -$\frac{1}{1\times365}$ Assumed
$\mu_A$ $\frac{1}{40\times365}$ $\frac{1}{60\times365}$ -$\frac{1}{18\times365}$ Assumed
$\mu_S$ $\frac{1}{70\times365}$ $\frac{1}{80\times365}$ -$\frac{1}{60\times365}$ Assumed
$\varepsilon_J, \varepsilon_A, \varepsilon_S$ 0.155 0.03-0.28 [48]
$\sigma_J, \sigma_S$ $\frac{1}{2\times3}$ $\frac{1}{2\times4}$ -$\frac{1}{2\times2}$ Assumed
$\sigma_A$ $\frac{1}{3}$ $\frac{1}{4}$ -$\frac{1}{2}$ [17,27,30,38,46,45]
$\gamma_{AJ},\gamma_{SJ}$ $\frac{1}{1.5\times4.5}$ $\frac{1}{1.5\times8}$ -$\frac{1}{1.5\times3}$ Assumed
$\gamma_{AA},\gamma_{SA}$ $\frac{1}{4.5}$ $\frac{1}{7}$ -$\frac{1}{3}$ [30,34,46,45]
$\gamma_{AS},\gamma_{SS}$ $\frac{1}{2.5\times4.5}$ $\frac{1}{2.5\times8}$ -$\frac{1}{2.5\times3}$ Assumed
$\pi_M$ 0.24 0.015 -0.32 [10,11,30,35]
$\beta_M$ 0.24 0.005 -0.35 [16,31,37,40,50]
$\sigma_M$ $\frac{1}{3.5}$ $\frac{1}{6}$ -$\frac{1}{2}$ [15,17,34,46]
$\mu_M$ $\frac{1}{14}$ $\frac{1}{42}$ -$\frac{1}{14}$ [17,27,34,46,45]
Table 3.  Incremental cost-effectiveness ratio in increasing order of total infection averted
Strategies Total infection averted Total Cost ICER
Strategy C $2.4390\times 10^{6}$ $9.3014\times 10^{7}$ $38.1361$
Strategy A $2.7536\times 10^{6}$ $ 6.5306\times 10^{7}$ $-88.0737$
Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
Strategies Total infection averted Total Cost ICER
Strategy C $2.4390\times 10^{6}$ $9.3014\times 10^{7}$ $38.1361$
Strategy A $2.7536\times 10^{6}$ $ 6.5306\times 10^{7}$ $-88.0737$
Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
Table 4.  Incremental cost-effectiveness ratio in increasing order of total infection averted
Strategies Total infection averted Total Cost ICER
Strategy A $2.7536\times 10^{6}$ $ 6.5306\times 10^{7}$ $23.7166$
Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
Strategies Total infection averted Total Cost ICER
Strategy A $2.7536\times 10^{6}$ $ 6.5306\times 10^{7}$ $23.7166$
Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $-76.1028$
Table 5.  Incremental cost-effectiveness ratio in increasing order of total infection averted
Strategies Total infection averted Total Cost ICER
Strategy F $2.7536\times 10^{6}$ $6.1476\times 10^{7}$ $22.32536$
Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-98.3986$
Strategy D $3.3475\times 10^{6}$ $6.5525\times 10^{6}$ $1035.6129$
Strategies Total infection averted Total Cost ICER
Strategy F $2.7536\times 10^{6}$ $6.1476\times 10^{7}$ $22.32536$
Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-98.3986$
Strategy D $3.3475\times 10^{6}$ $6.5525\times 10^{6}$ $1035.6129$
Table 6.  Incremental cost-effectiveness ratio in increasing order of total infection averted
Strategies Total infection averted Total Cost ICER
Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $6.7470$
Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-710.5187$
Strategy G $3.3475\times 10^{6}$ $2.7131\times 10^{6}$ $-202.9032$
Strategies Total infection averted Total Cost ICER
Strategy B $3.3176\times 10^{6}$ $2.2384\times 10^{7}$ $6.7470$
Strategy E $3.3444\times 10^{6}$ $3.3421\times 10^{6}$ $-710.5187$
Strategy G $3.3475\times 10^{6}$ $2.7131\times 10^{6}$ $-202.9032$
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