Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach.

Gani Stamov,Ivanka Stamova,Cvetelina Spirova

doi:10.3390/e23121631

Gani Stamov, Ivanka Stamova + Show 1 more

Open Access

https://doi.org/10.3390/e23121631

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Abstract

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

Highlights

The dynamics of reaction-diffusion and related equations are traditionally applied to problems in biology, population dynamics, ecology, and neurosciences [1,2,3,4]
After considering an impulsive control approach to existing delayed systems with reaction-diffusion terms applied in epidemiology, we introduce the concept of integral manifolds to the model under consideration
New sufficient conditions for existence, boundedness, permanence, and uniform global asymptotic stability of integral manifolds related to the introduced model are established

Summary

Introduction

The dynamics of reaction-diffusion and related equations are traditionally applied to problems in biology, population dynamics, ecology, and neurosciences [1,2,3,4]. An impulsive control strategy is considered for a class of delayed reaction-diffusion models in biology, which arises naturally in a wide variety of biological applications and allows for synchronization of a complex system by using only small control impulses, even though the system’s behavior may follow unpredictable patterns; The integral manifold notion is introduced for the first time for the model under consideration, which generalizes the single state concepts and is very effective in systems with several equilibria; New existence, boundedness, and permanence results are established with respect to integral manifolds; Criteria for asymptotic stability of an integral manifold related to the impulsive model under consideration are proved; We apply a Poincarè-type integral inequality, which allows for more accurate estimation of the reaction diffusion terms, and leads to a better exploration of the diffusion effect.

Model Description and Preliminaries

Asymptotic Stability of Integral Manifolds

Conclusions