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Unstable Dynamics of Vector-Borne Diseases: Modeling Through Delay-Differential Equations

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Dynamic Models of Infectious Diseases

Abstract

Vector-borne diseases provide unique challenges to public health because the epidemiology is so closely tied to external environmental factors such as climate, landscape, and population migration, as well as the complicated biology of vector-transmitted pathogens. In particular, this close link between the epidemiology, the environment, and pathogen biology means that the traditional view that many vector-borne diseases are relatively stable in numerous regions does not provide a complete picture of their complexity. In fact, several regions exist with low levels of endemicity most of the time, punctuated by severe, often explosive, epidemics. These regions are considered unstable transmission settings. Ordinary differential equation (ODE) models have thus far dominated the study of vector-borne disease and have provided considerable insight into our understanding of transmission and effective control in stable transmission settings. To address the short-­comings of autonomous ODE models, we present a class of models, differential-delay equation (DDE) models, that have the potential to better describe unstable endemic settings for vector-borne disease. These models develop naturally out of the biology of diseases transmitted by vectors because of the extrinsic and intrinsic incubation periods and vector maturation process necessary for successful transmission of vector-transmitted pathogens. In this chapter, we introduce five examples of vector-borne diseases that span the globe, and discuss the clinical implications of unstable transmission of these diseases. Next, we present the original ODE version of the Ross-Macdonald model for vector-borne diseases, modify this model by introducing different types of naturally occurring delays, then illustrate how these models can exhibit more complex behavior such as oscillations via Hopf bifurcation and chaos via period-doubling, that the ODE model cannot produce. Finally, we explore the possibility for delay-models to contribute to our understanding of unstable transmission settings, which in turn will inform the development of effective control strategies for these epidemic-prone regions.

AMS Subject Classification: 92D30, 92D40

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Acknowledgements

Maia Martcheva acknowledges partial support from NSF grant DMS-0817789. Olivia Prosper acknowledges support from IGERT grant NSF DGE-0801544.

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Correspondence to Maia Martcheva .

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Martcheva, M., Prosper, O. (2013). Unstable Dynamics of Vector-Borne Diseases: Modeling Through Delay-Differential Equations. In: Rao, V., Durvasula, R. (eds) Dynamic Models of Infectious Diseases. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3961-5_2

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