Impulsive Differential Equations with Applications to Infectious Diseases
TLDR
Three biological applications showing the use of impulsive differential equations in real-world problems and the existence and uniqueness of T-periodic solutions are presented, and how stability changes when varying the immune response rate, the impulses and a certain nonlinear infection term are shown.Abstract:
Impulsive differential equations are useful for modelling certain biological events. We present three biological applications showing the use of impulsive differential equations in real-world problems. We also look at the effects of stability on a reduced two-dimensional impulsive HIV system. The first application is a system describing HIV induction-maintenance therapy, which shows how the solution to an impulsive system is used in order to find biological results (adherence, etc). A second application is an HIV system describing the interaction between T-cells, virus and drugs. Stability of the system is determined for a fixed drug level in three specific regions: low, intermediate and high drug levels. Numerical simulations show the effects of varying drug levels on the stability of a system by including an impulse. We reduce these two models to a two-dimensional impulsive model. We show analytically the existence and uniqueness of T-periodic solutions, and show how stability changes when varying the immune response rate, the impulses and a certain nonlinear infection term. The third application shows how seasonal changes can be incorporated into an impulsive differential system of Rift Valley Fever, and looks at how stability may differ when impulses are included. The analysis of impulsive differential systems is crucial in developing more realistic mathematical models for infectious diseases.read more
Citations
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Journal ArticleDOI
Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses
Akbar Zada,Wajid Ali,Syed Farina +2 more
Journal ArticleDOI
Ulam’s-Type Stability of First-Order Impulsive Differential Equations with Variable Delay in Quasi–Banach Spaces
TL;DR: In this article, Ulam's-type stabilities for a class of first-order impulsive differential equations with bounded variable delays on compact interval with finite number of impulses are studied via newly established integral inequality of Bellman-Grönwall-Bihari type with delay for discontinuous functions.
Journal ArticleDOI
On Periodic Solutions of Delay Differential Equations with Impulses
TL;DR: The existence of the periodic solution of impulsive delay differential equations is obtained by using the Schaffer fixed point theorem in regulated space R ( [ − r , 0 ] , R n ) .
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References
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TL;DR: Impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.
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HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time
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