Journal of Mathematical Biology

, Volume 70, Issue 7, pp 1581–1622

Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito populations: implications for control

  • Calistus N. Ngonghala
  • Miranda I. Teboh-Ewungkem
  • Gideon A. Ngwa

DOI: 10.1007/s00285-014-0804-9

Cite this article as:
Ngonghala, C.N., Teboh-Ewungkem, M.I. & Ngwa, G.A. J. Math. Biol. (2015) 70: 1581. doi:10.1007/s00285-014-0804-9


We derive and study a deterministic compartmental model for malaria transmission with varying human and mosquito populations. Our model considers disease-related deaths, asymptomatic immune humans who are also infectious, as well as mosquito demography, reproduction and feeding habits. Analysis of the model reveals the existence of a backward bifurcation and persistent limit cycles whose period and size is determined by two threshold parameters: the vectorial basic reproduction number \(\fancyscript{R}_{m}\), and the disease basic reproduction number \(\fancyscript{R}_0\), whose size can be reduced by reducing \(\fancyscript{R}_{m}\). We conclude that malaria dynamics are indeed oscillatory when the methodology of explicitly incorporating the mosquito’s demography, feeding and reproductive patterns is considered in modeling the mosquito population dynamics. A sensitivity analysis reveals important control parameters that can affect the magnitudes of \(\fancyscript{R}_{m}\) and \(\fancyscript{R}_0\), threshold quantities to be taken into consideration when designing control strategies. Both \(\fancyscript{R}_{m}\) and the intrinsic period of oscillation are shown to be highly sensitive to the mosquito’s birth constant \(\lambda _{m}\) and the mosquito’s feeding success probability \(p_{w}\). Control of \(\lambda _{m}\) can be achieved by spraying, eliminating breeding sites or moving them away from human habitats, while \(p_{w}\) can be controlled via the use of mosquito repellant and insecticide-treated bed-nets. The disease threshold parameter \(\fancyscript{R}_0\) is shown to be highly sensitive to \(p_{w}\), and the intrinsic period of oscillation is also sensitive to the rate at which reproducing mosquitoes return to breeding sites. A global sensitivity and uncertainty analysis reveals that the ability of the mosquito to reproduce and uncertainties in the estimations of the rates at which exposed humans become infectious and infectious humans recover from malaria are critical in generating uncertainties in the disease classes.


Malaria transmission and control Mosquito demography  Backward bifurcation Oscillatory dynamics Sensitivity analysis Reproduction numbers 

Mathematics Subject Classification

34A 34C 37G 92B 92D 

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Calistus N. Ngonghala
    • 1
    • 2
  • Miranda I. Teboh-Ewungkem
    • 3
  • Gideon A. Ngwa
    • 4
  1. 1.Department of Global Health and Social MedicineHarvard Medical SchoolBostonUSA
  2. 2.National Institute for Mathematical and Biological Synthesis (NIMBioS)KnoxvilleUSA
  3. 3.Department of Mathematics and Institute for Biomedical Engineering and Mathematical BiologyLehigh UniversityBethlehemUSA
  4. 4.Department of MathematicsUniversity of BueaBueaCameroon