Bulletin of Mathematical Biology

, Volume 76, Issue 7, pp 1607–1641 | Cite as

Malaria Drug Resistance: The Impact of Human Movement and Spatial Heterogeneity

Original Article

Abstract

Human habitat connectivity, movement rates, and spatial heterogeneity have tremendous impact on malaria transmission. In this paper, a deterministic system of differential equations for malaria transmission incorporating human movements and the development of drug resistance malaria in an \(n\) patch system is presented. The disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. For a two patch case, the boundary equilibria (drug sensitive-only and drug resistance-only boundary equilibria) when there is no movement between the patches are shown to be locally asymptotically stable when they exist; the co-existence equilibrium is locally asymptotically stable whenever the reproduction number for the drug sensitive malaria is greater than the reproduction number for the resistance malaria. Furthermore, numerical simulations of the connected two patch model (when there is movement between the patches) suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity. With slow movement (or low migration) between the patches, the drug sensitive strain dominates the drug resistance strain. However, with fast movement (or high migration) between the patches, the drug resistance strain dominates the drug sensitive strain.

Keywords

Malaria Drug resistance Human movement Spatial heterogeneity 

Mathematics Subject Classifications

92B05 93A30 93C15 

References

  1. Adams B, Kapan DD (2009) Man bites mosquito: understanding the contribution of human movement to vector-borne disease dynamics. PLoS One 4(8):e6763CrossRefGoogle Scholar
  2. Anderson RM, May R (1991) Infectious diseases of humans. Oxford University Press, New YorkGoogle Scholar
  3. Aneke SJ (2002) Mathematical modelling of drug resistant malaria parasites and vector populations. Math Methods Appl Sci 25:335–346Google Scholar
  4. Ariey F, Robert V (2003) The puzzling links between malaria transmission and drug resistance. Trends Parasitol 19(4):158–160CrossRefGoogle Scholar
  5. Ariey F, Duchemin JB, Robert V (2003) Metapopulation concepts applied to falciparum malaria and their impact on the emergence and spread of chloroquine resistance. Infect Genet Evol 2:185–192CrossRefGoogle Scholar
  6. Arino J, van den Driessche P (2003) A multi-city epidemic model. Math Popul Stud 10:175–193MathSciNetCrossRefMATHGoogle Scholar
  7. Arino J, van den Driessche P (2003) The basic reproducton number in a multi-city compartment model. LNCIS 294:135–142MATHGoogle Scholar
  8. Arino J, Ducrot A, Zongo P (2012) A metapopulation model for malaria with transmission-blocking partial immunity in hosts. J Math Biol 64(3):423–448MathSciNetCrossRefMATHGoogle Scholar
  9. Auger P, Kouokam E, Sallet G, Tchuente M, Tsanou B (2008) The Ross–Macdonald model in a patchy environment. Math Biosci 216:123–131MathSciNetCrossRefMATHGoogle Scholar
  10. Bacaer N, Sokna C (2005) A reaction–diffusion system modeling the spread of resistance to an antimalarial drug. Math Biosci Eng 2:227–238MathSciNetCrossRefGoogle Scholar
  11. Bowman C, Gumel AB, van den Driessche P, Wu J, Zhu H (2005) A mathematical model for assessing control strategies against West Nile virus. Bull Math Biol 67:1107–1133MathSciNetCrossRefGoogle Scholar
  12. Breman JG, Holloway CN (2007) Malaria surveillance counts. Am J Trop Med Hyg 77:36–47Google Scholar
  13. Bush AO, Fernandez JC, Esch GW, Seedv JR (2001) Parasitism: the diversity and ecology of animal parasites, 1st edn. Cambridge University Press, CambridgeGoogle Scholar
  14. Carrara VI, Sirilak S, Thonglairuam J, Rojanawatsirivet C (2006) Deployment of early diagnosis and mefloquine–artesunate treatment of falciparum malaria in Thailand: The Tak malaria initiative. PLoS Med 3(6):e183CrossRefGoogle Scholar
  15. Carrara VI, Zwang J, Ashley EA, Price RN et al (2009) Changes in the treatment responses to artesunate–mefloquine on the northwestern border of Thailand during 13 Years of continuous deployment. PLoS One 4:e4551CrossRefGoogle Scholar
  16. Cheeseman IH, Miller BA, Nair S, Nkhoma S et al (2012) A major genome region underlying artemisinin resistance in malaria. Science 336:79–82. doi:10.1126/science.1215966 CrossRefGoogle Scholar
  17. Chitnis N, Cushing JM, Hyman JM (2006) Bifurcation analysis of a mathematical model for malaria transmission. SIAM J Appl Math 67:24–45MathSciNetCrossRefMATHGoogle Scholar
  18. Chiyaka C, Tchuenche JM, Garira W, Dube S (2008) A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria. Appl Math Comput 195:641–662MathSciNetCrossRefMATHGoogle Scholar
  19. Cosner C, Beier JC, Cantrell RS, Impoinvil D, Kapitanski L, Potts MD, Troyo A, Ruan S (2009) The effects of human movement on the persistence of vector-borne diseases. J Theor Biol 258(4):550–560. doi:10.1016/j.jtbi.2009.02.016 MathSciNetCrossRefGoogle Scholar
  20. Denis MB, Tsuyuoka R, Lim P, Lindegardh N et al (2006) Efficacy of artemetherlumefantrine for the treatment of uncomplicated falciparum malaria in northwest Cambodia. Trop Med Int Health 11(12):1800–1807. doi:10.1111/j.1365-3156.2006.01739.x CrossRefGoogle Scholar
  21. Diekmann O, Heesterbeek JAP, Metz JAP (1990) On the definition and computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28:503–522MathSciNetCrossRefMATHGoogle Scholar
  22. Dietz K (1988) Mathematical models for transmission and control of malaria. In: Wensdorfer WH, McGregor I (eds) Malaria. Churchill Livingstone, Edinburgh, pp 1091–1133Google Scholar
  23. Dondorp AM, Nosten F, Yi P, Das D et al (2009) Artemisinin resistance in Plasmodium falciparum malaria. The N Engl J Med 361(5):455–467CrossRefGoogle Scholar
  24. Esteva L, Vargas C (2000) Influence of vertical and mechanical transmission on the dynamics of dengue disease. Math Biosci 167:51–64CrossRefMATHGoogle Scholar
  25. Esteva L, Gumel AB (2009) Qualitative study of transmission dynamics of drug-resistant malaria. Math Comput Model 50:611–630MathSciNetCrossRefMATHGoogle Scholar
  26. Feng Z, Yinfei Y, Zhu H (2004) Fast and slow dynamics of malaria and the s-gene frequency. J Dyn Differ Equ 16:869–895CrossRefMathSciNetMATHGoogle Scholar
  27. Flahault A, Le Menach A, McKenzie EF, Smith DL (2005) The unexpected importance of mosquito oviposition behaviour for malaria: non-productive larval habitats can be sources for malaria transmissionn. Malar J 4(1):23CrossRefGoogle Scholar
  28. Grimwade K, French N, Mbatha DD, Zungu DD, Dedicoat M, Gilks CF (2004) HIV infection as a cofactor for severe falciparum malaria in adults living in a region of unstable malaria transmission in South Africa. AIDS 18:547–554CrossRefGoogle Scholar
  29. Hastings IM (1997) A model for the origins and spread of drug resistant malaria. Parasitol 115:133–141CrossRefGoogle Scholar
  30. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599–653MathSciNetCrossRefMATHGoogle Scholar
  31. Hsieh Y, van den Driessche P, Wang L (2007) Impact of travel between patches for spatial spread of disease. Bull Math Biol 69:1355–1375MathSciNetCrossRefMATHGoogle Scholar
  32. Koella JC, Antia R (2003) Epidemiological models for the spread of antimalarial resistance. Malar J 2:3CrossRefGoogle Scholar
  33. Lakshmikantham V, Leela S, Martynyuk AA (1989) Stability analysis of nonlinear systems. Marcel Dekker, New York and BaselMATHGoogle Scholar
  34. Le Menach A, Ellis Mckenzie F (2005) The unexpected importance of mosquito oviposition behaviour for malaria: non-producive larval habitats can be sources for malaria transmission. Malar J 4(1):23CrossRefGoogle Scholar
  35. Lindsay SW, Martens WJM (1998) Malaria in the African highlands: past, present and future. Bull WHO 76:33–45Google Scholar
  36. Mackinnon MJ (2005) Drug resistance models for malaria. Acta Trop 94:207–217CrossRefGoogle Scholar
  37. Mbogob CM, Gu W, Killeena GF (2003) An individual-based model of Plasmodium falciparum malaria transmission on the coast of Kenya. Trans R Soc Trop Med Hyg 97:43–50CrossRefGoogle Scholar
  38. Molineaux L, Gramiccia G (1980) The Garki project. World Health Organization, GenevaGoogle Scholar
  39. Niger AM, Gumel AB (2008) Mathematical analysis of the role of repeated exposure on malaria transmission dynamics. Differ Equ Dyn Syst 16(3):251–287MathSciNetCrossRefMATHGoogle Scholar
  40. Phyo AP, Nkhoma S, Stepniewska K, Ashley EA et al (2012) Emergence of artemisinin-resistant malaria on the western border of Thailand: a longitudinal study. Lancet 379:1960–1966. doi:10.1016/S0140-6736(12)60484-X CrossRefGoogle Scholar
  41. Pongtavornpinyo W, Yeung S, Hastings IM, Dondorp AM, Day NPJ, White NJ (2008) Spread of anti-malarial drug resistance: mathematical model with implications for ACT drug policies. Malar J 7:229CrossRefGoogle Scholar
  42. Prosper OF, Ruktanoncha N, Martcheva M (2012) Assessing the role of spatial heterogeneity and human movement in malaria dynamics and control. J Theor Biol 303:1–14CrossRefMathSciNetGoogle Scholar
  43. Rodrguez DJ, Torres-Sorando L (2001) Models of infectious diseases in spatially heterogeneous environments. Bull Math Biol 63:547–571CrossRefMATHGoogle Scholar
  44. Ross R (1911) The prevention of malaria. John Murray, LondonGoogle Scholar
  45. Salmani M, van den Driessche P (2006) A model for disease transmission in a patchy environment. Discret Contin Dyn Syst Ser B 6:185–202MathSciNetMATHGoogle Scholar
  46. Smith HL, Waltman P (1995) The theory of the chemostat. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  47. Smith DL, Mckenzie EF (2004) Statics and dynamics of malaria infection in anopheles mosquito. Malar J 3:13CrossRefGoogle Scholar
  48. Smith DL, Dushoff J, Ellis Mckenzie F (2005) The risk of a mosquito-borne infection in a heterogeneous environnement. PLoS Biol 2:1957–1964Google Scholar
  49. Smith T, Killen GF, Maire N, Ross A, Molineaux L, Tediosi F, Hutton G, Utzinger J, Dietz K, Tanner M (2006) Mathematical modelling of the impact of malaria vaccines on the clinical epidemiology and natural history of Plasmodium falciparum malaria: Overview. Am J Trop Med Hyg 75:1–10Google Scholar
  50. Snow RW, Omumbo J (2006) In: Jamison DT et al (eds) Malaria, in diseases and mortality in Sub-Saharan Africa. The World Bank, WashingtonGoogle Scholar
  51. US Census Bureau International database (2010)Google Scholar
  52. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetCrossRefMATHGoogle Scholar
  53. World Health Organization (WHO) Malaria (2010) http://www.who.int/mediacentre/factsheets/fs094/en/
  54. World Health Organization (WHO) World malaria report 2009Google Scholar
  55. Zhou G, Minakawa N, Githeko AK, Yan G (2004) Association between climate variability and malaria epidemics in the east African highlands. Proc Natl Acad Sci USA 101:2375–2380CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAustin Peay State UniversityClarksvilleUSA

Personalised recommendations