Advertisement

Bulletin of Mathematical Biology

, Volume 81, Issue 11, pp 4366–4411 | Cite as

Analysis of Malaria Control Measures’ Effectiveness Using Multistage Vector Model

  • Jean Claude KamgangEmail author
  • Christopher Penniman Thron
Special Issue: Mathematical Epidemiology

Abstract

We analyze an epidemiological model to evaluate the effectiveness of multiple means of control in malaria-endemic areas. The mathematical model consists of a system of several ordinary differential equations and is based on a multi-compartment representation of the system. The model takes into account the multiple resting–questing stages undergone by adult female mosquitoes during the period in which they function as disease vectors. We compute the basic reproduction number \(\mathcal R_0\) and show that if \(\mathcal R_0\le 1\), the disease-free equilibrium is globally asymptotically stable (GAS) on the nonnegative orthant. If \(\mathcal R_0>1\), the system admits a unique endemic equilibrium (EE) that is GAS. We perform a sensitivity analysis of the dependence of \(\mathcal R_0\) and the EE on parameters related to control measures, such as killing effectiveness and bite prevention. Finally, we discuss the implications for a comprehensive, cost-effective strategy for malaria control.

Keywords

Epidemiological model Malaria Basic reproduction number Lyapunov function Global asymptotic stability Control strategies Sensitivity analysis 

Mathematics Subject Classification

34C60 34D20 34D23 92D30 

Notes

Acknowledgements

The second author would like to thank the Fulbright U.S. Scholar program and the International Mathematicians Union’s VLP program for supporting visits to ENSAI which made this research possible. The authors would also like to thank the reviewers for helpful comments that led to significant improvements in the quality and clarity of this paper.

References

  1. Akhavan D, Musgrove P, Abrantes A, d’A Gusmão R (1999) Cost-effective malaria control in Brazil: cost-effectiveness of a malaria control program in the Amazon basin of Brazil, 1988–1996. Soc Sci Med 49(10):1385–1399Google Scholar
  2. Atieli HE, Zhou G, Afrane Y, Lee M-C, Mwanzo I, Githeko AK, Yan G (2011) Insecticide-treated net (ITN) ownership, usage, and malaria transmission in the highlands of western Kenya. Parasites Vectors 4(1):113Google Scholar
  3. Awoleye OJ, Thron C (2016) Improving access to malaria rapid diagnostic test in Niger state, Nigeria: an assessment of implementation up to 2013. Malar Res Treat 2016:7436265.  https://doi.org/10.1155/2016/7436265 CrossRefGoogle Scholar
  4. Barbour AD (1978) Macdonald’s model and the transmission of bilharzia. Trans R Soc Trop Med Hyg 72(1):6–15Google Scholar
  5. Bayoh MN, Mathias DK, Odiere MR, Mutuku FM, Kamau L, Gimnig JE, Vulule JM, Hawley WA, Hamel MJ, Walker ED (2010) Anopheles gambiae: historical population decline associated with regional distribution of insecticide-treated bed nets in western Nyanza province, Kenya. Malar J 9(1):62Google Scholar
  6. Beier JC, Keating J, Githure JI, Macdonald MB, Impoinvil DE, Novak RJ (2008) Integrated vector management for malaria control. Malar J 7(1):S4Google Scholar
  7. Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences, volume 9 of classics in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Revised reprint of the 1979 originalGoogle Scholar
  8. Besansky NJ, Hill CA, Costantini C (2004) No accounting for taste: host preference in malaria vectors. Trends Parasitol 20(6):249–251Google Scholar
  9. Bhatia NP, Szegö GP (1970) Stability theory of dynamical systems. Springer, BerlinzbMATHGoogle Scholar
  10. Carnevale P, Vincent R (2009) Les anophèles, Biologie, transmission du Paludisme et lutte antivectorielle. IRD, WellingtonGoogle Scholar
  11. Chitnis N (2005) Using mathematical models in controlling the spread of malaria. PhD thesis, University of ArizonaGoogle Scholar
  12. Chitnis N, Hyman JM, Cushing JM (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull Math Biol 70(5):1272MathSciNetzbMATHGoogle Scholar
  13. Ehiri JE, Anyanwu EC (2004) Mass use of insecticide-treated bednets in malaria endemic poor countries: public health concerns and remedies. J Public Health Policy 25(1):9–22Google Scholar
  14. Floore TG (2006) Mosquito larval control practices: past and present. J Am Mosq Control Assoc 22(3):527–533Google Scholar
  15. Fontenille D, Lochouarn L, Diagne N, Sokhna C, Lemasson JJ, Diatta M, Konate L, Faye F, Rogier C, Trape JF (1997) High annual and seasonal variations in malaria transmission by anophelines and vector species composition in Dielmo, a holoendemic area in Senegal. Am J Trop Med Hyg 56:247–53Google Scholar
  16. Gollin D, Zimmermann C (2007) Malaria: disease impacts and long-run income differences. IZA Discussion Papers 2997, Institution for the Study of Labor (IZA), August 2007Google Scholar
  17. Guo H, Li MY, Shuai Z (2006) Global stability of the endemic equilibrium of multigroup models. Can Appl Math Q 14(3):259–284MathSciNetzbMATHGoogle Scholar
  18. Guo H, Li MY, Shuai Z (2008) A graph-theoretic approach to the method of global Lyapunov functions. Proc Am Math Soc 136:2793–2802 MathSciNetzbMATHGoogle Scholar
  19. Hawley WA, Phillips-Howard PA, ter Kuile FO, Terlouw DJ, Vulule JM, Ombok M, Nahlen BL, Gimnig JE, Kariuki SK, Kolczak MS et al (2003) Community-wide effects of permethrin-treated bed nets on child mortality and malaria morbidity in western Kenya. Am J Trop Med Hyg 68(4-suppl):121–127Google Scholar
  20. Jacquez JA, Simon CP (1993) Qualitative theory of compartmental systems. SIAM Rev 35(1):43–79MathSciNetzbMATHGoogle Scholar
  21. Kamgang JC, Sallet G (2008) Computation of threshold conditions for epidemiological models and global stability of the disease free equilibrium. Math Biosci 213(1):1–12MathSciNetzbMATHGoogle Scholar
  22. Kamgang JC, Kamla VC, Tchoumi SY (2014) Modeling the dynamics of malaria transmission with bed net protection perspective. Appl Math 5(19):3156–3205, 11Google Scholar
  23. Keiser J, Singer BH, Utzinger J (2005) Reducing the burden of malaria in different eco-epidemiological settings with environmental management: a systematic review. Lancet Infect Dis 5(11):695–708Google Scholar
  24. Korobeinikov A (2001) A Lyapunov function for Leslie–Gower predator–prey models. Appl Math Lett 14(6):697–699MathSciNetzbMATHGoogle Scholar
  25. Korobeinikov A (2004) Lyapunov functions and global properties for SEIR and SEIS models. Math Med Biol 21:75–83zbMATHGoogle Scholar
  26. Korobeinikov A, Maini PK (2004) A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math Biosci Eng 1(1):57–60MathSciNetzbMATHGoogle Scholar
  27. Korobeinikov A, Wake GC (2002) Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models. Appl Math Lett 15(8):955–960MathSciNetzbMATHGoogle Scholar
  28. LaSalle JP (1968) Stability theory for ordinary differential equations. J Differ Equ 41:57–65MathSciNetzbMATHGoogle Scholar
  29. LaSalle JP (1976a) The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied MathematicsGoogle Scholar
  30. LaSalle JP (1976b) Stability theory and invariance principles. In: Dynamical systems (Proceedings of the international Symposium, Brown University, Providence, RI, 1974), vol I. Academic Press, New York, pp 211–222Google Scholar
  31. Lawrance CE, Croft AM (2004) Do mosquito coils prevent malaria? A systematic review of trials. J Travel Med 11(2):92–96Google Scholar
  32. Li J, Blakeley D, Smith RJ (2011) The failure of \(R_0\). Comput Math Method Med 2011:1–17zbMATHGoogle Scholar
  33. Luenberger DG (1979) Introduction to dynamic systems. Theory, models, and applications. Wiley, New YorkzbMATHGoogle Scholar
  34. Ma Z, Liu J, Li J (2003) Stability analysis for differential infectivity epidemic models. Nonlinear Anal Real World Appl 4(5):841–856MathSciNetzbMATHGoogle Scholar
  35. Maia MF, Kliner M, Richardson M, Lengeler C, Moore SJ (2015) Mosquito repellents for malaria prevention. Cochrane Database Syst Rev 2018(2):CD011595.  https://doi.org/10.1002/14651858.CD011595.pub2
  36. McCluskey CC (2003) A model of HIV/AIDS with staged progression and amelioration. Math Biosci 181(1):1–16MathSciNetzbMATHGoogle Scholar
  37. McCluskey CC (2005) A strategy for constructing Lyapunov functions for non-autonomous linear differential equations. Linear Algebra Appl 409:100–110MathSciNetzbMATHGoogle Scholar
  38. McCluskey CC (2006) Lyapunov functions for tuberculosis models with fast and slow progression. Math Biosci Eng 3(4):603–614MathSciNetzbMATHGoogle Scholar
  39. McCluskey CC (2008) Global stability fo a class of mass action systems allowing for latency in tuberculosis. J Math Anal Appl 338:518–535MathSciNetzbMATHGoogle Scholar
  40. McCluskey CC, van den Driessche P (2004) Global analysis of two tuberculosis models. J Dyn Differ Equ 16(1):139–166MathSciNetzbMATHGoogle Scholar
  41. Menze BD, Riveron JM, Ibrahim SS, Irving H, Antonio-Nkondjio C, Awono-Ambene PH, Wondji CS (2016) Multiple insecticide resistance in the malaria vector anopheles funestus from Northern Cameroon is mediated by metabolic resistance alongside potential target site insensitivity mutations. PLoS One 11(10):e0163261Google Scholar
  42. Morel CM, Lauer JA, Evans DB (2005) Cost effectiveness analysis of strategies to combat malaria in developing countries. Bmj 331(7528):1299Google Scholar
  43. Ngwa AG, Shu WS (2000) A mathematical model for endemic malaria with variable human and mosquito populations. Math Comput Model 32:747–763MathSciNetzbMATHGoogle Scholar
  44. Pluess B, Tanser FC, Lengeler C, Sharp BL (2010) Indoor residual spraying for preventing malaria. Cochrane Database Syst Rev 4(4):Google Scholar
  45. Ranson H, N’Guessan R, Lines J, Moiroux N, Nkuni Z, Corbel V (2011) Pyrethroid resistance in African anopheline mosquitoes: what are the implications for malaria control? Trends Parasitol 27(2):91–98Google Scholar
  46. Rogier C, Tall A, Diagne N, Fontenille D, Spiegel A, Trape JF (2000) Plasmodium falciparum clinical malaria: lessons from longitudinal studies in Senegal. Parassitologia 41(1–3):255–259Google Scholar
  47. Ross R (1911) The prevention of malaria. John Murray, LondonGoogle Scholar
  48. Russell TL, Govella NJ, Azizi S, Drakeley CJ, Kachur SP, Killeen GF (2011) Increased proportions of outdoor feeding among residual malaria vector populations following increased use of insecticide-treated nets in rural Tanzania. Malar J 10(1):80Google Scholar
  49. Russell CL, Sallau A, Emukah E, Graves PM, Noland GS, Ngondi JM, Ozaki M, Nwankwo L, Miri E, McFarland DA et al (2015) Determinants of bed net use in southeast Nigeria following mass distribution of LLINs: implications for social behavior change interventions. PLoS One 10(10):e0139447Google Scholar
  50. Sharp BL, Kleinschmidt I, Streat E, Maharaj R, Barnes KI, Durrheim DN, Ridl FC, Morris N, Seocharan I, Kunene S et al (2007) Seven years of regional malaria control collaboration Mozambique, South Africa, and Swaziland. Am J Trop Med Hyg 76(1):42–47Google Scholar
  51. Shillcutt S, Morel C, Goodman C, Coleman P, Bell D, Whitty CJM, Mills A (2008) Cost-effectiveness of malaria diagnostic methods in sub-Saharan Africa in an era of combination therapy. Bull World Health Organ 86:101–110Google Scholar
  52. Tewa JJ, Dimi JL, Bowong S (2009) Lyapunov functions for a dengue disease transmission model. Chaos Solitons Fractals 39(2):936–941MathSciNetzbMATHGoogle Scholar
  53. Tewa JJ, Fokouop R, Mewoli B, Bowong S (2012) Mathematical analysis of a general class of ordinary differential equations coming from within-hosts models of malaria with immune effectors. Appl Math Comput 218(14):7347–7361MathSciNetzbMATHGoogle Scholar
  54. Utzinger J, Tozan Y, Singer BH (2001) Efficacy and cost-effectiveness of environmental management for malaria control. Trop Med Int Health 6(9):677–687Google Scholar
  55. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetzbMATHGoogle Scholar
  56. Walker K, Lynch M (2007) Contributions of anopheles larval control to malaria suppression in tropical Africa: review of achievements and potential. Med Vet Entomol 21(1):2–21Google Scholar
  57. White MT, Conteh L, Cibulskis R, Ghani AC (2011) Costs and cost-effectiveness of malaria control interventions—a systematic review. Malar J 10(1):337Google Scholar
  58. WHO (2013) World malaria report 2013. Technical report, WHO, Dec 2013 Google Scholar
  59. Wilson AL et al (2011) A systematic review and meta-analysis of the efficacy and safety of intermittent preventive treatment of malaria in children (IPTc). PloS ONE 6(2):e16976Google Scholar
  60. World Health Organization (2015) World malaria report 2014. World Health Organization, GenevaGoogle Scholar
  61. Worrall E, Fillinger U (2011) Large-scale use of mosquito larval source management for malaria control in Africa: a cost analysis. Malar J 10(1):338Google Scholar
  62. Yohannes M, Haile M, Ghebreyesus TA, Witten KH, Getachew A, Byass P, Lindsay SW (2005) Can source reduction of mosquito larval habitat reduce malaria transmission in Tigray, Ethiopia? Trop Med Int Health 10(12):1274–1285Google Scholar
  63. Zhu L, Müller GC, Marshall JM, Arheart KL, Qualls WA, Hlaing WM, Schlein Y, Traore SF, Doumbia S, Beier JC (2017) Is outdoor vector control needed for malaria elimination? An individual-based modelling study. Malar J 16(1):266Google Scholar
  64. Zongo P (2009) Modélisation mathématique de la dynamique de transmission du paludisme. PhD thesis, Universite de OuagadougouGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesENSAI – University of N’GaoundéréN’GaoundéréCameroon
  2. 2.Department of Sciences and MathematicsTexas A&M University – Central TexasKilleenUSA

Personalised recommendations