Global dynamics of tick-borne diseases

Ardak Kashkynbayev; Daiana Koptleuova; Ardak Kashkynbayev; Daiana Koptleuova

doi:10.3934/mbe.2020225

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 4: 4064-4079. doi: 10.3934/mbe.2020225

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Research article Special Issues

Global dynamics of tick-borne diseases

Department of Mathematics, Nazarbayev University, 53 Kabanbay batyr avenue, Nur-Sultan 010000, Kazakhstan

Received: 11 March 2020 Accepted: 29 May 2019 Published: 05 June 2020

A tick-borne disease model is considered with nonlinear incidence rate and piecewise constant delay of generalized type. It is known that the tick-borne diseases have their peak during certain periods due to the life cycle of ticks. Only adult ticks can bite and transmit disease. Thus, we use a piecewise constant delay to model this phenomena. The global asymptotic stability of the disease-free and endemic equilibrium is shown by constructing suitable Lyapunov functions and Lyapunov-LaSalle technique. The theoretical findings are illustrated through numerical simulations.
- epidemic model,
- the basic reproduction number,
- piecewise constant argument of generalized type,
- Lyapunov functions,
- global stability
Citation: Ardak Kashkynbayev, Daiana Koptleuova. Global dynamics of tick-borne diseases[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 4064-4079. doi: 10.3934/mbe.2020225

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Abstract

A tick-borne disease model is considered with nonlinear incidence rate and piecewise constant delay of generalized type. It is known that the tick-borne diseases have their peak during certain periods due to the life cycle of ticks. Only adult ticks can bite and transmit disease. Thus, we use a piecewise constant delay to model this phenomena. The global asymptotic stability of the disease-free and endemic equilibrium is shown by constructing suitable Lyapunov functions and Lyapunov-LaSalle technique. The theoretical findings are illustrated through numerical simulations.

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