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

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## 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## Notes

### Acknowledgments

CNN acknowledges the support of the National Institute for Mathematical and Biological Synthesis (NIMBioS), an Institute sponsored by the National Science Foundation, 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. All three authors (CNN, MIT-E and GAN) acknowledge the support of NIMBioS through the investigative workshop grant that supported the 2011 workshop: Malaria Modeling and Control (organized my MIT-E et al.), where all three authors met. They also acknowledge support of the NSF grant DMS-1261662 that supported a 2013 School on Stochastic Analysis, Financial, and Actuarial Mathematics with Applications, that enabled all three authors to continue the work on this manuscript. GAN acknowledges support of the Cameroon Ministry of Higher Education through the initiative for the modernization of research in Cameroon’s Higher Education granting scheme for 2014.

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