Optimal control of the spread of malaria superinfectivity

Journal of Biological Systems

Volumn 21, Issue 4, 2013, Pages

  Agusto, Folashade ^a   Lenhart, Suzanne ^b  

^a AUSTIN PEAY STATE UNIVERSITY   (United States)

^b UNIVERSITY OF TENNESSEE   (United States)

Author keywords

Differential Equations; Malaria; Optimal Control; Optimal System; Superinfection

Indexed keywords

EID: 84881013165     PISSN: 02183390     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218339013400020     Document Type: Article

References (82)

1. World Health Organization (WHO) Malaria, 2010, Available at http://www.who.int/mediacentre/factsheets/fs094/en/.
2. World Health Organization (WHO) World malaria report, 2009.
3. Lindsay SW, Martens WJM, Malaria in the African highlands: Past, present and future, Bull WHO 76:33-45, 1998.
4. Zhou G, Minakawa N, Githeko AK, Yan G, Association between climate variability and malaria epidemics in the east African highlands, Proc Natl Acad Sci USA 101(8):2375-2380, 2004.
5. Bush AO, Fernandez JC, Esch GW, Seedv JR, Parasitism: The Diversity and Ecology of Animal Parasites, 1st ed., Cambridge University Press, Cambridge, 2001.
6. Grimwade K, French N,Mbatha DD, Zungu DD, Dedicoat M, Gilks CF, HIV infection as a cofactor for severe falciparum malaria in adults living in a region of unstable malaria transmission in South Africa, Aids 18:547-554, 2004.
7. Cancr'e N, Tall A, Rogier C, Faye J, Sarr O, Trape J, Spiegel A, Bois F, Bayesian analysis of an epidemiologic model of Plasmodium falciparum malaria infection in ndiop, senegal, Am J Epidemiol 152(8):760-770, 2000.
8. Rosenberg R, Andre RG, Ketrangse S, Seasonal fluctuation of Plasmodium falciparum gametocytaemia, Trans R Soc Trop Med Hyg 84:29-33, 1990.
9. Smith DL, Dushoff J, Snow RW, Hay SI, The entomological inoculation rate and Plasmodium falciparum infection in African children, Nature 438(7067):492-495, 2005.
10. Smith DL, Simon IH, Endemicity response timelines for Plasmodium falciparum elimination, Malar J 8:87, 2009.
11. Dietz K, Molineaux L, Thomas A, A malaria model tested in the African savannah, Bull World Health Org 50:347-357, 1974.
12. Nedelman J, Introductory review: Some new thoughts about some old malaria models, Math Biosci 73:159-182, 1985.
13. Nedelman J, Inoculation and recovery rates in the malaria model of Dietz, Molineaux, and Thomas, Math Biosci 69:209-233, 1984.
14. Parthasarathy PR, The effect of superinfection on the distribution of the infectious period - A continued fraction approximation, IMA J Math Appl Med Biol 14:113- 123, 1997.
15. MacDonald G, The analysis of infection rates in diseases in which superinfection occurs, Trop Dis Bull 47:907-914, 1950.
16. Niger AM, Gumel AB, Mathematical analysis of the role of repeated exposure on malaria transmission dynamics, Differ Equ Dyn Syst 16(3):251-287, 2008.
17. Rutledge LC, Gould DJ, Tantichareon B, Factors affecting the infection of anophelines with human malaria in Thailand, Trans R Soc Trop Med Hyg 63:613-619, 1969.
18. Macdonald G, Majid A, Report on an intensive malaria survey in the karmal district, Punjab Rec Malar Surv India 2:423, 1931.
19. Drakeley CJ, Secka1 I, Correa1 S, Greenwood BM, Targett GAT, Host haematological factors influencing the transmission of Plasmodium falciparum gametocytes to Anopheles gambiae s.s. mosquitoes, Trop Med Int Health 4(2):131-138, 1999.
20. Wilson DB, Report of the malaria unit, Tonga, 1933-34, together with a report on a study of malaria in India, Government Printer, Dar-es-Salaam, 1936.
21. Greenwood BM, Bradley AK, Greenwood AM, Byass P, Jammeh K, Marsh K et al., Mortality and morbidity from malaria among children in a rural area of the Gambia, West Africa, Trans R Soc Trop Med Hyg 81:478-486, 1987.
22. Molineaux L, Gramiccia G, The Garki Project, World Health Organization, 1980.
23. Coggeshall L, Kumm HW, Effect of repeated superinfection upon the potency of immune serum of monkeys harboring chronic infections of plasmiodium knowlesi, J Exp Med 68(1):17-27, 1938.
24. Portugal S, Drakesmith H, Mota MM, Superinfection in malaria: Plasmodium shows its iron will, EMBO Rep 12(12):1233-1242, 2011.
25. van Santen S, de Mast Q, Swinkels DW, van der Ven AJAM, Hepcidin in malaria superinfection: Can findings be translated to humans? Nature Med 17(11):1341, 2011.
26. Portugal S, Carret C, Recker M, Armitage AE, Gonalves LA, Epiphanio S, Sullivan D, Roy C, Newbold CI, Drakesmith H, Mota MM, Host-mediated regulation of superinfection in malaria, Nat Med 17(6):732-738, 2011.
27. Ṅ asell I, On superinfection in malaria, IMA J Math Appl Med Biol 3:211-227, 1986.
28. Aneke SJ, Mathematical modelling of drug resistant malaria parasites and vector populations, Math Meth Appl Sci 25:335-346, 2002.
29. Bacae N, Sokna RC, A, reaction-diffusion system modeling the spread of resistance to an antimalarial drug, Math Biosci Eng 22:227-238, 2005.
30. Nakul C, Cushing JM, Hyman JM, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J Appl Math 67:24-45, 2006.
31. Chiyaka C, Garira W, Dube S, Transmission model of endemic human malaria in a partially immune population, Math Comput Modelling 46:806-822, 2007.
32. Chiyaka C, Tchuenche JM, Garira W, Dube S, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria, Appl Math Comput 195:641-662, 2008.
33. Koella JC, Antia R, Epidemiological models for the spread of antimalarial resistance, Malar J 2:3, 2003.
34. Mukandavire Z, Gumel AB, Garira W, Tchuenche JM, Mathematical analysis of a model for HIV-Malaria co-infection, Math Biosci Eng 6(2):333-362, 2009.
35. Blayneh K, Cao Y, Kwon H-D, Optimal control of vector-borne diseases: Treatment and prevention, Discrete Continuous Dyn Syst-Ser B 11(3):587-411, 2009.
36. Felippe de Souza JAM, Caetano MAL, Yoneyama T, Optimal control theory applied to the anti-viral treatment of AIDS, Proc 39th IEEE Conf Decision and Control Sydney, Australia, Vol. 5, pp. 4839-4844, 2000.
37. Fister KR, Lenhart S, McNally JS, Optimizing chemotherapy in an HIV model, Electron J Diff Equat 32:1-12, 1998.
38. Kwon H-D, Optimal treatment strategies derived from a HIV model with drugresistant mutants, Appl Math Comput 188:1193-1204, 2007.
39. Joshi HR, Optimal control of an HIV immunology model, Optim Control Appl Math 23:199-213, 2002.
40. Karrakchou J, Rachik M, Gourari S, Optimal control and infectiology: Application to an HIV/AIDS model, Appl Math Comput 177:807-818, 2006.
41. Kirschner D, Lenhart S, Serbin S, Optimal control of the chemotherapy of HIV,J Math Biol 35:775-792, 1997.
42. Jung E, Lenhart S, Feng Z, Optimal control of treatments in a two-strain tuberculosis model, Discrete Continuous Dyn Syst B 2(4):473-482, 2002.
43. Yan X, Zou Y, Li J, Optimal quarantine and isolation strategies in epidemics control, World J Modell Simul 3(3):202-211, 2007.
44. Yan X, Zou Y, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math Comput Modell 47:235-245, 2008.
45. Caetano MAL, Yoneyama T, Optimal and sub-optimal control in Dengue epidemics, Optim Control Appl Methods 22:63-73, 2001.
46. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF, The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.
47. Aron JL, May RM, The population dynamics of malaria, in Anderson RM (ed.) The Population Dynamics of Infectious Diseases: Theory and Application, Halstead, New York, pp. 139-179, 1980.
48. Bailey NTJ, The Biomathematics of Malaria, Griffin, London, 1982.
49. Covell G, Relationship between malaria paresitaemia and symptoms of the disease: A review of the literature, Bull World Health Org 22:605-619, 1960.
50. Lakshmikantham V, Leela S, Martynyuk AA, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York and Basel, 1989.
51. Hethcote HW, The mathematics of infectious diseases, SIAM Rev 42(4):599-653, 2000.
52. van den Driessche P, Watmough J, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci 180:29-48, 2002.
53. Anderson RM, May R, Infectious Diseases of Humans, Oxford University Press, New York, 1991.
54. Diekmann O, Heesterbeek JAP, Metz JAP, On the definition and computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J Math Biol 28:503-522, 1990.
55. Agusto FB, Gumel AB, Theoretical assessment of avian influenza vaccine, Discrete Continuous Dyn Syst B 13(1):1-25, 2010, doi:10.3934/dcdsb.2009.13.1.
56. Agusto FB, Gumel AB, Quantitative analysis for a model for the transmission dynamics of low and highly pathogenic avian influenza, Math Biosci 243(2):147-162, 2013.
57. Agusto FB, Del Valle SY, Blayneh KW, Ngonghala CN, Goncalves MJ, Li N, Zhao R, Gong H, The impact of bed-net use on malaria prevalence, J Theor Biol 320(7):58-65, 2013.
58. Brauer F, Backward bifurcations in simple vaccination models, J Math Anal Appl 298(2):418-431, 2004.
59. Dushoff J, Wenzhang H, Castillo-Chavez C, Backwards bifurcations and catastrophe in simple models of fatal diseases, J Math Biol 36:227-248, 1998.
60. Elbasha EH, Gumel AB, Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits, Bull Math Biol 68:577-614, 2006.
61. Garba SM, Gumel AB, Abu Bakar MR, Backward bifurcations in dengue transmission dynamics, Math Biosci 215(1):11-25, 2008.
62. Sharomi O, Podder CN, Gumel AB, Elbasha EH, Watmough J, Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Math Biosci 210:436-463, 2007.
63. Sharomi O, Podder CN, Gumel AB, Song B, Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment, Math Biosci Eng 5(1):145-174, 2008.
64. Carr J, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981.
65. Castillo-Chavez C, Song B, Dynamical models of tuberculosis and their applications, Math Biosci Eng 1(2):361-404, 2004.
66. U.S. Census Bureau, International Database. 2007.
67. Flahault A, Le Menach A, McKenzie EF, Smith DL, The unexpected importance of mosquito oviposition behaviour for malaria: Non-productive larval habitats can be sources for malaria transmission, Malar J 4:23, 2005.
68. Ariey F, Robert V, The puzzling links between malaria transmission and drug resistance, Trends Parasitol 19:158-160, 2003.
69. Mbogob CM, Gu W, Killeena GF, An individual-based model of plasmodium falciparum malaria transmission on the coast of kenya, Trans R Soc Trop Med Hyg 97:43-50, 2003.
70. Snow RW, Omumbo J, Malaria, in Disease and Mortality in Sub-Saharan Africa, 2nd edn. Jamison DT, Feachem RG, Makgoba MW et al., eds. World Bank, Washington (DC), pp. 195-213, 2006.
71. Smith DL, Mckenzie EF, Statics and dynamics of malaria infection in anopheles mosquito, Malar J 3:13, 2004.
72. Sherman IW, Malaria: Parasite Biology, Pathogenesis, and Protection, AMS Press, Washington DC, 1998.
73. Mitchell VS, Oaks Jr. SC, Pearson GW, Malaria: Obstacles and Opportunities, National Academy Press, Washington DC, 1991.
74. Bowman C, Gumel AB, van den Driessche P, Wu J, Zhu H, A mathematical model for assessing control strategies against West Nile virus, Bull Math Biol 67:1107-1133, 2005.
75. Breman JG, Holloway CN, Malaria surveillance counts, Am J Trop Med Hyg 77:36-47, 2007.
76. Ruiz D, Poveda G, Vlez ID et al., Modelling entomological-climatic interactions of plasmodium falciparum malaria transmission in two colombian endemic-regions: Contributions to a national malaria early warning system, Malar J 5:66, 2006.
77. Deressa W, Ali A, Enqusellassie F, Self-treatment of malaria in rural communities, Butajira, southern Ethiopia, Bull World Health Org 81:261-268, 2003.
78. Deressa W, Chibsa S, Olana D, Treatment seeking for malaria patients in East Shewa Zone of Oromia, Ethiopia, Ethiopian J Health Dev 17:9-16, 2003.
79. Deressa W, Treatment-seeking behaviour for febrile illness in an area of seasonal malaria transmission in rural Ethiopia, Malar J 6:49, 2007.
80. Fleming WH, Rishel RW, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975.
81. Krener AJ, The high order maximal principle and its application to singular extremals, SIAM J Control Optim 15:156-293, 1977.
82. Lenhart S, Workman JT, Optimal Control Applied to Biological Models, Chapman and Hall, 2007.