Full article: Integral manifolds for impulsive HCV conformable neural network models

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ABSTRACT

In this paper, we offer a new modelling approach for hepatitis C virus (HCV) infection models with cytotoxic T lymphocytes (CTL) immune responses. The standard neural network models with reaction-diffusion terms are generalized to the discontinuous case considering impulsive controllers. Also, the conformable approach is proposed as a more adjustable modelling apparatus which overcomes existing difficulties in the application of the fractional-order modelling technique. In addition, the existence of integral manifolds for the proposed model is investigated and efficient criteria are established. Since stability and control of a neural network model are essential for its proper application, stability results for the constructed integral manifold are presented. These results extend and generalize the stability results for steady states of proposed HCV models with CTL immune responses and contribute to the development of new innovative modelling approaches in biology. The qualitative analysis is performed via the conformable Lyapunov function method. The established results are demonstrated through an example.

KEYWORDS:

MATHEMATICS SUBJECT CLASSIFICATIONS:

1. Introduction

Neural network systems are a well-known modelling approach in biology. Particularly, they are used as adequate mathematical models in the analysis of the spread and control of HCV virus infection [Citation1, Citation2]. The interest of scientists in developing such models is motivated by the fact that the HCV epidemic is a major global health problem.

In addition, neural network models are used to study the complex relation between HCV and the immune system [Citation3–5]. However, the mechanism of the immune response during viral infection is still an object of intensive investigations. For example, the model presented in [Citation6], which contains five variables namely, susceptible host cells ( $ξ_{1}$ ), infected cells ( $ξ_{2}$ ), free virus ( $ξ_{3}$ ), an antibody response ( $ξ_{4}$ ) and a CTL response ( $ξ_{5}$ ), is given by the neural network (1) ${\begin{cases} {\dot{ξ}}_{1} (t) = λ - d ξ_{1} (t) - β ξ_{1} (t) ξ_{3} (t), \\ {\dot{ξ}}_{2} (t) = β ξ_{3} (t) ξ_{1} (t) - a ξ_{2} (t) - p ξ_{2} (t) ξ_{5} (t), \\ {\dot{ξ}}_{3} (t) = κ ξ_{2} (t) - m ξ_{3} (t) - q ξ_{3} (t) ξ_{4} (t), \\ {\dot{ξ}}_{4} (t) = g ξ_{3} (t) ξ_{4} (t) - h ξ_{4} (t), \\ {\dot{ξ}}_{5} (t) = c ξ_{2} (t) ξ_{5} (t) - b ξ_{5} (t), \end{cases}$ (1) where t>0, λ is the production rate of the susceptible host cells, $d ξ_{1}$ represents the death rate, $β ξ_{1} (t) ξ_{3} (t)$ denotes the rate with which the susceptible host cells become infected by virus. It is also assumed that the infected cells death rate is $a ξ_{2}$ , the CTL response rate is $p ξ_{2} ξ_{5}$ , the infected cells produced free virus at a rate $κ ξ_{2}$ , $m ξ_{3}$ denotes the decay rate of free virus, free virus is neutralized by antibodies at a rate $q ξ_{3} ξ_{4}$ , antibodies develop in response to free virus at a rate $g ξ_{3} ξ_{4}$ and decay at a rate $h ξ_{4}$ , CTLs expand in response to viral antigen derived from infected cells at a rate $c ξ_{2} ξ_{5}$ and decay in the absence of antigenic stimulation at a rate $b ξ_{5}$ .

Neural network models of type (Equation1(1) ${\begin{cases} {\dot{ξ}}_{1} (t) = λ - d ξ_{1} (t) - β ξ_{1} (t) ξ_{3} (t), \\ {\dot{ξ}}_{2} (t) = β ξ_{3} (t) ξ_{1} (t) - a ξ_{2} (t) - p ξ_{2} (t) ξ_{5} (t), \\ {\dot{ξ}}_{3} (t) = κ ξ_{2} (t) - m ξ_{3} (t) - q ξ_{3} (t) ξ_{4} (t), \\ {\dot{ξ}}_{4} (t) = g ξ_{3} (t) ξ_{4} (t) - h ξ_{4} (t), \\ {\dot{ξ}}_{5} (t) = c ξ_{2} (t) ξ_{5} (t) - b ξ_{5} (t), \end{cases}$ (1) ) have been studied by various authors. However, the authors in [Citation7–9] found that the spatial dependence cannot be ignored in the formulation of the model. Therefore, they considered a diffused HCV model with CTL and antibody responses represented by the neural network system (2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) where $ξ_{1} (x, t)$ , $ξ_{2} (x, t)$ , $ξ_{3} (x, t)$ , $ξ_{4} (x, t)$ and $ξ_{5} (x, t)$ denote the densities of susceptible host cells, infected cells, free virus, the antibody response and the CTL response at location x and time t, respectively, $x \in Θ$ , Θ is a bounded domain in $R^{n}$ with a smooth boundary $∂Θ$ and a positive measure $mes Θ > 0$ containing the origin, $d_{ξ_{3}} \geq 0$ is the diffusion coefficient and $Δ ξ_{3} = \sum_{i = 1}^{n} \frac{\partial^{2} ξ_{3} (x, t)}{\partial x_{i}}$ .

Indeed, the spatial dynamics of HCV virus transmission attracted the attention of researchers, since, for example, the spatial heterogeneity seriously affected the spread of infectious diseases. Hence, in order to take this into account, the process can be modelled by neural network systems with reaction-diffusion terms. For some important results on complex disease and related models that include partial derivatives, diffusion and cross-diffusion coefficients to reflect the effect of the spatial spreading of infection, we refer to [Citation10–16] and the references therein.

On the other side, numerous impulsive control strategies have been recently proposed to some epidemic models that are of a great importance for contemporary society [Citation17–21], including existing HCV disease models [Citation22, Citation23]. In fact, impulsive state controllers have some important advantages which make them preferable control strategies. They are applied only at discrete times and as such can reduce the control cost and the amount of transmitted information drastically. Also, impulsive-type perturbations are very common in modelling biological processes since the states of a biological neural network model can often be affected by the environmental variation and random births and deaths of cells. As a response of environmental noises, the qualitative behaviour of the model can be affected significantly. Hence, it is highly important to study impulsive generalizations of the HCV models. Such impulsive generalizations are represented by systems of impulsive differential equations [Citation24–28]. Impulsive control systems [Citation29, Citation30] may offer a convenient tool for designing efficient therapy for HCV disease. However, for diffused HCV models with CTL responses of type (Equation2(2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) ), in our knowledge, there are no mathematical models proposed with impulsive control therapeutic approach.

In addition, a lot of researchers have recently worked on the development of fractional mathematical models in biology. Fractional-order modelling approach has been applied to model the infection dynamics of several infectious diseases [Citation31–35]. The employment of fractional calculus in mathematical modelling of biological processes is justified by the benefits of fractional-order derivatives, which include more degrees of freedom and memory features [Citation36–38]. Some fractional-order HCV models have also been developed recently [Citation39–41].

However, the use of most of the classical fractional derivatives leads to some complexities in their applications to real-world problems due to more complicated chain rule formulas and some singularity properties. Therefore, researchers have been motivated to introduce new definitions to avoid the limitations of fractional-order derivatives that allow running of easy simulations in the analysis and parameter identification. One such concept is that of conformable derivatives defined in [Citation42, Citation43]. The definition of a conformable derivative is limit-based and it offers great computational simplifications in the application of chain rules. That is why various researchers have started research in the development of the theory of systems with conformable derivatives [Citation44–48]. In recent times, conformable calculus is an area of growing research. In addition, some existing integer-order and fractional-order mathematical models have been extended to the conformable case. For example, the conformable time-fractional version of the generalized Pochhammer–Chree equation is analysed in [Citation49]. The cobweb model has been reformulated in terms of conformable fractional-order derivatives in [Citation50] and some stability results have been established. The paper [Citation51] investigates exact solutions of a Wu–Zhang system described by conformable fractional derivative. The solutions of a conformable fractional-order SIR epidemic model are studied in [Citation52]. The recently published paper [Citation53] is devoted to the formulation of a SIR model and a SEIRD model based on conformable space-time partial differential derivatives. In [Citation54], to further improve the performance of the grey-based model, it has been enhanced using the continuous conformable fractional derivative. Indeed, the various uses of conformable derivatives show that there is a substantial requirement for further numerical and analytical studies, including real objects and methods.

Mathematical models using impulsive conformable systems have also received some attention [Citation55–58]. However, compared with continuous models, the discontinuous cases need more investigation. In particular, the research on conformable infection diseases models under impulsive perturbations is still under development. Hence, it is very meaningful to formulate and study an impulsive conformable extension of the HCV model (Equation2(2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) ), which is the main goal of our paper.

The stable behaviour is very important to be achieved for the desired performance of a neural network model. Thus numerous stability results about integer-order and fractional-order infectious diseases models have been provided [Citation1, Citation2, Citation5, Citation7, Citation9, Citation12, Citation13, Citation16, Citation20, Citation22, Citation31, Citation32, Citation34, Citation35, Citation41]. The most studied stable behaviours are those of the global and local asymptotic stability of the steady states of the models. Correspondingly, there are not much stability results for conformable systems [Citation57].

However, in many practical problems, including infection diseases models, the considered mathematical model has more than a single one equilibrium of interest. In such cases, the concept of integral manifolds stability is more relevant to be considered. The integral manifolds notion, which generalizes the isolate steady state concept, has been applied by several authors [Citation17, Citation59–61]. Although the integral manifold approach is a convenient method to study the behaviour of several states of biological models, it is not yet applied to HCV models.

In this article, the targets of the research work are:

development of a neural network mathematical model for the HCV virus infection using the conformable setting, which extends the existing type (Equation2(2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) ) models;
introduction of appropriate control impulses to implement the impulsive control approach;
defining of integral manifolds of neuronal states, which generalize the single state concept;
establishing of efficient existence and stability criteria for the introduced integral manifold;
applying the Lyapunov stability method which does not require a specific knowledge of the solutions.

The article is organized in sections. In Section 2, the main definitions and properties from conformable calculus are given. The impulsive conformable diffused HCV model with CTL and antibody responses is formulated and the integral manifold concept is defined for the introduced model. Some basic notes related to the Lyapunov stability approach are also presented. The main existence and stability results for the integral manifolds related to the impulsive conformable model are developed in Section 3. An example is established in Section 4 to illustrate the main results. Finally, the last Section 5 is devoted to concluding remarks and perspectives.

2. Preliminaries

2.1. Notes on conformable calculus

In this section, some basic definitions and notes on conformable calculus will be given. We will use the following notations: the norm of a vector $x = (x_{1}, x_{2}, \dots, x_{n})^{T} \in R^{n}$ will be defined by $| | x | | = \sqrt{x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2}}$ , $R_{+} = [0, \infty)$ , and $t_{0} \in R_{+}$ .

Consider the sequence of impulsive instances $τ_{k}$ , $k \in N$ defined as (3) $t_{0} < τ_{1} < τ_{2} < \dots, lim_{k \to \infty} τ_{k} = \infty .$ (3) Following [Citation55, Citation58], for any $\bar{t} \geq t_{0}$ and $y : [\bar{t}, \infty) \to R^{n}$ , the limit $D_{\bar{t}}^{α} y (t) = lim {\frac{y (t + θ (t - \bar{t})^{1 - α}) - y (t)}{θ}, θ \to 0}$ will be called a generalized conformable derivative of order α, $0 < α \leq 1$ with the lower limit $\bar{t}$ of the function y.

Note that some authors [Citation46, Citation57] also used the name ‘fractional-like’ instead of ‘conformable’ derivative.

For $\bar{t} = τ_{k}$ , $k \in N$ , we have $D_{t_{k}}^{α} y (τ_{k}) = lim_{t \to τ_{k}^{+}} D_{τ_{k}}^{α} y (t) .$ The notation $y \in C^{α} ((\bar{t}, \infty), R^{n})$ means that [Citation55] the α-generalized conformable derivatives for the continuous function y exists at any point $t \in (\bar{t}, \infty)$ , and such functions are called α-generalized conformable differentiable on $(\bar{t}, \infty)$ .

Similar to the above, the generalized conformable integral of order $0 < α \leq 1$ with a lower limit $\bar{t} \geq t_{0}$ of a function $y : [\bar{t}, \infty) \to R^{n}$ is defined as $I_{\bar{t}}^{α} y (t) = \int_{\bar{t}}^{t} (σ - \bar{t})^{α - 1} y (σ) d σ .$ We will also list some important properties of the α-generalized conformable derivatives [Citation55, Citation57, Citation58, Citation62].

Lemma 2.1

Let the function $y : (\bar{t}, \infty) \to R$ be α-generalized conformable differentiable on $(\bar{t}, \infty)$ for $0 < α \leq 1$ . Then

(i)	$I_{\bar{t}}^{α} (D_{\bar{t}}^{α} y (t)) = y (t) - y (\bar{t}), t > \bar{t}$ ;
(ii)	if $u (y (t)) : (\bar{t}, \infty) \to R$ is differentiable with respect to $y (t)$ , then for any $t \in [\bar{t}, \infty)$ and $y (t) \neq 0$ , we have $D_{\bar{t}}^{α} u (y (t)) = u^{'} (y (t)) D_{\bar{t}}^{α} y (t),$ where $u^{'}$ is the derivative of $u (\cdot)$ .

Remark 2.1

Part (ii) of Lemma 2.1 shows the rationality of the use of the conformable fractional approach. For classical fractional-order derivatives, such a simple chain rule does not exist.

Consider the vector-function defined by $ξ (x, t) = (ξ_{1} (x, t), ξ_{2} (x, t), \dots, ξ_{5} (x, t))^{T} \in R^{5}$ . Throughout this paper, we will use the norm $| | ξ (\cdot, t) | |_{2} = {[\int_{Θ} \sum_{j = 1}^{5} ξ_{j}^{2} (x, t) d x]}^{1 / 2} .$

Definition 2.2

[Citation62]

For a function $ξ_{j} (x, t)$ , $ξ_{j} : Θ \times (\bar{t}, \infty) \to R$ , $j = 1, 2, \dots, 5$ , the limit $\frac{\partial^{α} ξ_{j} (x, t)}{\partial \bar{t}} = lim_{θ \to 0} {(ξ_{j} (x, t + θ (t - \bar{t})^{1 - α}) - ξ_{j} (x, t)) θ^{- 1}}, t > \bar{t}$ is called a generalized conformable partial derivative along t of order α, $0 < α \leq 1$ with the lower limit $\bar{t}$ .

For $\bar{t} = τ_{k}$ , $k \in N$ , we have [Citation62] $\frac{\partial^{α} ξ_{j} (x, t)}{\partial τ_{k}} = lim_{\bar{t} \to τ_{k}^{+}} \frac{\partial^{α} ξ_{j} (x, t)}{\partial \bar{t}} .$ For more results related to generalized conformable partial derivatives, we refer to [Citation55–58, Citation62].

2.2. Development of the impulsive conformable HCV virus infection model

In order to apply the advantages of the generalized conformable derivatives, the extended impulsive conformable diffused HCV model with CTL and antibody responses represented is given by the neural network system (4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) where $k \in N$ , $0 < α \leq 1$ , $ξ_{1} (x, t)$ , $ξ_{2} (x, t)$ , $ξ_{3} (x, t)$ , $ξ_{4} (x, t)$ and $ξ_{5} (x, t)$ are the same five compartments describing the interaction between susceptible host cells, infected cells, free virus, the antibody response and the CTL response at location x and time t, the impulsive points $τ_{k}$ are defined by (Equation3(3) $t_{0} < τ_{1} < τ_{2} < \dots, lim_{k \to \infty} τ_{k} = \infty .$ (3) ), $J_{jk}$ are the impulsive control functions for j = 2, 3 and $k \in N$ , $ξ_{j} (x, τ_{k})$ and $ξ_{j} (x, τ_{k}^{+})$ are the states of the jth compartment at $τ_{k}$ and $τ_{k}^{+}$ , respectively, i.e. before and after an impulsive jump at $τ_{k}$ .

The last two equations in the proposed model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) represent the impulsive control conditions. The functions $J_{jk}$ , $j = 2, 3, k \in N$ , determine the controlled outputs $ξ_{j} (\cdot, τ_{k}^{+})$ at the instances $τ_{k}$ for the infected cells and free virus, respectively, where the following relations hold: $ξ_{j} (x, τ_{k}^{-}) = ξ_{j} (x, τ_{k}), ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3,$ $k \in N$ .

Next, we denote by $ξ (x, t) = ξ (x, t; t_{0}, ξ_{0})$ the solution of the impulsive control conformable model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) that satisfies the boundary and initial conditions of the type (5) $\begin{aligned} \frac{\partial ξ (x, t)}{\partial N} & = 0, x \in ∂Θ, t \geq t_{0}, \end{aligned}$ (5) (6) $\begin{aligned} ξ (x, t_{0}^{+}) & = ξ_{0} (x) \geq 0, x \in Θ, t_{0} \in R_{+}, \end{aligned}$ (6) where $ξ (x, t) = {(ξ_{1} (x, t), ξ_{2} (x, t), ξ_{3} (x, t), ξ_{4} (x, t), ξ_{5} (x, t))}^{T},$ $ξ_{0} = (ξ_{10}, ξ_{20}, ξ_{30}, ξ_{40}, ξ_{50})^{T}$ , $ξ_{i 0} = ξ_{i 0} (x)$ are continuous in Θ, and $\frac{\partial}{\partial N}$ is the outward normal derivative on $∂Θ$ .

Remark 2.2

The introduced impulsive conformable model extends the model (Equation2(2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) ) proposed in [Citation7] to the impulsive conformable case. The proposed impulsive controllers may allow the use of an appropriate impulsive control therapy. The $τ_{k}$ values are the impulse instances of the therapy (drug or vaccination) by means of which the infected cells and free virus would be reduced. The use of generalized conformable derivatives makes the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) a more flexible modelling mechanism which also includes the case when $α = 1$ as represented in [Citation7]. The impulsive conformable modelling approach can also be applied to numerous neural network infection disease models existing in the research literature.

In the proofs of our main results, we will apply the following Poincarè-type integral inequality [Citation17].

Lemma 2.3

Let Θ be a bounded domain in $R^{n}$ with a smooth boundary $∂Θ$ , and $V : Θ \to R_{+}$ be a real-valued function belonging to $H^{1} (Θ)$ which satisfies $\frac{\partial V (x)}{\partial N} |_{∂Θ} = 0$ . Then $ζ_{1} \int_{Θ} | V (x) |^{2} d x \leq \int_{Θ} | \nabla V (x) |^{2} d x,$ where $ζ_{1}$ is the smallest positive eigenvalue of the Neumann boundary problem ${\begin{cases} - Δ χ (x) = ζχ (x), x \in Θ, \\ \frac{\partial χ (x)}{\partial N} = 0, x \in ∂Θ . \end{cases}$

2.3. Integral manifolds

In this section we will introduce the integral manifolds concept to the proposed impulsive conformable model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ). The following definition will be adopted from [Citation59–61] and some of the references cited therein.

Consider the extended phase space $X = Θ \times [t_{0}, \infty) \times R^{5}$ of the system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ).

Definition 2.4

An arbitrary manifold $Y$ in $X$ is called an integral manifold for the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ), if for any solution $ξ (x, t) = ξ (x, t; t_{0}, ξ_{0})$ , $(x, t_{0}^{+}, ξ_{0}) \in Y$ implies $(x, t, ξ (x, t)) \in Y$ , $x \in Θ$ , $t \geq t_{0}$ .

Remark 2.3

The concept of integral manifolds generalizes that of considering a single state of a model. We propose it as particularly appropriate to be applied to infectious disease models where the dynamics of several steady states are investigated [Citation1, Citation2, Citation5, Citation7, Citation9, Citation12, Citation13, Citation16, Citation20, Citation22, Citation31, Citation32, Citation34, Citation35, Citation41]. Numerous definitions and properties for integral manifolds of impulsive systems can be found in [Citation59–61].

For the manifold $Y \subset X$ , we define the following sets and distances: $\begin{aligned} Y (x, t) & = {ξ \in R^{5} : (x, t, ξ) \in Y, (x, t) \in Θ \times R_{+}}; \\ Y (x, t) (ε) & = {ξ \in R^{5} : d (ξ, Y (x, t)) < ε}, ε > 0, \end{aligned}$ where $d (ξ, Y (x, t)) = inf_{η \in Y (x, t)} | | ξ - η | |_{2}$ is the distance between $ξ \in R^{5}$ and $Y (x, t)$ ; $\bar{Y} (x, t) (ε) = {ξ \in R^{5} : d (ξ, Y (x, t)) \leq ε} .$ Throughout this paper, we will also assume that the set $Y (x, t)$ is nonempty for $(x, t) \in Θ \times R_{+}$ and the distance $d (ξ, Y (x, t))$ is Lipschitz with respect to t on any compact subset of $Θ \times [t_{0}, \infty) \times R_{+}$ .

Since we aim to generalize stability results for the steady states of existing infection disease models to the integral manifold case using the conformable approach, we will introduce the following definitions for exponential conformable stability of an integral manifold $Y$ with respect to the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ).

For $0 < α \leq 1$ , consider the conformable exponential function $E_{α} (ν, σ)$ defined as [Citation42] $E_{α} (ν, σ) = \exp (ν \frac{σ^{α}}{α}), ν \in R, σ \in R_{+} .$ Note that Mittag–Leffler functions play a significant role in the classical fractional calculus as a generalization of exponential functions [Citation36, Citation38], while the conformable exponential function appears in case of conformable fractional calculus [Citation42].

Definition 2.5

We will say that an integral manifold $Y$ of system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) is said to be globally exponentially conformable stable, if there exist K>0 and $γ \geq 0$ such that for any $ξ_{0} \in Y (x, t_{0}^{+})$ , we have $ξ (x, t; t_{0}, ξ_{0}) \in \bar{Y} (x, t) (Kd (ξ_{0}, Y (x, t_{0}^{+})) E_{α}^{ω} (- γ, (t - t_{0})))$ for $t \geq t_{0}$ , $x \in Θ$ , $ω > 0$ .

Remark 2.4

Definition 2.5 again shows that the stability of an integral manifold notion extends and generalizes the stability of a single solution concept for conformable systems. In the cases when the integral manifold consists of a single system's state, Definition 2.5 reduces to the definition used in [Citation48].

2.4. Lyapunov function method's fundamentals

In our analysis, we will apply the conformable Lyapunov function method [Citation55, Citation57, Citation62]. The Lyapunov function method and its modifications offer a very powerful mechanism in the study of the qualitative properties of infection diseases neural network models and is used by numerous authors [Citation7, Citation13, Citation17, Citation32, Citation34, Citation63].

Some fundamentals of the Lyapunov function method for impulsive conformable systems will be given below.

Define the sets $ϕ_{k} = {(ξ, t) : ξ \in R^{5}, t \in (τ_{k - 1}, τ_{k}), k \in N, τ_{0} = t_{0}} .$ We will also need the set $ϕ = ⋃_{k \in N} ϕ_{k} .$ We will use the class $W_{τ_{k}}^{α}$ of nonnegative Lyapunov functions $W$ for any $τ_{k} \in R_{+}, k \in N$ , such that $W (ξ, t) = 0$ for $ξ \in Y (x, t)$ , $(x, t) \in Θ \times R_{+}$ , $t \geq τ_{k}$ , $W$ is continuous in ϕ, α-generalized conformable differentiable in t and locally Lipschitz continuous with respect to ξ on each of the sets $ϕ_{k}$ , and for each $k \in N$ and $ξ \in R^{5}$ , there exist the finite limits $W (ξ, τ_{k}^{-}) = lim_{\overset{t \to τ_{k}}{t < τ_{k}}} W (ξ, t) = W (ξ, τ_{k}), W (ξ, τ_{k}^{+}) = lim_{\overset{t \to τ_{k}}{t > τ_{k}}} W (ξ, t) .$ For a function $W \in W_{τ_{k}}^{q}$ , the following derivative is defined [Citation55, Citation57, Citation62] as (7) $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) = lim sup {(W (ξ (x, t + θ (t - τ_{k})^{1 - α}), t + θ (t - τ_{k})^{1 - α}) \\ - W (ξ, t)) θ^{- 1}, θ \to 0^{+}} . \end{aligned}$ (7) Analogously to part (ii) of Lemma 2.1, if $W (ξ (\cdot, t), t) = W (ξ (\cdot, t)), 0 < α \leq 1$ , $W$ is differentiable on ξ, and $ξ (\cdot, t)$ is generalized conformable α-differentiable with respect to t for $t > τ_{k}$ , then (8) $^{+} D_{τ_{k}}^{α} W (ξ, t) = W^{'} (ξ (\cdot, t)) D_{τ_{k}}^{α} ξ (\cdot, t) .$ (8) We will also need the following result from [Citation55, Citation57, Citation62].

Lemma 2.6

For a function $W \in W_{τ_{k}}^{α}$ such that for $t \in R_{+}$ , $ξ \in R^{5}$ , $\begin{aligned} W (ξ + J_{k} (ξ), τ_{k}^{+}) & \leq W (ξ, τ_{k}), k \in N, J_{k} = (0, J_{2 k}, J_{3 k}, 0, 0)^{T}, \\ ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq - γ W (ξ, t), t \neq τ_{k}, k \in N, \end{aligned}$ where $γ = const \geq 0$ , we have $W (ξ (\cdot, t), t) \leq W (ξ_{0}, t_{0}^{+}) E_{α} (- γ, t - t_{0}), t \geq t_{0} .$

3. Results

We will consider the following hypotheses:

H3.1.	The functions $ξ_{j} (x, t)$ , $j = 1, 2, \dots, 5$ are all bounded on $Θ \times R_{+}$ , $t \neq τ_{k}$ , $k \in N$ , and $ξ_{j} (x, t) \leq {\hat{ξ}}_{j}, {\hat{ξ}}_{j} = const > 0.$
H3.2.	The impulsive functions $J_{jk}$ are continuous, $J_{jk} (0) = 0$ for all j = 2, 3, $k \in N$ and $ξ_{j} (x, τ_{k}) \geq 0$ implies $ξ_{j} (x, τ_{k}^{+}) \geq 0$ for all j = 2, 3, $k \in N$ .
H3.3.	$ξ_{j} (x, τ_{k}^{+}) \leq ξ_{j} (x, τ_{k}), x \in Θ, j = 2, 3, k \in N$ .

Remark 3.1

The boundedness hypothesis H3.1 is reasonable from the biological point of view, since all the cell populations are bounded [Citation6, Citation7, Citation32] due to the capacity of the environment. Also, the boundedness of the solutions can be easily shown using the function $F = F (ξ, t) = g ξ_{1} + g ξ_{2} + \frac{ag}{2 κ} ξ_{3} + \frac{aq}{2 κ} ξ_{4} + \frac{gp}{c} ξ_{5},$ for which we have $^{+} D_{τ_{k}}^{α} F (ξ, t) = λg + \frac{ag}{2 κ} d_{ξ_{3}} Δ ξ_{3} - dg ξ_{1} - \frac{ag}{2} ξ_{2} - \frac{ag}{2 κ} ξ_{3} - \frac{aqh}{2 κ} ξ_{4} - \frac{gpb}{c} ξ_{5}, t \neq τ_{k},$ i.e. $^{+} D_{τ_{k}}^{α} F (ξ, t) \leq A - ρF (ξ, t), t \neq τ_{k},$ where $\begin{aligned} A & = λg + \frac{ag}{2 κ} d_{ξ_{3}} Δ ξ_{3}, ρ = min {d, \frac{a}{2}, h, b}, \\ F (ξ, τ_{k}^{+}) & \leq F (ξ, τ_{k}), k \in N, \end{aligned}$ by using Lemma 2.6.

Remark 3.2

The hypothesis H3.2 guarantees nonnegativity of the solutions for any nonnegative initial conditions (Equation6(6) $\begin{aligned} ξ (x, t_{0}^{+}) & = ξ_{0} (x) \geq 0, x \in Θ, t_{0} \in R_{+}, \end{aligned}$ (6) ), which is also biologically reasonable.

3.1. Existence of integral manifolds

We need the next conditions:

H3.4.	The impulsive controllers $J_{jk}, j = 2, 3$ in the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) are such that $J_{jk} (ξ_{j} (x, τ_{k})) = - φ_{jk} ξ_{j} (x, τ_{k}), 0 < φ_{jk} < 2, j = 2, 3, k \in N .$
H3.5.	The diffusion coefficient $d_{ξ_{3}}$ is such that $0 \leq \underline{d} \leq d_{ξ_{3}} .$

Let $ξ (x, t; t_{0}, ξ_{0}) = ξ (x, t)$ be the solution of (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) for $0 \leq t_{0} < τ_{1}$ and $ξ_{0} \in Θ$ . To prove the main existence result of an integral manifold, we use a Lyapunov-type function $W \in W_{τ_{k}}^{α}$ defined as (9) $\begin{aligned} W (ξ, t) & = \frac{1}{2} d^{2} (ξ, Y (x, t)) \\ = \frac{1}{2} inf_{η \in Y (x, t)} | | ξ - η | |_{2}^{2} = inf_{η \in Y (x, t)} \frac{1}{2} \int_{Θ} \sum_{j = 1}^{5} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x, \end{aligned}$ (9) where $t \in R_{+}$ .

Theorem 3.1

A manifold $Y$ in the extended phase space of the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) is an integral manifold of (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ), if hypotheses H3.1, H3.2, H3.4 and H3.5 hold and the system's parameters are such that $\begin{aligned} d & \geq \frac{β}{2} ({\hat{ξ}}_{1} + {\hat{ξ}}_{3}), \\ a & \geq \frac{1}{2} (κ + β ({\hat{ξ}}_{1} + {\hat{ξ}}_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5}), \\ m + D & \geq β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ h & \geq g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ b & \geq c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5}), \end{aligned}$ where $D = n ζ_{1} \underline{d}$ .

Proof.

As a first step, we will suppose that $t = τ_{k}$ , $k \in N$ , and then hypotheses H3.2 and H3.4 imply $\begin{aligned} \frac{1}{2} \int_{Θ} {(ξ_{j} (x, t^{+}) - η_{j} (x, t^{+}))}^{2} d x & = \frac{1}{2} \int_{Θ} (1 - φ_{jk})^{2} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x \\ < \frac{1}{2} \int_{Θ} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x, j = 2, 3, \end{aligned}$ and $\frac{1}{2} \int_{Θ} {(ξ_{j} (x, t^{+}) - η_{j} (x, t^{+}))}^{2} d x = \frac{1}{2} \int_{Θ} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x, j = 1, 4, 5.$ Hence, for any $ξ_{0} \in R^{5}$ , we have (10) $W (ξ + J_{k} (ξ), t^{+}) \leq W (ξ, t), t = τ_{k}, k \in N .$ (10) Next is to calculate the time generalized conformable derivative of $W$ of order α along the solution of model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) for $t \neq τ_{k}, k \in N$ , and we get from (Equation8(8) $^{+} D_{τ_{k}}^{α} W (ξ, t) = W^{'} (ξ (\cdot, t)) D_{τ_{k}}^{α} ξ (\cdot, t) .$ (8) ) and (Equation9(9) $\begin{aligned} W (ξ, t) & = \frac{1}{2} d^{2} (ξ, Y (x, t)) \\ = \frac{1}{2} inf_{η \in Y (x, t)} | | ξ - η | |_{2}^{2} = inf_{η \in Y (x, t)} \frac{1}{2} \int_{Θ} \sum_{j = 1}^{5} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x, \end{aligned}$ (9) ) that (11) $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq \frac{1}{2} \int_{Θ} \sum_{j = 1}^{5} D_{τ_{k}}^{α} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x \\ = \int_{Θ} \sum_{j = 1}^{5} (ξ_{j} (x, t) - η_{j} (x, t)) D_{τ_{k}}^{α} (ξ_{j} (x, t) - η_{j} (x, t)) d x . \end{aligned}$ (11) Now, we have $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq \int_{Θ} (ξ_{1} (x, t) - η_{1} (x, t)) [- d (ξ_{1} (x, t) - η_{1} (x, t)) \\ - β (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t))] d x \\ + \int_{Θ} (ξ_{2} (x, t) - η_{2} (x, t)) [- a (ξ_{2} (x, t) - η_{2} (x, t)) \\ + β (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t)) \\ - p (ξ_{2} (x, t) ξ_{5} (x, t) - η_{2} (x, t) η_{5} (x, t))] d x \\ + \int_{Θ} (ξ_{3} (x, t) - η_{3} (x, t)) [d_{ξ_{3}} Δ (ξ_{3} (x, t) - η_{3} (x, t)) \\ + κ (ξ_{2} (x, t) - η_{2} (x, t)) - m (ξ_{3} (x, t) - η_{3} (x, t)) \\ - q (ξ_{3} (x, t) ξ_{4} (x, t) - η_{3} (x, t) η_{4} (x, t))] d x \\ + \int_{Θ} (ξ_{4} (x, t) - η_{4} (x, t)) [g (ξ_{3} (x, t) ξ_{4} (x, t) - η_{3} (x, t) η_{4} (x, t)) \\ - h (ξ_{4} (x, t) - η_{4} (x, t))] d x \\ + \int_{Θ} (ξ_{5} (x, t) - η_{5} (x, t)) [c (ξ_{2} (x, t) ξ_{5} (x, t) - η_{2} (x, t) η_{5} (x, t)) \\ - b (ξ_{5} (x, t) - η_{5} (x, t)] d x . \end{aligned}$ By the Neumann boundary conditions (Equation5(5) $\begin{aligned} \frac{\partial ξ (x, t)}{\partial N} & = 0, x \in ∂Θ, t \geq t_{0}, \end{aligned}$ (5) ) of model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ), the Gauss formula and Lemma 2.3, we obtain (12) $\begin{aligned} \int_{Θ} (ξ_{3} (x, t) - η_{3} (x, t)) [d_{ξ_{3}} Δ (ξ_{3} (x, t) - η_{3} (x, t))] d x \\ \leq - \bar{d} \sum_{i = 1}^{n} \int_{Θ} {(\frac{ξ_{3} (x, t) - η_{3} (x, t)}{\partial x_{i}})}^{2} d x = - n \bar{d} \int_{Θ} {(\frac{ξ_{3} (x, t) - η_{3} (x, t)}{\partial x_{i}})}^{2} d x \\ \leq - D \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x . \end{aligned}$ (12) From H3.1 and the fact that $η \in Y (x, t)$ , for each of the terms in $^{+} D_{τ_{k}}^{α} W (ξ, t)$ , we have (13) $\begin{aligned} \int_{Θ} (ξ_{1} (x, t) - η_{1} (x, t)) [- d (ξ_{1} (x, t) - η_{1} (x, t)) \\ - β (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t))] d x \\ = - d \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} \\ - β \int_{Θ} (ξ_{1} (x, t) - η_{1} (x, t)) (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t))] d x \\ \leq - d \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x - β \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} ξ_{3} (x, t) d x \\ + β \int_{Θ} | ξ_{1} (x, t) - η_{1} (x, t) | | η_{1} (x, t) | | ξ_{3} (x, t) - η_{3} (x, t) | d x \\ \leq - d \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x \\ + \frac{β}{2} {\hat{ξ}}_{1} \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x + \frac{β}{2} {\hat{ξ}}_{1} \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x, \end{aligned}$ (13) (14) $\begin{aligned} \int_{Θ} (ξ_{2} (x, t) - η_{2} (x, t)) [- a (ξ_{2} (x, t) - η_{2} (x, t)) \\ + β (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t)) \\ - p (ξ_{2} (x, t) ξ_{5} (x, t) - η_{2} (x, t) η_{5} (x, t))] d x \\ = - a \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} \\ + β \int_{Θ} (ξ_{2} (x, t) - η_{2} (x, t)) (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t))] d x \\ - p \int_{Θ} (ξ_{2} (x, t) - η_{2} (x, t)) (ξ_{2} (x, t) ξ_{1} (x, t) - η_{2} (x, t) η_{5} (x, t))] d x \\ \leq - a \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} \\ + \frac{β}{2} {\hat{ξ}}_{3} \int_{Θ} [{(ξ_{2} (x, t) - η_{2} (x, t))}^{2} + {(ξ_{1} (x, t) - η_{1} (x, t))}^{2}] d x \\ + \frac{β}{2} {\hat{ξ}}_{1} \int_{Θ} [{(ξ_{2} (x, t) - η_{2} (x, t))}^{2} + {(ξ_{3} (x, t) - η_{3} (x, t))}^{2}] d x \\ + \frac{p}{2} {\hat{ξ}}_{2} \int_{Θ} [{(ξ_{2} (x, t) - η_{2} (x, t))}^{2} + {(ξ_{5} (x, t) - η_{5} (x, t))}^{2}] d x \\ = (- a + \frac{β}{2} {\hat{ξ}}_{3} + \frac{β}{2} {\hat{ξ}}_{1} + \frac{p}{2} {\hat{ξ}}_{2}) \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} d x \\ + \frac{β}{2} {\hat{ξ}}_{3} \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x + \frac{β}{2} {\hat{ξ}}_{1} \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x \\ + \frac{p}{2} {\hat{ξ}}_{2} \int_{Θ} {(ξ_{5} (x, t) - η_{5} (x, t))}^{2} d x . \end{aligned}$ (14) Using (Equation12(12) $\begin{aligned} \int_{Θ} (ξ_{3} (x, t) - η_{3} (x, t)) [d_{ξ_{3}} Δ (ξ_{3} (x, t) - η_{3} (x, t))] d x \\ \leq - \bar{d} \sum_{i = 1}^{n} \int_{Θ} {(\frac{ξ_{3} (x, t) - η_{3} (x, t)}{\partial x_{i}})}^{2} d x = - n \bar{d} \int_{Θ} {(\frac{ξ_{3} (x, t) - η_{3} (x, t)}{\partial x_{i}})}^{2} d x \\ \leq - D \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x . \end{aligned}$ (12) ), for the next term in $^{+} D_{τ_{k}}^{α} W (ξ, t)$ , we have (15) $\begin{aligned} \int_{Θ} (ξ_{3} (x, t) - η_{3} (x, t)) [d_{ξ_{3}} Δ (ξ_{3} (x, t) - η_{3} (x, t)) + κ (ξ_{2} (x, t) - η_{2} (x, t)) \\ - m (ξ_{3} (x, t) - η_{3} (x, t)) - q (ξ_{3} (x, t) ξ_{4} (x, t) - η_{3} (x, t) η_{4} (x, t))] d x \\ \leq - (m + D) \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x \\ + κ \int_{Θ} | ξ_{3} (x, t) - η_{3} (x, t) | | ξ_{2} (x, t) - η_{2} (x, t) | d x \\ - q \int_{Θ} (ξ_{3} (x, t) - η_{3} (x, t)) (ξ_{3} (x, t) ξ_{4} (x, t) - η_{3} (x, t) η_{4} (x, t))] d x \\ \leq - (m + D) \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x \\ + \frac{κ}{2} \int_{Θ} [{(ξ_{2} (x, t) - η_{2} (x, t))}^{2} + {(ξ_{3} (x, t) - η_{3} (x, t))}^{2}] d x \\ + \frac{q}{2} {\hat{ξ}}_{3} \int_{Θ} [{(ξ_{3} (x, t) - η_{3} (x, t))}^{2} + {(ξ_{4} (x, t) - η_{4} (x, t))}^{2}] d x \\ = (- m + D + \frac{κ}{2} + \frac{q}{2} {\hat{ξ}}_{3}) \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x \\ + \frac{κ}{2} \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} d x + \frac{q}{2} {\hat{ξ}}_{3} \int_{Θ} {(ξ_{4} (x, t) - η_{4} (x, t))}^{2} d x . \end{aligned}$ (15) Also, (16) $\begin{aligned} \int_{Θ} (ξ_{4} (x, t) - η_{4} (x, t)) [g (ξ_{3} (x, t) ξ_{4} (x, t) - η_{3} (x, t) η_{4} (x, t)) \\ - h (ξ_{4} (x, t) - η_{4} (x, t))] d x \\ = (- h + g {\hat{ξ}}_{3} + \frac{g}{2} {\hat{ξ}}_{4}) \int_{Θ} {(ξ_{4} (x, t) - η_{4} (x, t))}^{2} d x \\ + \frac{g}{2} {\hat{ξ}}_{4} \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x \end{aligned}$ (16) and (17) $\begin{aligned} \int_{Θ} (ξ_{5} (x, t) - η_{5} (x, t)) [c (ξ_{2} (x, t) ξ_{5} (x, t) - η_{2} (x, t) η_{5} (x, t)) \\ - b (ξ_{5} (x, t) - η_{5} (x, t)] d x \\ = (- b + c {\hat{ξ}}_{2} + \frac{c}{2} {\hat{ξ}}_{5}) \int_{Θ} {(ξ_{5} (x, t) - η_{5} (x, t))}^{2} d x \\ + \frac{c}{2} {\hat{ξ}}_{5} \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} d x . \end{aligned}$ (17) Combining (Equation13(13) $\begin{aligned} \int_{Θ} (ξ_{1} (x, t) - η_{1} (x, t)) [- d (ξ_{1} (x, t) - η_{1} (x, t)) \\ - β (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t))] d x \\ = - d \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} \\ - β \int_{Θ} (ξ_{1} (x, t) - η_{1} (x, t)) (ξ_{3} (x, t) ξ_{1} (x, t) - η_{3} (x, t) η_{1} (x, t))] d x \\ \leq - d \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x - β \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} ξ_{3} (x, t) d x \\ + β \int_{Θ} | ξ_{1} (x, t) - η_{1} (x, t) | | η_{1} (x, t) | | ξ_{3} (x, t) - η_{3} (x, t) | d x \\ \leq - d \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x \\ + \frac{β}{2} {\hat{ξ}}_{1} \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x + \frac{β}{2} {\hat{ξ}}_{1} \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x, \end{aligned}$ (13) )–(Equation17(17) $\begin{aligned} \int_{Θ} (ξ_{5} (x, t) - η_{5} (x, t)) [c (ξ_{2} (x, t) ξ_{5} (x, t) - η_{2} (x, t) η_{5} (x, t)) \\ - b (ξ_{5} (x, t) - η_{5} (x, t)] d x \\ = (- b + c {\hat{ξ}}_{2} + \frac{c}{2} {\hat{ξ}}_{5}) \int_{Θ} {(ξ_{5} (x, t) - η_{5} (x, t))}^{2} d x \\ + \frac{c}{2} {\hat{ξ}}_{5} \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} d x . \end{aligned}$ (17) ), for the time generalized conformable derivative of $W$ of order α along the solution of model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) for $t \neq τ_{k}, k \in N$ , we obtain $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq (- d + \frac{β}{2} ({\hat{ξ}}_{1} + {\hat{ξ}}_{3})) \int_{Θ} {(ξ_{1} (x, t) - η_{1} (x, t))}^{2} d x \\ + (- a + \frac{1}{2} (κ + β ({\hat{ξ}}_{1} + {\hat{ξ}}_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5})) \\ \times \int_{Θ} {(ξ_{2} (x, t) - η_{2} (x, t))}^{2} d x \\ + (- m - D + β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4})) \int_{Θ} {(ξ_{3} (x, t) - η_{3} (x, t))}^{2} d x \\ + (- h + g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4})) \int_{Θ} {(ξ_{4} (x, t) - η_{4} (x, t))}^{2} d x \\ + (- b + c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5})) \int_{Θ} {(ξ_{5} (x, t) - η_{5} (x, t))}^{2} d x \end{aligned}$ to which we apply the system's parameters restrictions and get (18) $^{+} D_{τ_{k}}^{α} W (ξ, t) \leq 0,$ (18) for $t \neq τ_{k}, k \in N$ .

From (Equation11(11) $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq \frac{1}{2} \int_{Θ} \sum_{j = 1}^{5} D_{τ_{k}}^{α} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x \\ = \int_{Θ} \sum_{j = 1}^{5} (ξ_{j} (x, t) - η_{j} (x, t)) D_{τ_{k}}^{α} (ξ_{j} (x, t) - η_{j} (x, t)) d x . \end{aligned}$ (11) ), (Equation18(18) $^{+} D_{τ_{k}}^{α} W (ξ, t) \leq 0,$ (18) ) and Lemma 2.6, we obtain (19) $W (ξ (\cdot, t), t) \leq W (ξ_{0}, t_{0}^{+}), t \geq t_{0} .$ (19) The second step is to consider the solution $ξ (x, t) = ξ (x, t; t_{0}, ξ_{0})$ of problem (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) –( Equation6(6) $\begin{aligned} ξ (x, t_{0}^{+}) & = ξ_{0} (x) \geq 0, x \in Θ, t_{0} \in R_{+}, \end{aligned}$ (6) ), and let us suppose that $(x, t_{0}^{+}, ξ_{0}) \in Y$ for some $ξ_{0} \in R^{5}$ . We will prove that $Y$ is an integral manifold of (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ), i.e. $(x, t_{0}^{+}, ξ_{0}) \in Y$ , implies $(x, t, ξ (x, t)) \in Y$ , $x \in Θ$ , $t \geq t_{0}$ .

If we assume that the above is not true, then there exists $\bar{t} > t_{0}$ with $(x, t, ξ (x, t)) \in Y$ for $t_{0} < t \leq \bar{t}$ and $(x, t, ξ (x, t)) \notin Y$ for $t > \bar{t}$ and $x \in Θ$ .

There are two possibilities for $\bar{t}$ :

$t^{'} \neq τ_{k}$ for $k \in N$ and then there exists $\tilde{t}$ , $τ_{k} < \tilde{t} < τ_{k + 1}$ such that $(x, \tilde{t}, ξ (x, \tilde{t})) \notin Y$ and $W (ξ (x, \tilde{t}), \tilde{t}) > 0$ . On the other hand, from (Equation19(19) $W (ξ (\cdot, t), t) \leq W (ξ_{0}, t_{0}^{+}), t \geq t_{0} .$ (19) ) for $t = \tilde{t}$ and $x \in Θ$ , since $(x, t_{0}^{+}, ξ_{0}) \in Y$ , we have $W (ξ (x, \tilde{t}), \tilde{t}) \leq 0$ which contradicts the fact that $W (ξ (x, \tilde{t}), \tilde{t}) > 0$ .
If $\bar{t} = τ_{k}$ for some $k = l, l + 1, \dots$ , $l \geq 1$ , then by (Equation11(11) $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq \frac{1}{2} \int_{Θ} \sum_{j = 1}^{5} D_{τ_{k}}^{α} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x \\ = \int_{Θ} \sum_{j = 1}^{5} (ξ_{j} (x, t) - η_{j} (x, t)) D_{τ_{k}}^{α} (ξ_{j} (x, t) - η_{j} (x, t)) d x . \end{aligned}$ (11) ), we get $0 \leq W (ξ (\cdot, {\bar{t}}^{+}), {\bar{t}}^{+}) < W (ξ (\cdot, \bar{t}), \bar{t}) = 0$ since $(x, \bar{t}, ξ (x, \bar{t})) \in Y$ .

The contradictions obtained prove that $Y$ is an integral manifold for the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ), and this completes the proof.

3.2. Exponential conformable stability results

Theorem 3.2

If the hypotheses H3.1, H3.2, H3.4 and H3.5 hold, and the system's parameters are such that (20) $\begin{aligned} min {d, a, m + D, h, b} \\ \geq max {\frac{1}{2} (κ + β ({\hat{ξ}}_{1} + {\hat{ξ}}_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5}), β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5})} + γ, γ \geq 0, \end{aligned}$ (20) then the integral manifold $Y$ of system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) is globally exponentially conformable stable.

Proof.

Since the conditions of Theorem 3.2 imply those of Theorem 3.1, the manifold $Y$ is an integral manifold for the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ).

Using the same Lyapunov-type function, we can prove that for any $ξ_{0}$ , $(x, t_{0}^{+}, ξ_{0}) \in Y$ , we have (Equation11(11) $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq \frac{1}{2} \int_{Θ} \sum_{j = 1}^{5} D_{τ_{k}}^{α} {(ξ_{j} (x, t) - η_{j} (x, t))}^{2} d x \\ = \int_{Θ} \sum_{j = 1}^{5} (ξ_{j} (x, t) - η_{j} (x, t)) D_{τ_{k}}^{α} (ξ_{j} (x, t) - η_{j} (x, t)) d x . \end{aligned}$ (11) ) and for $t \neq τ_{k}, k \in N$ , we obtain $\begin{aligned} ^{+} D_{τ_{k}}^{α} W (ξ, t) & \leq - min {d, a, m + D, h, b} W (ξ, t) \\ + max {\frac{1}{2} (κ + β ({\hat{ξ}}_{1} + ξ_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5}), \\ β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5})} W (ξ, t) . \end{aligned}$ Denote $γ_{1} = min {d, a, m + D, h, b}$ and $\begin{aligned} γ_{2} & = max {\frac{1}{2} (κ + β ({\hat{ξ}}_{1} + ξ_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5}), β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5})} . \end{aligned}$ By (Equation20(20) $\begin{aligned} min {d, a, m + D, h, b} \\ \geq max {\frac{1}{2} (κ + β ({\hat{ξ}}_{1} + {\hat{ξ}}_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5}), β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5})} + γ, γ \geq 0, \end{aligned}$ (20) ), we obtain (21) $^{+} D_{τ_{k}}^{α} W (ξ, t) \leq - γ W (ξ, t), t \neq τ_{k}, k \in N,$ (21) where $γ \geq 0$ .

Remark 3.3

The integral manifold's exponential conformable stability result established in Theorem 3.2 generalizes and extends the stability results proposed in [Citation7] for the diffused HCV model with CTL and antibody responses (Equation2(2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) ) to the conformable case considering impulsive control mechanism. Also, instead of considering single states of the neural network models, the generalization offered is concerned with integral manifolds of states. The proposed impulsive conformable approach taking into consideration integral manifolds can be also applied to various infectious diseases models of integer and fractional order [Citation1, Citation2, Citation5, Citation7, Citation9, Citation12, Citation13, Citation16, Citation20, Citation22, Citation31, Citation32, Citation34, Citation35, Citation41]. In addition, the result demonstrated in Theorem 3.2 extends the exponential conformable result in [Citation48] to the integral manifold case, including impulsive effects at some discrete instances.

Remark 3.4

The exponential stability of an integer-order system guarantees its asymptotic stability. Using properties of the conformable exponential function $E_{α} (- γ, t - t_{0})$ [Citation42, Citation48] for $t \to \infty$ , a similar conclusion can be made for the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ). Hence, the conditions of Theorem 3.2 also imply the global asymptotic stability of the integral manifold $Y$ of system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ).

Remark 3.5

Theorem 3.2 also can be applied in the study of local asymptotic stability of the integral manifold $Y$ of system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ), if we denote by $B_{ε} = {ξ \in R^{5} : | | ξ | |_{2} < ε} .$ Then for any $μ > 0$ and $ε > 0$ , from the fact that $W (ξ, t_{0}) = 0$ for $ξ \in Y (x, t_{0})$ it follows that there exists a $δ = δ (t_{0}, μ, ε) > 0$ such that if $ξ \in B_{μ} \cap \bar{Y} (x, t_{0}) (δ)$ , then $W (ξ, t_{0}^{+}) < \frac{1}{2} ε^{2}$ . If we let $ξ_{0} \in B_{μ} \cap \bar{Y} (x, t_{0}) (δ)$ , then we will have from (Equation22(22) $W (ξ (\cdot, t), t) \leq W (ξ_{0}, t_{0}^{+}) E_{α} (- γ, t - t_{0}), t \geq t_{0},$ (22) ) that $d (ξ (x, t; t_{0}, ξ_{0}), Y (x, t)) < ε, t \geq t_{0},$ which shows the stability (local) of the integral manifold $Y$ . In addition, since the system's solutions are bounded and $lim_{t \to \infty} | | ξ (\cdot, t) | |_{2} = 0,$ it follows that the integral manifold $Y$ is asymptotically stable.

4. An example

Example 4.1

To illustrate the main existence and stability integral manifold results, we consider the model (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) with $t \geq 0$ , $x \in Θ \subset R^{3}$ , Θ is a bounded set, $λ = 2$ , d = 3.1, a = 4.5, m = 1.3, h = 4.7, b = 5.1, $\underline{d} = 10$ , $κ = 1.2$ , $β = 1.3$ , p = 1.3, q = 1.1, g = 1.4, c = 1.2, $φ_{jk} = 2 - \frac{1}{k}$ , j = 2, 3, $k \in N$ .

Define the manifold $Y \subset Θ \times R_{+} \times R^{5}$ such that $Y (x, t) = {ξ \in R^{5} : ξ \leq 0.7, (x, t, ξ) \in Y, (x, t) \in Θ \times R_{+}} .$ Hence, we have that ${\hat{ξ}}_{j} = 0.7$ , $j = 1, 2, \dots, 5$ , and since all conditions of Theorem 3.1 are satisfied, the manifold $Y$ is an integral manifold of the model.

In addition, we have that $γ_{1} = min {d, a, m + D, h, b} = 3.1$ and $\begin{aligned} γ_{2} & = max {\frac{1}{2} (κ + β ({\hat{ξ}}_{1} + ξ_{3}) + p {\hat{ξ}}_{2} + g {\hat{ξ}}_{4} + c {\hat{ξ}}_{5}), β {\hat{ξ}}_{1} + \frac{1}{2} (κ + q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), \\ g {\hat{ξ}}_{3} + \frac{1}{2} (q {\hat{ξ}}_{3} + g {\hat{ξ}}_{4}), c {\hat{ξ}}_{2} + \frac{1}{2} (p {\hat{ξ}}_{2} + c {\hat{ξ}}_{5})} = 2.875 \end{aligned}$ Therefore, all the requirement of Theorem 3.2 are satisfied for $0 < γ \leq 0.225$ , and the integral manifold $Y$ of system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) is globally exponentially conformable stable.

Remark 4.1

According to [Citation7] and some of the references cited therein, when the basic reproductive number $R_{0} = \frac{λβκ}{dam}$ satisfies $R_{0} < 1$ , the equilibrium $E_{0} = (\frac{λ}{d}, 0, 0, 0, 0)$ of model (Equation2(2) ${\begin{cases} \frac{\partial ξ_{1} (x, t)}{\partial t} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), \\ \frac{\partial ξ_{2} (x, t)}{\partial t} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), \\ \frac{\partial ξ_{3} (x, t)}{\partial t} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), \\ \frac{\partial ξ_{4} (x, t)}{\partial t} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), \\ \frac{\partial ξ_{5} (x, t)}{\partial t} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), \end{cases}$ (2) ) is globally asymptotically stable. For the particular choice of the parameters, the integral manifold defined in Example 4.1 contains the equilibrium $E_{0}$ and $R_{0} < 1$ . Therefore, in the absence of impulses and for $α = 1$ , our results imply the results in [Citation7]. However, besides the state $E_{0}$ , the integral manifold introduced for the system (Equation4(4) ${\begin{cases} \frac{\partial^{α} ξ_{1} (x, t)}{\partial τ_{k}} = λ - d ξ_{1} (x, t) - β ξ_{3} (x, t) ξ_{1} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{2} (x, t)}{\partial τ_{k}} = β ξ_{3} (x, t) ξ_{1} (x, t) - a ξ_{2} (x, t) - p ξ_{2} (x, t) ξ_{5} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{3} (x, t)}{\partial τ_{k}} = d_{ξ_{3}} Δ ξ_{3} + κ ξ_{2} (x, t) - m ξ_{3} (x, t) - q ξ_{3} (x, t) ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{4} (x, t)}{\partial τ_{k}} = g ξ_{3} (x, t) ξ_{4} (x, t) - h ξ_{4} (x, t), t \neq τ_{k}, \\ \frac{\partial^{α} ξ_{5} (x, t)}{\partial τ_{k}} = c ξ_{2} (x, t) ξ_{5} (x, t) - b ξ_{5} (x, t), t \neq τ_{k}, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}) + J_{jk} (ξ_{j} (x, τ_{k})), j = 2, 3, \\ ξ_{j} (x, τ_{k}^{+}) = ξ_{j} (x, τ_{k}), j = 1, 4, 5, \end{cases}$ (4) ) contains other states of the generalized model. Thus, the proposed research extends the existing results for stability of separate system's states.

Remark 4.2

The example provided also demonstrates that an efficient impulsive control strategy requires appropriate controllers. If the impulsive controllers do not satisfy H3.4, then a conclusion about the integral manifold stability cannot be made.

5. Concluding remarks

In this research, the authors propose an impulsive conformable approach for representing a diffused HCV model with CTL and antibody responses. The proposed modelling perspective leads to the development of more generalized and flexible model, which includes as a specific case the existing models proposed in the literature. The impulsive controllers that can be used in the implementation of an impulsive therapy to the model are considered at fixed instances. Since the stability behaviour of the system's states is essential for the construction and operation of the model, the concept for global exponential conformable stability is introduced. However, instead of considering a separate neuronal state, the integral manifolds notion is adopted. Using the conformable Lyapunov function method, efficient existence and stability criteria for the integral manifold are established. The efficiency of the obtained result is demonstrated by an example. Given the advantages of the developed method, it would provide a convenient tool of extending various other biological models to the impulsive conformable case. As future development perspectives, we note the consideration of delay terms, which is biologically motivated.

Disclosure statement

No potential conflict of interest was reported by the author(s).

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Integral manifolds for impulsive HCV conformable neural network models

ABSTRACT

1. Introduction

2. Preliminaries

2.1. Notes on conformable calculus

[Citation62]

2.2. Development of the impulsive conformable HCV virus infection model

2.3. Integral manifolds

2.4. Lyapunov function method's fundamentals

3. Results

3.1. Existence of integral manifolds

3.2. Exponential conformable stability results

4. An example

5. Concluding remarks

Disclosure statement

References

Information for

Open access

Opportunities

Help and information

Integral manifolds for impulsive HCV conformable neural network models

ABSTRACT

1. Introduction

2. Preliminaries

2.1. Notes on conformable calculus

[Citation62]

2.2. Development of the impulsive conformable HCV virus infection model

2.3. Integral manifolds

2.4. Lyapunov function method's fundamentals

3. Results

3.1. Existence of integral manifolds

3.2. Exponential conformable stability results

4. An example

5. Concluding remarks

Disclosure statement

References

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