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

Abstract

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.

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Notes

  1. 1.

    A common assumption made in the analysis of epidemic models is that the duration of immunity is independent of exposure to infection (Anderson and May 1979; Hethcote et al. 1982; Anderson 1991). However, immunity to malaria is sustained by continuous exposure (Aron 1983), and such acquired immunity may confer protection against severe clinical illness without eliminating chronic or mild infections. Thus, asymptomatic immune malaria carriers can be infective (Aron 1988), which we’ve assumed here, that it is at a lower rate.

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Acknowledgments

CNN acknowledges the support of the National Institute for Mathematical and Biological Synthesis (NIMBioS), an Institute sponsored by the National Science Foundation (NSF), the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #EF-0832858, with additional support from the University of Tennessee, Knoxville, and NSF Award #OISE-0855380. Additional funding for CNN comes from a Scholar Award in Complex Systems Science to MHB from the James S. McDonnell Foundation. MIT-E acknowledges support of NSF Award #OISE-0855380. GAN acknowledges the grants and support of the Cameroon Ministry of Higher Education through the initiative for the mordenization of research in Cameroon’s Higher Education.

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Correspondence to Miranda I. Teboh-Ewungkem.

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Ngonghala, C.N., Teboh-Ewungkem, M.I. & Ngwa, G.A. Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito populations: implications for control. J. Math. Biol. 70, 1581–1622 (2015). https://doi.org/10.1007/s00285-014-0804-9

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Keywords

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

Mathematics Subject Classification

  • 34A
  • 34C
  • 37G
  • 92B
  • 92D