Theoretical Ecology

, Volume 9, Issue 3, pp 337–351 | Cite as

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

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

Abstract

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.

Keywords

Period doubling bifurcation Backward bifurcation Malaria transmission Lyapunov exponent 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Global Health and Social MedicineHarvard Medical SchoolBostonUSA
  2. 2.Department of MathematicsLehigh UniversityBethlehemUSA
  3. 3.Department of MathematicsUniversity of BueaBueaCameroon

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