Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity

BioSystems

Volumn 106, Issue 2-3, 2011, Pages 136-145

  Okosun, K O ^a,c   Ouifki, Rachid ^b   Marcus, Nizar ^c  

^a UNIVERSITY OF THE WESTERN CAPE   (South Africa)

^b STELLENBOSCH UNIVERSITY   (South Africa)

^c UNIVERSITY OF CAPE TOWN   (South Africa)

Author keywords

Bifurcation; Cost effectiveness; Malaria; Optimal control; Stability

Indexed keywords

DISEASE SPREAD; DISEASE TRANSMISSION; EPIDEMIOLOGY; IMMUNITY; INFECTIOUS DISEASE; MALARIA; NUMERICAL MODEL; OPTIMIZATION; VACCINATION;

ANALYSIS; ARTICLE; BASIC REPRODUCTION NUMBER; DISEASE MODEL; DISEASE TRANSMISSION; EPIDEMIOLOGICAL DATA; HUMAN; IMMUNITY; MALARIA; MALARIA CONTROL; VACCINATION;

EID: 80054851833     PISSN: 03032647     EISSN: 18728324     Source Type: Journal    
DOI: 10.1016/j.biosystems.2011.07.006     Document Type: Article

References (64)

1. Ashrafi M.N., Gumel A.B. Mathematical analysis of the role of repeated exposure on malaria transmission dynamics. Differ. Equat. Dyn. Syst. 2008, 16(3):251-287.
2. Abdulla S., Oberholzer R., Juma O., Kubhoja S., Machera F., Membi C., Omari S., Urassa A., Mshinda H., Jumanne A., Salim N., Shomari M., Aebi T., Schellenberg D.M., Carter T., Villafana T., Demoiti M.A., Dubois M.C., Leach M., Lievens A., Vekemans J., Cohen J., Ballou W.R., Tanner Safety and immunogenicity of RTS,S/AS02 malaria vaccine in infants. N. Engl. J. Med. 2008, 359:2533-2544.
3. Adams B.M., Banks H.T., Kwon H., Tran H.T. Dynamic multidrug therapies for HIV: optimal and STI control approaches. Math. Biosci. Eng. 2004, 1(2):223-241.
4. Ainseba, B., Benosman, C., 2010. Optimal control for resistance and suboptimal response in CML. Math. Biosci., 227, 81-93.
5. Alonso P.L., Sacarlal J., Aponte J.J., Leach A., Macete E., Milman J., Mandomando I., Spiessens B., Guinovart C., Espasa M., Bassat Q., Aide P., Ofori-Anyinam O., Navia M.M., Corachan S., Ceuppens M., Dubois M.C., Demoiti M.A., Dubovsky F., Menndez C., Tornieporth N., Ballou W.R., Thompson R., Cohen J. Efficacy of the RTS,S/AS02A vaccine against Plasmodium falciparum infection and disease in young African children: randomized controlled trial. Lancet 2004, 364(9443):1411-1420. 16-22.
6. Alonso P.L., Sacarlal J., Aponte J.J., Leach A., Macete E., Aide P., Sigauque B., Milman J., Mandomando I., Bassat Q., Guinovart C., Espasa M., Corachan S., Lievens M., Navia N.M., Dubois M.C., Menendez C., Dubovsky F., Cohen J., Thompson R., Ballou W.R. Duration of protection with RTS,S/AS02A malaria vaccine in prevention of Plasmodium falciparum disease in Mozambican children: single-blind extended follow-up of a randomised controlled trial. Lancet 2005, 366(9502):2012-2018. 10.
7. Aponte J.J., Aide P., Renom M., Mandomando I., Bassat Q., Sacarlal J., Manaca M.N., Lafuente S., Barbosa A., Leach A., Lievens M., Vekemans J., Sigauque B., Dubois M.C., Demoiti M.A., Sillman M., Savarese B., McNeil J.G., Macete E., Ballou W.R., Cohen J., Alonso P.L. Safety of the RTS,S/AS02D candidate malaria vaccine in infants living in a highly endemic area of Mozambique: a double blind randomised controlled phase I/IIb trial. Lancet 2007, 370(9598):1543-1551.
8. Augas R., White L.J., Snow R.W., Gabriela M., Gomes M. Prospects for malaria eradication in sub-Saharan Africa. PLoS One 2008, 3(3):e1767.
9. Barrett S., Hoel M. Optimal disease eradication. Environ. Dev. Econ. 2007, 12:627-652.
10. Bejon P., Lusingu J., Olotu A., Leach A., Lievens M., Vekemans J., Mshamu S., Lang J., Gould J., Dubois M.C., Demoiti M.A., Stallaert J.F., Vansadia P., Carter T., Njuguna P., Awuondo K.O., Malabeja A., Abdul O., Gesase S., Mturi N., Drakeley C.J., Savarese B., Villafana T., Ballou W.R., Cohen J., Riley E.M., Lemnge M.M., Marsh K., Von Seidlein L. Efficacy of RTS,S/AS01E vaccine against malaria in children 5 to 17 months of age. N. Engl. J. Med. 2008, 359:2521-2532.
11. Blayneh K., Cao Y., Hee-Dae K. Optimal control of vector-borne diseases: treatment and prevention. Discrete Contin. Dyn. Syst. Ser. B 2009, 11:587-611.
12. Bojang K.A., Milligan P.J.M., Pinder M., Vigneron L., Alloueche A., Kester K.E., Ballou W.R., Conway D.J., Reece W.H.H., Gothard P., Yamuah L., Delchambre M., Voss G., Greenwood B.M., Hill A., McAdam K.P.W.J., Tornieporth N., Cohen J.D., Doherty T., for the RTS,S Malaria Vaccine Trial Team Efficacy of RTS,S/AS02 malaria vaccine against Plasmodium falciparum infection in semi-immune adult men in The Gambia: a randomised trial. Lancet 2001, 358(9297):1927-1934.
13. Bojang K., Milligan P., Pinder M., Doherty T., Leach A., Ofori-Anyinam O., Lievens M., Kester K., Schaecher K., Ballou W.R., Cohen J. Five-year safety and immunogenicity of GlaxoSmithKline's candidate malaria vaccine RTS,S/AS02 following administration to semi-immune adult men living in a malaria-endemic region of The Gambia. Hum. Vaccine 2009, 5(4):242-247.
14. Cesar C. Optimal control of an epidemic through educational campaigns. Electron. J. Differ. Equat. 2006, 125:1-11.
15. Chiyaka C., Tchuenche J.M., Garira W., Dube S. A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria. Appl. Math. Comput. 2008, 195:641-662.
16. Chiyaka C., Garira W., Dube S. Effects of treatment and drug resistance on the transmission dynamics of malaria in endemic areas. Theor. Popul. Biol. 2009, 75:14-29.
17. de Souza F., Antonio Leonel Caetano J.A.M., Yoneyama T. Optimal control theory applied to the anti-viral treatment of AIDS. Proceeding of Conference on Decision and Control 2000.
18. Denis K., Lenhart S., Steve S. Optimal control of the chemotherapy of HIV. J. Math. Biol. 1997, 35:775-792.
19. Dietz K., Molineaux L., Thomas A. A malaria model tested in the African savannah. Bull. World Health Organ. 1974, 50:347-357.
20. Doedel, E.J., Paffenroth, R.C., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X., 2002. AUTO 2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), v.0.9.7, 2002, online at http://sourceforge.net/projects/auto2000/.
21. Fleming W.H., Rishel R.W. Deterministic and Stochastic Optimal Control 1975, Springer Verlag, New York.
22. Garba S.M., Gumel A.B., Abu B. Backward bifurcations in dengue transmission dynamics. Math. Biosci. 2008, 215:11-25.
23. Goldman S.M., Lightwood J. Cost optimisation in the SIS model of infectious disease with treatment. Top. Econ. Anal. Policy 2002, 2. article 4.
24. Gumel A.B., Song B. Existence of multiple-stable equilibria for a multi-drug-resistant model of Mycobacterium tuberculosis. Math. Biosci. Eng. 2008, 5:437-455.
25. Gupta N.K., Rink R.E. Optimal control of epidemics. Math. Biosci. 1973, 18:383-396.
26. Isao K., Akira S., Motoyoshi M. Combining zooprophylaxis and insecticide spraying: a malaria-control strategy limiting the development of insecticide resistance in vector mosquitoes. Proc. R. Soc. Lond. 2004, 271:301-309. 10.1098/rspb.2003.2575.
27. Joshi H.R., Lenhart S., Li M.Y., Wang L. Optimal control methods applied to disease models. Contemp. Math. 2006, 410:187-207.
28. Joshi H.R. Optimal control of an HIV immunology model. Optim. Control Appl. Math. 2002, 23:199-213.
29. Kar T.K., Batabyal A. Stability analysis and optimal control of an SIR epidemic model with vaccination. Biosystems 2011, 104(2/3):127-135.
30. Karrakchou J., Rachik M., Gourari S. Optimal control and infectiology: application to an HIV/AIDS model. Appl. Math. Comput. 2006, 177:807-818.
31. Kirschner D., Lenhart S., Serbin S. Optimal control of the chemotherapy of HIV. J. Math. Biol. 1997, 35:775-792.
32. Koella J.C., Anita R. Epidemiological models for the spread of anti-malaria resistance. Malaria J. 2003, 2:1-11.
33. Lenhart S., Bhat M.G. Application of distributed parameter control model in wildlife damage management. Math. Models Methods Appl. Sci. 1992, 2:423-439.
34. Lenhart S., Workman J.T. Optimal Control Applied to Biological Models 2007, Chapman and Hall.
35. Lenhart S., Yong J. Optimal Control for Degenerate Parabolic Equations with Logistic Growth 1997, Preprint Institute for Mathematics and Application.
36. Li J. A malaria model with partial immunity in humans. Math. Biosci. Eng. 2008, 5:789-801.
37. Lindsay S.W., Martens W.J.M. Malaria in the African highlands: past, present and future. Bull. WHO 1998, 76:33-45.
38. Lord C.C., Woolhouse M.E.J., Heesterbeek J.A.P. Vector-borne diseases and the basic reproduction number: a case study of African horse sickness. Med. Vet. Entomol. 1996, 10:19-28.
39. Maire N., Aponte J.J., Ross A., Thompson R., Alonso P., Utzinger J., Tanner M., Smith T. Modeling a field trial of the RTS,S/AS02A malaria vaccine. Am. J. Trop. Med. Hyg. 2006, 75(2 Suppl.):104-110.
40. Makinde O.D., Okosun K.O. Impact of chemo-therapy on optimal control of malaria disease with infected immigrants. Biosystems 2011, 104(1):32-41. 10.1016/j.biosystems.2010.12.010.
41. Marsh K. Malaria disaster in Africa. Lancet 1998, 352:924-925.
42. Mukandavire Z., Gumel A.B., Garira W., Tchuenche J.M. Mathematical analysis of a model for HIV-malaria co-infection. Math. Biosci. Eng. 2009, 6:333-362.
43. Mushayabasa S., Tchuenche J.M., Bhunu C.P., Ngarakana-Gwasira E. Modelling gonorrhea and HIV co-interaction. J. Biosyst. 2011, 103:27-37.
44. Nakul C., Cushing J.M., Hyman J.M. Bifurcation analysis of a mathematical model for malaria transmission. SIAM J. Appl. Math. 2006, 67(1):24-45.
45. Nakul C., Hyman J.M., Cushing J.M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 2008, 70:1272-1296.
46. Nanda S., Moore H., Lenhart S. Optimal control of treatment in a mathematical model of chronic myelogenous leukemia. Math. Biosci. 2007, 210:143.
47. Owusu-Agyei S., Ansong D., Asante K., Kwarteng Owusu S., Owusu R., Wireko Brobby N.A., Dosoo D., Osei Akoto A., Osei-Kwakye K., Adjei E.A., Boahen K.O., Sylverken J., Adjei G., Sambian D., Apanga S., Kayan K., Vekemans J., Ofori-Anyinam O., Leach A., Lievens M., Demoitie M.A., Dubois M.C., Cohen J., Ballou W.R., Savarese B., Chandramohan D., Gyapong J.O., Milligan P., Antwi S., Agbenyega T., Greenwood B., Evans J. Randomized controlled trial of RTS,S/AS02D and RTS,S/AS01E malaria candidate vaccines given according to different schedules in Ghanaian children. PLoS One 2009, 4(10):e7302.
48. Polhemus M.E., Remich S.A., Ogutu B.R., Waitumbi J.N., Otieno L., Apollo S., Cummings J.F., Kester K.E., Ockenhouse C.F., Stewart A., Ofori-Anyinam O., Ramboer I., Cahill C.P., Lievens M., Dubois M.C., Demoitie M.A., Leach A., Cohen J., Ballou W.R., Heppner D.G. Evaluation of RTS,S/AS02A and RTS,S/AS01B in adults in a high malaria transmission area. PLoS One 2009, 4(7):e6465.
49. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The Mathematical Theory of Optimal Processes 1962, Wiley, New York.
50. Rafikov M., Bevilacqua L., Wyse A.P.P. Optimal control strategy of malaria vector using genetically modified mosquitoes. J. Theor. Biol. 2009, 258:418-425.
51. Sacarlal J., Aponte J.J., Aide P., Mandomando I., Bassat Q., Guinovart C., Leach A., Milman J., Macete E., Espasa M., Ofori-Anyinam O., Thonnard J., Corachan S., Dubois M.C., Lievens M., Dubovsky F., Ballou W.R., Cohen J., Alonso P.L. Safety of the RTS,S/AS02A malaria vaccine in Mozambican children during a Phase IIb trial. Vaccine 2008, 26(2):174-184.
52. Sacarlal J., Aide P., Aponte J.J., Renom M., Leach A., Mandomando I., Lievens M., Bassat Q., Lafuente S., Macete E., Vekemans J., Guinovart C., Sigaque B., Sillman M., Milman J., Dubois M.C., Demoiti M.A., Thonnard J., Menndez C., Ballou W.R., Cohen J., Alonso P.L. Long-term safety and efficacy of the RTS,S/AS02A malaria vaccine in Mozambican children. J. Infect. Dis. 2009, 200:329-336.
53. Sachs J.D. A new infected global effort to control malaria. Science 2002, 298:122-124.
54. Seierstad A., Sydsaeter K. Optimal Control Theory with Economic Applications 1987, North-Holland, Amsterdam.
55. Sethi S.P. Optimal quarantine programmes for controlling an epidemic spread. J. Opl. Res. Soc. 1978, 29:265-268.
56. Sethi S.P., Staats W.P. Optimal control of some simple deterministic epidemic models. J. Opl. Res. Soc. 1978, 29:129-136.
57. Tumwiine J., Mugisha J.Y.Y., Luboobi L.S. Threshhold and stability results for a malaria model in a population with protective intervention among high-risk groups. Math. Modell. Anal. 2008, 13:443-460.
58. Van den Driessche P., Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 2002, 180:29-48.
59. Vásquez Pinzón J.U., Méndez-Sánchez S.C., Duque Luna J.E. Mathematical model for the analysis of vaccination and vector control in malaria incidence. Rev. Salud Publica (Bogota) 2003, 5(2):172-179.
60. WHO expert committee on malaria, 20th report, WHO Regional Office of Africa, 2003.
61. Wickwire K. A note on the optimal control of carrier-borne epidemic. J. Appl. Probab. 1975, 12:565-568.
62. Yang H.M. Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector). Rev. Saude Publica 2000, 34(3):223-231.
63. Yang H.M. A mathematical model for malaria transmision relating global warming and local socioeconomic conditions. Rev. Saude Publica 2001, 35(3):224-231.
64. Zhixing H., Sheng L., Hui W. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate. Nonlinear Anal: Real World Appl. 2008, 9:2302-2312.