Abstract
We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.
Highlights
Plant epidemiology is the branch of science in which we study various diseases in different families of plants
Mathematical models are becoming very effective to utilize the dynamics of vector-borne plant epidemic transmission in host plants
As we know, when we study any real-world phenomena or, the dynamics of any epidemic, some common concerns always exist, for example, how the disease will behave for the long time interval or how we can project the real data for future predictions? To fulfil these requirements, the graphical interpretations are very important
Summary
Plant epidemiology is the branch of science in which we study various diseases in different families of plants. We apply the well-known Caputo fractional derivatives with singular type memory for studying the proposed time-delay plant epidemic model. Lemma 1 ([44]) The mapping ∈ C([–τ , T]; Rn), where C([–τ , T]; Rn) is a space of continuous mappings from [–τ , T] to Rn with the supremum norm · ∞), is a solution to the IVP (14)–(15) on the interval [–τ , T] if and only if it solves the fractional-order time-delay integral equation. 4.2 Derivation of the solution Here we establish the solution of the given fractional time-delay model (5) by using the well-known Adams–Bashforth–Moulton P-C scheme specified in [46]. We simulate some graphs at different values of infection rate to explore the role of parameter on the classes of the model
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