==A fractional numerical study on a plant disease model with replanting and preventive treatment==

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'''

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Zuhur americanAlqahtani<math>^{1}</math>,  Ahmed Hagagamerican<math>^{2,*}</math>

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american<math>^{1}</math>Department of Mathematical Science, College of Science, Princess Nourah Bint Abdulrahman University,

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americanP.O. Box 84428, Riyadh 11671, Saudi Arabia

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american<math>^{2}</math>Department of Basic Science, Faculty of Engineering, Sinai University, Ismailia, Egypt

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american<math>^{2,*}</math>E-mail:ahmed.shehata@su.edu.eg'''

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==Abstract==

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Food security has become a significant issue due to the growing human population. In this case, a significant role is played by agriculture. The essential foods are obtained mainly from plants. Plant diseases can, however, decrease both food production and its quality. Therefore, it is substantial to comprehend the dynamics of plant diseases as they can provide insightful information about the dispersal of plant diseases. In order to investigate the dynamics of plant disease and analyze the effects of strategies of disease control, a mathematical model can be applied. We show that this model provides the non-negative solutions that population dynamics requires. The model was investigated by using the Atangana-Baleanu in Caputo sense (ABC) operator which is symmetrical to the Caputo-Fabrizio (CF) operator with a different function. Whereas the ABC operator uses the generalized Mittag-Leffler function while the CF operator employs the exponential kernel. For the proposed model, we have displayed the local and global stability of a nonendemic and an endemic equilibrium, existence and uniqueness theorems. By applying the fractional Adams-Bashforth-Moulton method, we have implemented numerical solutions to illustrate the theoretical analysis.

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'''Keywords''': Adams-Bashforth-Moulton method, local and global asymptotic equilibrium stability, sensitivity analysis, existence theorems, uniqueness theorems, plant diseases model, numerical simulations

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==1. Introduction==

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Plants are an incredibly valuable component of our world. The earth, due to the presence of plants, is known as a green planet. They are perhaps the most important component of the life of all the earth's living beings. Some of the plant's essential functions are food, reducing the level of pollution, supplying fresh oxygen, medications, furniture and refuge. Plant disease, however, triggers a decline in food production and efficiency, which can lead to numerous health and social issues. Moreover, it can cause considerable economic ramifications.

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Plant disease epidemiology studies the evolution of populations of plant diseases in time and space. Usually, the execution of the techniques is accomplished by roguing and then replanting, which offers two possible advantages. Initially, infected plants could be removed and could inoculum sources could be reduced, possibly slowing the dispersal of pathogens. Then, the infected plant may die or suffer a reduction in yield. Consequently, substituting diseased plants with healthful plants can indemnity crop casualties. The exposed plant can also be able to spread disease in some cases [1-3]. Since the exposed plant is capable of spreading disease, the propagation of plant disease can be more rapid. It is therefore important to realize the impact of plants uncovered on the dynamics of plant disease contamination. Their simulation revealed that the application of fungicides is efficient in minimizing population infection [4-6].

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Mathematical modeling is useful in explaining how diseases dispersal and different factors involved in the dispersal of the disease have been specified [7-11]. The defensive and curative fungicide model was introduced in Anggriani et al. [12], where it has been split into three ingredients: Infectious, protected and susceptible. Their simulation revealed that the implementation of fungicides is active in minimizing population infection. Model plant diseases with replanting, roguing and preventive care have been introduced in Anggriani et al. [13] without consideration to curative therapy. In 2017, Anggriani et al. [14] created a plant disease mathematical model that includes five ingredients: Susceptible, Protected, Infectious, Exposed and Post-Infectious with protective curative therapies. They observed that by using curative and preventative care, the transmission of plant disease can be minimized. However, where only one therapy is offered, preventive therapy is favored over curative therapy.

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In actuality, there are various meanings of fractional derivatives which in general do not necessarily correspond. One of these definitions which is often utilized in different implementations of fractional differential equations is Caputo fractional derivative [15-18]. There is also a novel fractional derivative definition, named the Caputo-Fabrizio fractional operator, which centered on the exponential function [19-21]. And in the same context, there is a novel fractional derivative definition, named the Atangana-Baleanu fractional operator, which centered on the Mittag-Leffler function [22-27]. Many authors have successfully attempted to model actual processes utilizing this fractional derivative operator [28-29].

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We consider a plant disease spread model within fractional calculus, where a susceptible person crosses an exposed stage prior to becoming an infectious person and diseases may also be passed on by exposed plants. The major purpose for this protraction is that the plant diseases of the classical case [12-14] do not load any acquaintance about the memory and learning techniques that impact the propagation of a disease [25]. Now, we regard the plant diseases model with fractional order as follows:

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<span id="eq-1"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }S_{1}\left(t\right) & =  r(M-N)-\gamma S_{1}\left(t\right)-\frac{a_{1}}{M}S_{1}\left(t\right)I_{1}\left(t\right)-\alpha S_{1}\left(t\right)+\beta P_{1}\left(t\right),\\

^{ABC}D_{*}^{\upsilon }P_{1}\left(t\right) & =  \alpha S_{1}\left(t\right)+\beta P_{1}\left(t\right)-\gamma P_{1}\left(t\right),\\

^{ABC}D_{*}^{\upsilon }E_{1}\left(t\right) & =\frac{a_{1}}{M}S_{1}\left(t\right)I_{1}\left(t\right) -\left(\gamma + a_{2}+b_{1}\right) E_{1}\left(t\right),\\

^{ABC}D_{*}^{\upsilon }I_{1}\left(t\right) & =  a_{2}E_{1}\left(t\right)-\left(\gamma{+}a_{3}+b_{2}+\eta \right)I_{1}\left(t\right),\\

^{ABC}D_{*}^{\upsilon }R_{1}\left(t\right) &=  a_{3}I_{1}\left(t\right)-\left(\gamma{+}b_{3}\right)R_{1}\left(t\right). \end{align}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" |(1)

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With initial conditions <math>\left(S_{1}\right)_{0}\left(0\right)>0</math>, <math>\left(P_{1}\right)_{0}\left(0\right)\geq{0}</math>, <math>\left(E_{1}\right)_{0}\left(0\right)\geq{0}</math>, <math>\left(I_{1}\right)_{0}\left(0\right)\geq{0}</math> and <math>\left(R_{1}\right)_{0}\left(0\right)\geq{0}</math>, where <math>N</math> indicate to the overall population of the actual plant, <math>S_{1}\left(t\right)</math> is representing the Susceptible population, <math>P_{1}\left(t\right)</math> is representing the Protected population, <math>E_{1}\left(t\right)</math> is representing the Exposed population, <math>I_{1}\left(t\right)</math> is representing the Infectious/Removed population, <math>R_{1}\left(t\right)</math> is representing the Recovered population, and <math>N=S_{1}\left(t\right)+P_{1}\left(t\right)+E_{1}\left(t\right)+I_{1}\left(t\right)+R_{1}\left(t\right)</math>, this suggests that the size of population is not steady. (i.e. size of the population is variable). The actual meaning of each of the model's parameters, all of which have positive values, is as follows:

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* <math display="inline">r</math>: rate of replanting.

* <math display="inline">a_{1}</math>:  disease progression diversion rate for latent compartment.

* <math display="inline">a_{2}</math>: disease progression diversion rate for infected compartment.

* <math display="inline">a_{3}</math>: disease progression diversion rate for removed compartment.

* <math display="inline">M</math>: overall maximum plant population (agronomy).

* <math display="inline">b_{1}</math>: influence accumulative death rate for latent compartment.

* <math display="inline">b_{2}</math>: influence accumulative death rate for infected compartment.

* <math display="inline">b_{3}</math>: influence accumulative death rate for latent removed compartment.

* <math display="inline">\alpha </math>: efficacy preventive therapy.

* <math display="inline">\beta </math>: preventive therapy rate.

* <math display="inline">\gamma </math>: rate of natural death.

* <math display="inline">\eta </math>: rate of roguing.

​

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Some assumptions have been made to facilitate understanding this model [american14]:

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* There is no closure of the plant population due to replanting and natural death.

* The infected plant compartment comprises of two compartments, called, latent <math display="inline">E(t)</math> and infected <math display="inline">I(t)</math>.

* The infected plant is lifted if it shows comprehensive symptoms.

* The insect vector and environment factor are neglected.

* The Preventive therapy (insecticide) be presented to susceptible compartments.

* The Protected compartment <math display="inline">P_{1}(t)</math> contains the Susceptible plants <math display="inline">S_{1}(t)</math> that have received protective therapy.

* Protected plants have defensive or prevention impact, but are not immune to disease, therefore it is permitting re-entry into the susceptible compartment.

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This article is structured to have some significant preliminaries in Section 2. Section 3 transacts with studying the local and global asymptotic equilibrium stability (the disease-free case and the endemic case) and the sensibility analysis of reproduction number (<math display="inline">\mathrm{\mathcal{R}}_{0}</math>) without control of our model ([[#eq-1|1]]). Section 4 transacts with studying the existence and uniqueness theorems of our model ([[#eq-1|1]]). Then, computational technique (Adams-Bashforth-Moulton method) are graphically represented and covered in Sections 5 and 6. Finally, conclusions are drawn.

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==2. Preliminaries==

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Recently, many fractional calculus concepts and definitions have been developed [31-32].

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===Definition 1===

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The fractional derivative of the Atangana-Baleanu in Caputo sense (ABC) is denoted as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>_{0}^{ABC}D_{\varrho }^{\upsilon }\varphi \left(\varrho \right)=\frac{\chi \left(\upsilon \right)}{1-\upsilon }\stackrel=0\varphi ^{\prime }\left(\tau \right)E_{\upsilon }\left[\frac{-\upsilon }{1-\upsilon }\left(\varrho{-\tau}\right)^{\upsilon }\right]d\tau , </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (2)

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where <math display="inline">\chi \left(\upsilon \right)</math> is a normalized function with <math display="inline">\chi (0)=\chi (1)=1</math> which are symmetrical to the Caputo-Fabrizio (CF) case, <math display="inline">\varphi \left(\varrho \right)\in \mathcal{H}^{1}(a,b)</math>, <math display="inline">b>a</math>, <math display="inline">\upsilon \in (0,1]</math>, where <math display="inline">\mathcal{H}^{1}(a,b)</math> is the Sobolev space <math display="inline">(\mathcal{H})</math> of order 1 in <math display="inline">(a,b)</math> and <math display="inline">E_{\upsilon }</math> indicate to a Mittag-Leffler function expressed as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>E_{\upsilon }\left(-\varrho ^{\upsilon }\right) =  \stackrel={\scriptscriptstyle i=0}\frac{\left(-\varrho \right)^{i\upsilon }}{\Gamma \left(i\upsilon{+1}\right)}. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (3)

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===Definition 2===

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The related fractional integral to the AB Caputo operator is denoted as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} _{0}^{AB}J_{\varrho }^{\upsilon }\varphi \left(\varrho \right) & =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varphi \left(\varrho \right)+\frac{\upsilon }{\chi \left(\upsilon \right)\Gamma \left(\upsilon \right)}\stackrel=0\varphi \left(\tau \right)\left(\varrho{-\tau}\right)^{\upsilon{-1}}d\tau ,\\

& =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varphi \left(\varrho \right)+\frac{\upsilon }{\chi \left(\upsilon \right)\Gamma \left(\upsilon \right)}\left(I^{\upsilon }\varphi \left(\varrho \right)\right).\end{align} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (4)

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==3. Analysis of plant disease model==

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===3.1 The local asymptotic equilibrium stability===

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Since <math display="inline">^{ABC}D^{\upsilon }S_{1}\left(t\right)=0</math>, <math display="inline">^{ABC}D^{\upsilon }P_{1}\left(t\right)=0</math>, <math display="inline">^{ABC}D^{\upsilon }E_{1}\left(t\right)=0</math>, <math display="inline">^{ABC}D^{\upsilon }I_{1}\left(t\right)=0</math> and <math display="inline">^{ABC}D^{\upsilon }R_{1}\left(t\right)=0</math> when <math display="inline">t\rightarrow \infty </math>, we can use them to find two equilibrium points of the fractional system (Eq.[[#eq-1|(1)]]), by solving the above equations we get

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====3.1.1 Non endemic equilibrium (NEE) point====

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The NEE solution of the system ([[#eq-1|1]]) is

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\mathcal{E}_{0}=\left(\left(S_{1}\right)_{eq},\left(P_{1}\right)_{eq},\left(E_{1}\right)_{eq},\left(I_{1}\right)_{eq},\left(R_{1}\right)_{eq}\right)</math><math> =\left(\frac{\gamma A_{3}Mr}{(\gamma{+}r)(\alpha \beta{+}A_{3}(\alpha{+\gamma}))},-\frac{\alpha \gamma Mr}{(\gamma{+}r)(\alpha \beta{+}A_{3}(\alpha{+\gamma}))},0,0,0\right), </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (5)

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with <math display="inline">N_{E_{1}q}=\frac{rM}{\gamma{+}r}</math> where <math display="inline">A_{1}=a_{2}+b_{1}+\gamma </math>, <math display="inline">A_{2}=\gamma{+}a_{3}+b_{2}+\eta </math>, <math display="inline">A_{3}=\beta{-\gamma}</math>. The basic reproduction number <math display="inline">\mathrm{\mathcal{R}}_{0}</math> that is calculated utilizing the generation operator method [28,29] can be found in [14] as follows:

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\mathrm{\mathcal{R}}_{0}  =  \frac{ra_{1}a_{2}(\beta{+\gamma})}{A_{1}A_{2}(\gamma{+}r)\left(\alpha{+\beta}+\gamma \right)}. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (6)

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It is known as the number of secondary contagions induced by a single primary contagion in a completely susceptible population [35] and is generally expressed using model (Eq.[[#eq-1|(1)]]). The rate of the basic reproduction number counts on the replanting rate value. This is the justification that we should replant the plant if we want to control plant disease.

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====3.1.2 Endemic equilibrium (EE) point====

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The endemic equilibrium solution of the system (1) is

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" |<math>\mathcal{E}^{*}=\left(S_{1}^{*},P_{1}^{*},E_{1}^{*},I_{1}^{*},R_{1}^{*}\right),</math>

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where

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{cases}S_{1}^{*}= & \displaystyle\frac{A_{1}A_{2}M}{a_{1}a_{2}},\\ P_{1}^{*}= & -\displaystyle\frac{\alpha A_{1}A_{2}M}{a_{1}a_{2}A_{3}},\\ E_{1}^{*}= & \displaystyle\frac{A_{2}M\left(b_{3}+\gamma \right)\left(A_{1}A_{2}(\gamma{+}r)(\alpha \beta{+}A_{3}(\alpha{+\gamma}))-a_{1}a_{2}\gamma A_{3}r\right)}{a_{1}a_{2}A_{3}\left(a_{2}r\left(b_{3}\left(a_{3}+b_{2}+\eta \right)+\gamma \left(b_{2}+\eta \right)\right)-A_{2}\left(b_{3}+\gamma \right)\left(A_{1}(\gamma{+}r)-b_{1}r\right)\right)},\\ I_{1}^{*}= & -\displaystyle\frac{M\left(b_{3}+\gamma \right)\left(A_{1}A_{2}(\gamma{+}r)(\alpha \beta{+}A_{3}(\alpha{+\gamma}))-a_{1}a_{2}\gamma A_{3}r\right)}{a_{1}A_{3}\left(A_{2}\left(b_{3}+\gamma \right)\left(A_{1}(\gamma{+}r)-b_{1}r\right)-a_{2}r\left(b_{3}\left(a_{3}+b_{2}+\eta \right)+\gamma \left(b_{2}+\eta \right)\right)\right)},\\ R_{1}^{*}= & -\displaystyle\frac{a_{3}M\left(A_{1}A_{2}(\gamma{+}r)(\alpha \beta{+}A_{3}(\alpha{+\gamma}))-a_{1}a_{2}\gamma A_{3}r\right)}{a_{1}A_{3}\left(A_{2}\left(b_{3}+\gamma \right)\left(A_{1}(\gamma{+}r)-b_{1}r\right)-a_{2}r\left(b_{3}\left(a_{3}+b_{2}+\eta \right)+\gamma \left(b_{2}+\eta \right)\right)\right)}. \end{cases} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (7)

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with <math display="inline">N^{*}=\displaystyle\frac{rM-b_{1}E_{1}^{*}-\left(b_{2}+\eta \right)I_{1}^{*}-b_{3}R_{1}^{*}}{\gamma{+}r}</math>. Conclusion of the outcomes of the system's equilibrium points ([[#eq-1|1]]) with prepared to research analytically the stability of the equilibrium points.

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'''Theorem 3.1.2.1'''. The NEE point <math display="inline">\mathcal{E}_{0}</math> of model (Eq.[[#eq-1|(1)]]) is locally asymptotically stable if <math display="inline">\mathrm{\mathcal{R}}_{0}<1</math> and unstable if <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math>.

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'''Proof'''. The system's Jacobian matrix (Eq.[[#eq-1|(1)]]) at <math>\mathcal{E}_{0}</math> is

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>J_{\mathcal{E}_{0}}=\left(\begin{array}{ccccc}-\left(\alpha{+\gamma}\right)& \beta & 0 & \displaystyle\frac{ra_{1}\left(\beta{+\gamma}\right)}{\left(\gamma{+}r\right)\left(\alpha{+\beta}+\gamma \right)} & 0\\ \alpha & -\left(\beta{+\gamma}\right)& 0 & 0 & 0\\ 0 & 0 & -A_{1} & \displaystyle\frac{ra_{1}\left(\beta{+\gamma}\right)}{\left(\gamma{+}r\right)\left(\alpha{+\beta}+\gamma \right)} & 0\\ 0 & 0 & a_{2} & -A_{2} & 0\\ 0 & 0 & 0 & a_{3} & -\left(\gamma{+}b_{3}\right) \end{array}\right). </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (8)

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The eigenvalues of <math display="inline">J_{\mathcal{E}_{0}}</math>are given as <math display="inline">\lambda _{1}=-\gamma{<0},\quad \lambda _{2}=-\alpha{-\beta}-\gamma{<0},\quad \lambda _{3}=-b_{3}-\gamma{<0}</math> and the following quadratic equation gives <math display="inline">\lambda _{4}</math> and <math display="inline">\lambda _{5}</math> as

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<span id="eq-9"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\psi ^{2}+\left(A_{1}+A_{2}\right)\psi{+}A_{1}A_{2}\left(1-\mathrm{\mathcal{R}}_{0}\right) =  0. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (9)

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From Eq.([[#eq-9|9]]), we can notice that if <math display="inline">\mathrm{\mathcal{R}}_{0}\leq{1}</math>, then <math display="inline">\mathcal{E}_{0}</math> is locally asymptotically stable. This suggests that all polynomial coefficients ([[#eq-9|9]]) have the same signal, then the eigenvalues (roots) have negative real part. But if <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math>, the NEE is unstable and this would lead to a stable endemic equilibrium being present of <math display="inline">\mathcal{E}^{*}</math>. Now by proving the theorem of local stability of <math display="inline">\mathcal{E}^{*}</math> , we conclude this section.

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'''Theorem 3.1.2.2'''. If <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math>, the endemic equilibrium <math>\mathcal{E}^{*}</math> of model ([[#eq-1|1]]) is locally asymptotically stable.

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'''Proof'''. The system's Jacobian matrix (Eq.[[#eq-1|(1)]]) at <math display="inline">\mathcal{E}^{*}</math> is

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>J_{\mathcal{E}^{*}}=\left(\begin{array}{ccccc}-\left(\beta{+\gamma}\right)-\displaystyle\frac{\gamma \left(\mathrm{\mathcal{R}}_{0}-1\right)\left(\alpha{+\beta}+\gamma \right)}{\left(\beta{+\gamma}\right)} & \beta & 0 & \displaystyle\frac{ra_{1}\left(\beta{+\gamma}\right)}{\mathrm{\mathcal{R}}_{0}\left(\gamma{+}r\right)\left(\alpha{+\beta}+\gamma \right)} & 0\\ \alpha & -\left(\beta{+\gamma}\right)& 0 & 0 & 0\\ \displaystyle\frac{\gamma \left(\mathrm{\mathcal{R}}_{0}-1\right)\left(\alpha{+\beta}+\gamma \right)}{\left(\beta{+\gamma}\right)} & 0 & -A_{1} & \displaystyle\frac{ra_{1}\left(\beta{+\gamma}\right)}{\mathrm{\mathcal{R}}_{0}\left(\gamma{+}r\right)\left(\alpha{+\beta}+\gamma \right)} & 0\\ 0 & 0 & a_{2} & -A_{2} & 0\\ 0 & 0 & 0 & a_{3} & -\left(\gamma{+}b_{3}\right) \end{array}\right), </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (10)

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and its polynomial characteristic written as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>C_{J_{\mathcal{E}^{*}}}  =  \frac{\varphi{+\gamma}+b_{3}}{\mathrm{\mathcal{R}}_{0}\left(\gamma{+}r\right)\left(\alpha{+\beta}+\gamma \right)\left(\beta{+\gamma}\right)}\left(c_{0}\varphi ^{4}+c_{1}\varphi ^{3}+c_{2}\varphi ^{2}+c_{3}\varphi{+}c_{4}\right). </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (11)

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The eigenvalues of <math display="inline">C_{J_{\mathcal{E}^{*}}}</math> are <math display="inline">-\left(\gamma{+}b_{3}\right)</math> and the roots of the polynomial <math display="inline">c_{0}\varphi ^{4}+c_{1}\varphi ^{3}+c_{2}\varphi ^{2}+c_{3}\varphi{+}c_{4}</math>. By applying the Routh-Hortwitz criterian [31] we find that <math display="inline">c_{4},c_{3},c_{2},c_{1},c_{0}>0</math>, <math display="inline">c_{2}c_{3}-c_{1}c_{4}>0</math> and <math display="inline">c_{1}c_{2}c_{3}-c_{1}^{2}c_{4}-c_{0}c_{3}^{2}>0</math>. Therefore every the polynomial roots <math display="inline">C_{J_{\mathcal{E}^{*}}}</math> have a negative real part when <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math> [32,33]. This implies that <math display="inline">\mathcal{E}^{*}=\left(S_{1}^{*},P_{1}^{*},E_{1}^{*},I_{1}^{*},R_{1}^{*}\right)</math> is locally asymptotically stable when <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math> [30].

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===3.2 The global asymptotic equilibrium stability===

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'''Theorem 3.2.1'''. The disease-free equilibrium of the plant disease model is globally asymptotically stable in the suitable range if <math display="inline">\mathrm{\mathcal{R}}_{0}<1</math> and unstable if <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math> .

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'''Proof'''. We are applying the Lyapunov function [33], which is defined by

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\mathcal{L}  =  \frac{1}{\ell _{1}}S_{1}+\frac{1}{\ell _{2}}P_{1}+\frac{1}{\ell _{3}}E_{1}+\frac{1}{\ell _{4}}I_{1}+\frac{1}{\ell _{5}}R_{1}, </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (12)

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where

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\ell _{1}=\alpha{+\gamma},\quad \ell _{2}=\gamma{-\beta},\quad \ell _{3}=\gamma{+}a_{2}+b_{1},\quad \ell _{4}=\gamma{+}a_{3}+b_{2}+\eta ,\quad \ell _{5}=\gamma{+}b_{3}. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (13)

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englishConsequently, its derivative along the plant disease model's solutions

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>^{ABC}D_{*}^{\upsilon }\mathcal{L}  =  \frac{1}{\ell _{1}}{}^{ABC}D_{*}^{\upsilon }S_{1}+\frac{1}{\ell _{2}}{}^{ABC}D_{*}^{\upsilon }P_{1}+\frac{1}{\ell _{3}}{}^{ABC}D_{*}^{\upsilon }E_{1}</math> <math>+\frac{1}{\ell _{4}}{}^{ABC}D_{*}^{\upsilon }I_{1}+\frac{1}{\ell _{5}}{}^{ABC}D_{*}^{\upsilon }R_{1}. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (14)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }\mathcal{L}  & =  \frac{1}{\ell _{1}}\left[r(M-N)-\frac{a_{1}}{M}S_{1}I_{1}+\beta P_{1}-\left(\alpha{+\gamma}\right)S_{1}\right]\\

& \quad +\frac{1}{\ell _{2}}\left[\alpha S_{1}-\left(\gamma{-\beta}\right)P_{1}\right]\\

& \quad +\frac{1}{\ell_{3}}\left[\frac{a_{1}}{M}S_{1}I_{1}-\left(\gamma{+}a_{2}+b_{1}\right)E_{1}\right]\\

& \quad +\frac{1}{\ell _{4}}\left[a_{2}E_{1}-\left(\gamma{+}a_{3}+b_{2}+\eta \right)I_{1}\right]\\

& \quad +\frac{1}{\ell _{5}}\left[a_{3}I_{1}-\left(\gamma{+}b_{3}\right)R_{1}\right], \end{align}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (15)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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| 

{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }\mathcal{L}  &  = \,  \frac{1}{\ell _{1}}\left[r(M-N)-\frac{a_{1}}{M}S_{1}I_{1}+\beta P_{1}-\ell _{1}S_{1}\right]+\frac{1}{\ell _{2}}\left[\alpha S_{1}-\ell _{2}P_{1}\right]+\frac{1}{\ell _{3}}\left[\frac{a_{1}}{M}S_{1}I_{1}-\ell _{3}E_{1}\right]\\

& \quad +  \frac{1}{\ell _{4}}\left[a_{2}E_{1}-\ell _{4}I_{1}\right]+\frac{1}{\ell _{5}}\left[a_{3}I_{1}-\ell _{5}R_{1}\right]\\

 & = \,  \left\{\frac{1}{\ell _{1}}\left(r(M-N)-\frac{a_{1}}{M}S_{1}I_{1}+\beta P_{1}\right)+\frac{\alpha }{\ell _{2}}S_{1}+\frac{1}{\ell _{3}}\frac{a_{1}}{M}S_{1}I_{1}+\frac{a_{2}}{\ell _{4}}E_{1}+\frac{a_{3}}{\ell _{5}}I_{1}\right\}\\

&\quad -   \left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right)\\

  & =\,  \Bigg[\left\{\frac{1}{\ell _{1}}\left(r(M-N)-\frac{a_{1}}{M}S_{1}I_{1}+\beta P_{1}\right)+\frac{\alpha }{\ell _{2}}S_{1}+\frac{1}{\ell _{3}}\frac{a_{1}}{M}S_{1}I_{1}+\frac{a_{2}}{\ell _{4}}E_{1}+\frac{a_{3}}{\ell _{5}}I_{1}\right\} \\

  & \quad\frac{1}{\left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right)}-1\Bigg]\left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right). \end{align}</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (16)

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Now we divide by S to get

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>^{ABC}D_{*}^{\upsilon }\mathcal{L}  =  \Bigg[\left\{\frac{1}{\ell _{1}}\left(\frac{r(M-N)}{S_{1}}-\frac{a_{1}}{M}I_{1}+\beta \frac{P_{1}}{S_{1}}\right)+\frac{\alpha }{\ell _{2}}+\frac{1}{\ell _{3}}\frac{a_{1}}{M}I_{1}+\frac{a_{2}}{\ell _{4}}\frac{E_{1}}{S_{1}}+\frac{a_{3}}{\ell _{5}}\frac{I_{1}}{S_{1}}\right\}</math>

|-

| style="text-align: center;" |<math>\frac{1}{\left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right)}-1\Bigg]\left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right). </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (17)

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Since <math>S</math> is major than <math>P,\, E,\, I</math> and <math>R</math> categories, so

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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| 

{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align}^{ABC}D_{*}^{\upsilon }\mathcal{L}  & \leq  \left[\frac{ra_{1}a_{2}(\beta{+\gamma})}{\left(a_{2}+b_{1}+\gamma \right)\left(\gamma{+}a_{3}+b_{2}+\eta \right)(\gamma{+}r)\left(\alpha{+\beta}+\gamma \right)}-1\right]\left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right)\\

&\leq  \left[\mathrm{\mathcal{R}}_{0}-1\right]\left(S_{1}+P_{1}+E_{1}+I_{1}+R_{1}\right)\leq{0.} \end{align}</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (18)

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Since <math display="inline">S_{1}+P_{1}+E_{1}+I_{1}+R_{1}>0,\forall t</math>. According to the proposed model, plant disease model would therefore be eliminated if and only if <math display="inline">\mathrm{\mathcal{R}}_{0}<1</math>. In general, because all parameters are positive in the plant disease model, Lyapunov function <math display="inline">\left(^{ABC}D_{*}^{\upsilon }\mathcal{L}\right)</math> therefore decreases if <math display="inline">\mathrm{\mathcal{R}}_{0}<1</math> and increases if <math display="inline">\mathrm{\mathcal{R}}_{0}>1</math>, eventually <math display="inline">\mathcal{L}=0</math> if <math display="inline">S_{1}=P_{1}=E_{1}=I_{1}=R_{1}=0</math>. <math display="inline">\mathcal{L}</math> is therefore the function of Lyapunov within the practicable biological interval and the greater compact invariant set in <math display="inline">\left\{S_{1},P_{1},E_{1},I_{1},R_{1}\in \Theta :{}^{ABC}D_{*}^{\upsilon }\mathcal{L}\leq{0}\right\}</math> is the point <math display="inline">\mathcal{E_{1}}_{0}</math>. Every solution of the plant disease model proposed in this study with an initial term in <math display="inline">\Theta </math> tends to <math display="inline">\mathcal{E_{1}}_{0}</math> when <math display="inline">t\rightarrow \infty </math> if and only if <math display="inline">\mathrm{\mathcal{R}}_{0}\leq{1}</math> through the well-known Lasalles invariance principle [38]. In conclusion, the plant disease model's disease-free equilibrium <math display="inline">\mathcal{E_{1}}_{0}</math> presented here is globally asymptotically stable.

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'''Theorem 3.2.2'''. The endemic equilibrium point <math display="inline">\mathcal{E_{1}}^{*}</math> of the plant disease system is globally asymptotically stable if <math>\mathrm{\mathcal{R}}_{0}\leq{1}</math>.

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'''Proof'''. We use the Lyapunov function [38] to prove this

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align}\mathcal{L}\left(S_{1}^{*},P_{1}^{*},E_{1}^{*},I_{1}^{*},R_{1}^{*}\right)  &= \,   \left(S_{1}-S_{1}^{*}-S_{1}^{*}\log \frac{S_{1}^{*}}{S_{1}}\right)+\left(P_{1}-P_{1}^{*}-P_{1}^{*}\log \frac{P_{1}^{*}}{P_{1}}\right)+\left(E_{1}-E_{1}^{*}-E_{1}^{*}\log \frac{E_{1}^{*}}{E_{1}}\right)\\

&\quad +  \left(I_{1}-I_{1}^{*}-I_{1}^{*}\log \frac{I_{1}^{*}}{I_{1}}\right)+\left(R_{1}-R_{1}^{*}-R_{1}^{*}\log \frac{R_{1}^{*}}{R_{1}}\right). \end{align}</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (19)

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Consequently, applying the derivative to both sides gives

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }\mathcal{L} & =  \left(\frac{S_{1}-S_{1}^{*}}{S_{1}}\right){}^{ABC}D_{*}^{\upsilon }S_{1}+\left(\frac{P_{1}-P_{1}^{*}}{P_{1}}\right){}^{ABC}D_{*}^{\upsilon }P_{1}+\left(\frac{E_{1}-E_{1}^{*}}{E_{1}}\right){}^{ABC}D_{*}^{\upsilon }E_{1}\\

  &\quad +  \left(\frac{I_{1}-I_{1}^{*}}{I_{1}}\right){}^{ABC}D_{*}^{\upsilon }I_{1}+\left(\frac{R_{1}-R_{1}^{*}}{R_{1}}\right){}^{ABC}D_{*}^{\upsilon }R_{1}, \end{align}</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (20)

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replacing <math display="inline">^{ABC}D_{*}^{\upsilon }S_{1}</math>, <math display="inline">^{ABC}D_{*}^{\upsilon }P_{1}</math>, <math display="inline">^{ABC}D_{*}^{\upsilon }E_{1}</math>, <math display="inline">^{ABC}D_{*}^{\upsilon }I_{1}</math> and <math display="inline">^{ABC}D_{*}^{\upsilon }R_{1}</math> by their values, we obtain

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }\mathcal{L} & =  \left(\frac{S_{1}-S_{1}^{*}}{S_{1}}\right)\left(r(M-N)-\frac{a_{1}}{M}S_{1}I_{1}+\beta P_{1}-\ell _{1}S_{1}\right)\\

&\quad +\left(\frac{P_{1}-P_{1}^{*}}{P_{1}}\right)\left(\alpha S_{1}-\ell _{2}P_{1}\right)  +\left(\frac{E_{1}-E_{1}^{*}}{E_{1}}\right)\left(\frac{a_{1}}{M}S_{1}I_{1}-\ell _{3}E_{1}\right)\\

&\quad +\left(\frac{I_{1}-I_{1}^{*}}{I_{1}}\right)\left(a_{2}E_{1}-\ell _{4}I_{1}\right)+\left(\frac{R_{1}-R_{1}^{*}}{R_{1}}\right)\left(a_{3}I_{1}-\ell _{5}R_{1}\right).\end{align} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (21)

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Then we have

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }\mathcal{L} & =  \left(\frac{S_{1}-S_{1}^{*}}{S_{1}}\right)\left[r(M-N)-\frac{a_{1}}{M}\left(S_{1}-S_{1}^{*}\right)\left(I_{1}-I_{1}^{*}\right)+\beta \left(P_{1}-P_{1}^{*}\right)-\ell _{1}\left(S_{1}-S_{1}^{*}\right)\right]\\

&\quad +\left(\frac{P_{1}-P_{1}^{*}}{P_{1}}\right)\left[\alpha \left(S_{1}-S_{1}^{*}\right)-\ell _{2}\left(P_{1}-P_{1}^{*}\right)\right]\\

&\quad   +\left(\frac{E_{1}-E_{1}^{*}}{E_{1}}\right)\left[\frac{a_{1}}{M}\left(S_{1}-S_{1}^{*}\right)\left(I_{1}-I_{1}^{*}\right)-\ell _{3}\left(E_{1}-E_{1}^{*}\right)\right]\\

&\quad +\left(\frac{I_{1}-I_{1}^{*}}{I_{1}}\right)\left[a_{2}\left(E_{1}-E_{1}^{*}\right)-\ell _{4}\left(I_{1}-I_{1}^{*}\right)\right]\\

&\quad +\left(\frac{R_{1}-R_{1}^{*}}{R_{1}}\right)\left[a_{3}\left(I_{1}-I_{1}^{*}\right)-\ell _{5}\left(R_{1}-R_{1}^{*}\right)\right].\qquad  \end{align} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (22)

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They can be separated in two part as follows

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align}  ^{ABC}D_{*}^{\upsilon }\mathcal{L} &  =  r(M-N)\left(\frac{S_{1}-S_{1}^{*}}{S_{1}}\right)-\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)\left(\frac{a_{1}}{M}\left(I_{1}-I_{1}^{*}\right)+\ell _{1}\right)\\

&\quad +\beta \left(P_{1}-P_{1}^{*}\right)\left(\frac{S_{1}-S_{1}^{*}}{S_{1}}\right)+\alpha \left(S_{1}-S_{1}^{*}\right)\left(\frac{P_{1}-P_{1}^{*}}{P_{1}}\right)\\

&\quad -\ell _{2}\left(\frac{\left(P_{1}-P_{1}^{*}\right)^{2}}{P_{1}}\right)+\frac{a_{1}}{M}\left(S_{1}-S_{1}^{*}\right)\left(I_{1}-I_{1}^{*}\right)\left(\frac{E_{1}-E_{1}^{*}}{E_{1}}\right)\\

&\quad -\ell_{3}\left(\frac{\left(E_{1}-E_{1}^{*}\right)^{2}}{E_{1}}\right)+a_{2}\left(E_{1}-E_{1}^{*}\right) \left(\frac{I_{1}-I_{1}^{*}}{I_{1}}\right)\\

&\quad -\ell _{4}\left(\frac{\left(I_{1}-I_{1}^{*}\right)^{2}}{I_{1}}\right) + a_{3}\left(I_{1}-I_{1}^{*}\right)\left(\frac{R_{1}-R_{1}^{*}}{R_{1}}\right)\\

&\quad -\ell _{5}\left(\frac{\left(R_{1}-R_{1}^{*}\right)^{2}}{R_{1}}\right),\end{align} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (23)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} ^{ABC}D_{*}^{\upsilon }\mathcal{L}  & =  r(M-N)-r(M-N)\frac{S_{1}^{*}}{S_{1}}-\frac{a_{1}}{M}I_{1}\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)+\frac{a_{1}}{M}I_{1}^{*}\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)\\

&\quad -\ell _{1}\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)+\beta P_{1}-\beta P_{1}^{*}-\beta P_{1}\frac{S_{1}^{*}}{S_{1}}+\beta P_{1}^{*}\frac{S_{1}^{*}}{S_{1}} +\alpha S_{1}-\alpha S_{1}^{*}-\alpha S_{1}\frac{P_{1}^{*}}{P_{1}}\\

&\quad +\alpha S_{1}^{*}\frac{P_{1}^{*}}{P_{1}}-\ell _{2}\left(\frac{\left(P_{1}-P_{1}^{*}\right)^{2}}{P_{1}}\right)+\frac{a_{1}}{M}S_{1}I_{1}-\frac{a_{1}}{M}S_{1}I_{1}^{*}-\frac{a_{1}}{M}S_{1}I_{1}\frac{E_{1}^{*}}{E_{1}}\\

&\quad +\frac{a_{1}}{M}S_{1}I_{1}^{*}\frac{E_{1}^{*}}{E_{1}}-\frac{a_{1}}{M}S_{1}^{*}I_{1}+\frac{a_{1}}{M}S_{1}^{*}I_{1}^{*}+\frac{a_{1}}{M}S_{1}^{*}I_{1}\frac{E_{1}^{*}}{E_{1}}-\frac{a_{1}}{M}S_{1}^{*}I_{1}^{*}\frac{E_{1}^{*}}{E_{1}}\\

&\quad -\ell _{3}\left(\frac{\left(E_{1}-E_{1}^{*}\right)^{2}}{E_{1}}\right)+a_{2}E_{1}-a_{2}E_{1}^{*}-a_{2}E_{1}\frac{I_{1}^{*}}{I_{1}}+a_{2}E_{1}^{*}\frac{I_{1}^{*}}{I_{1}}-\ell _{4}\left(\frac{\left(I_{1}-I_{1}^{*}\right)^{2}}{I_{1}}\right)\\

&\quad +a_{3}I_{1}-a_{3}I_{1}^{*}-a_{3}I_{1}\frac{R_{1}^{*}}{R_{1}}+a_{3}I_{1}^{*}\frac{R_{1}^{*}}{R_{1}}-\ell _{5}\left(\frac{\left(R_{1}-R_{1}^{*}\right)^{2}}{R_{1}}\right).\end{align} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (24)

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This can be simplified as

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>^{ABC}D_{*}^{\upsilon }\mathcal{L}  =  \mathcal{L}_{1}-\mathcal{L}_{2}, </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (25)

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where

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align}\mathcal{L}_{1}  & =  r(M-N)+\frac{a_{1}}{M}I_{1}^{*}\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)+\beta P_{1}+\beta P_{1}^{*}\frac{S_{1}^{*}}{S_{1}}+\alpha S_{1}\\ 

&\quad +\alpha S_{1}^{*}\frac{P_{1}^{*}}{P_{1}}+\frac{a_{1}}{M}S_{1}I_{1}+\frac{a_{1}}{M}S_{1}I_{1}^{*}\frac{E_{1}^{*}}{E_{1}} +\frac{a_{1}}{M}S_{1}^{*}I_{1}^{*}\\

&\quad +\frac{a_{1}}{M}S_{1}^{*}I_{1}\frac{E_{1}^{*}}{E_{1}}+a_{2}E_{1}+a_{2}E_{1}^{*}\frac{I_{1}^{*}}{I_{1}}+a_{3}I_{1}+a_{3}I_{1}^{*}\frac{R_{1}^{*}}{R_{1}}, \end{align}</math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (26)

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and

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\begin{align} \mathcal{L}_{2} &  =  r(M-N)\frac{S_{1}^{*}}{S_{1}}+\frac{a_{1}}{M}I_{1}\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)+\ell _{1}\left(\frac{\left(S_{1}-S_{1}^{*}\right)^{2}}{S_{1}}\right)+\beta P_{1}^{*}+\beta P_{1}\frac{S_{1}^{*}}{S_{1}}\\

&\quad +\alpha S_{1}^{*}+\alpha S_{1}\frac{P_{1}^{*}}{P_{1}}+\ell _{2}\left(\frac{\left(P_{1}-P_{1}^{*}\right)^{2}}{P_{1}}\right)+\frac{a_{1}}{M}S_{1}I_{1}^{*}+\frac{a_{1}}{M}S_{1}I_{1}\frac{E_{1}^{*}}{E_{1}}\\

&\quad +\frac{a_{1}}{M}S_{1}^{*}I_{1}+\frac{a_{1}}{M}S_{1}^{*}I_{1}^{*}\frac{E_{1}^{*}}{E_{1}}+\ell _{3}\left(\frac{\left(E_{1}-E_{1}^{*}\right)^{2}}{E_{1}}\right)+a_{2}E_{1}^{*}+a_{2}E_{1}\frac{I_{1}^{*}}{I_{1}}\\

&\quad +\ell _{4}\left(\frac{\left(I_{1}-I_{1}^{*}\right)^{2}}{I_{1}}\right)+a_{3}I_{1}^{*}+a_{3}I_{1}\frac{R_{1}^{*}}{R_{1}}+\ell _{5}\left(\frac{\left(R_{1}-R_{1}^{*}\right)^{2}}{R_{1}}\right),\end{align} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (27)

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this implies <math display="inline">^{ABC}D_{*}^{\upsilon }\mathcal{L}>0</math> if <math display="inline">\mathcal{L}_{1}>\mathcal{L}_{2}</math>, <math display="inline">^{ABC}D_{*}^{\upsilon }\mathcal{L}<0</math> if <math display="inline">\mathcal{L}_{2}>\mathcal{L}_{1}</math> and <math display="inline">^{ABC}D_{*}^{\upsilon }\mathcal{L}=0</math> if <math display="inline">\mathcal{L}_{1}=\mathcal{L}_{2}</math>, this implies <math display="inline">S_{1}=S_{1}^{*}</math>, <math display="inline">P_{1}=P_{1}^{*}</math>, <math display="inline">E_{1}=E_{1}^{*}</math>, <math display="inline">I_{1}-I_{1}^{*}</math> and <math display="inline">R_{1}=R_{1}^{*}</math>.

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We can now conclude that the largest compact invariant set for the plant diseases model in <math display="inline">\left\{S_{1}^{*},P_{1}^{*},E_{1}^{*},I_{1}^{*},R_{1}^{*}\in \Theta :{}^{ABC}D_{*}^{\upsilon }\mathcal{L}=0\right\}</math> is the point <math display="inline">\mathcal{E}^{*}</math> the endemic equilibrium of the plant diseases model.

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===3.3 <span id='lb-3.3'></span>Sensibility analysis of reproduction number (\mathcalR₀ ) without control===

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Since the parameters of the epizootic system are either predestined or equipped, this raises some doubt as to their values used to draw a conclusion about the underlying epidemic. Therefore, it's critical to identify the specific effects of each element on the dynamics of the pestilence and group the variables that have the most impact to limit or spread the outbreak. Within this section, a sensitivity analysis is conducted for the disease parameters identified with the suggested SPEIR model via the sensitivity indicator. Quantifying the most sensitive aspects of the basal reproductive number <math display="inline">\mathrm{\mathcal{R}}_{0}</math> can be done with the help of the Sensitivity Indicator strategy. The following equations give the standardized sensitivity indicator <math display="inline">\Psi _{*}^{\mathrm{\mathcal{R}}_{0}}</math> of <math display="inline">\mathrm{\mathcal{R}}_{0}</math> for all the parameters <math display="inline">\left(r,a_{1},a_{2},a_{3},b_{1},b_{2},\alpha ,\beta ,\gamma ,\eta \right)</math> used within the SPEIR model in Table [[#table-1|1]] where <math display="inline">\Psi _{*}^{\mathrm{\mathcal{R}}_{0}}=\frac{\partial \mathrm{\mathcal{R}}_{0}}{\partial }\times \frac{*}{\left|\mathrm{\mathcal{R}}_{0}\right|}</math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{r}^{\mathrm{\mathcal{R}}_{0}}=\frac{a_{1}a_{2}\gamma (\beta{+\gamma})}{A_{1}A_{2}(\gamma{+}r)^{2}(\alpha{+\beta}+\gamma )}=0.266667>0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{a_{1}}^{\mathrm{\mathcal{R}}_{0}}=\frac{a_{2}r(\beta{+\gamma})}{A_{1}A_{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=1>0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{a_{2}}^{\mathrm{\mathcal{R}}_{0}}=\frac{a_{1}r(\beta{+\gamma})\left(b_{1}+\gamma \right)}{A_{1}^{2}A_{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=0.0229885>0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{a_{3}}^{\mathrm{\mathcal{R}}_{0}}=-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}A_{2}^{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=-0.300752<0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{b_{1}}^{\mathrm{\mathcal{R}}_{0}}=-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}^{2}A_{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{b_{2}}^{\mathrm{\mathcal{R}}_{0}}=-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}A_{2}^{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=0, </math>

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<span id="eq-28"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\Psi _{\alpha }^{\mathrm{\mathcal{R}}_{0}}=-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}A_{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )^{2}}=-0.0909091<0, </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (28)

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{\beta }^{\mathrm{\mathcal{R}}_{0}}=\frac{\alpha a_{1}a_{2}r}{A_{1}A_{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )^{2}}=0.0839161>0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \Psi _{\eta }^{\mathrm{\mathcal{R}}_{0}}=-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}A_{2}^{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=-0.669173<0, </math>

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math> \Psi _{\gamma }^{\mathrm{\mathcal{R}}_{0}}=\frac{a_{1}a_{2}r\left(-\alpha \beta{-}(\beta{+\gamma})^{2}+\alpha r\right)}{A_{1}A_{2}(\gamma{+}r)^{2}(\alpha{+\beta}+\gamma )^{2}}-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}^{2}A_{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}-\frac{a_{1}a_{2}r(\beta{+\gamma})}{A_{1}A_{2}^{2}(\gamma{+}r)(\alpha{+\beta}+\gamma )}=-0.312737<0, </math>

|}

|}

​

As shown in the previous calculations, some component of the sensitivity indicator are positive, like <math>r</math>, <math>a_{1}</math>, <math>a_{2}</math> and <math>\beta </math>, while others, like <math>a_{3}</math>, <math>\alpha </math>, <math>\gamma </math> and <math>\eta </math> are negative. Furthermore, the most important feature of these indicators is the functionality of the SPEIR model parameters. This means that getting a small amendment in one of the parameters will amendment the epidemic dynamics. The value <math>\Psi _{a_{3}}^{\mathrm{\mathcal{R}}_{0}}=-0.300752</math> displays that decreasing (increasing) <math>a_{3}</math> for example by 70% increases (decreases) the basic reproductive number <math>\mathrm{\mathcal{R}}_{0}</math> by about 70% A small change in a parameter can head to comparatively enormous quantitative changes, requiring these sensitive parameters to be understood. It can be shown from previous calculations that parameters <math>a_{1}</math> (disease progression diversion rate for Latent Compartmentenglish) and <math>\eta </math> (rate of roguingenglish), respectively, are the maximum and minimum sensitivity epidemical parameters <math>\mathrm{\mathcal{R}}_{0}</math>.

​

==4 Achieve existence and uniqueness==

​

In this section, we will prove that model 4 has a unique solution, that the kernel satisfies Lipschitz's condition and that the functions in this model is bounded.

​

Now, we will analyze the fractional model ([[#eq-1|1]]). Usage of an integral fractional operator on Eq. ([[#eq-1|1]]), we are gaining

​

<span id="eq-29"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>S_{1}\left(t\right)-S_{1}\left(0\right) =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{1}\left(t,S_{1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{1}\left(\theta ,S_{1}\right)d\theta ,</math>

|-

| style="text-align: center;" | <math> P_{1}\left(t\right)-P_{1}\left(0\right) =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{2}\left(t,P_{1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{2}\left(\theta ,P_{1}\right)d\theta ,</math>

|-

| style="text-align: center;" | <math> E_{1}\left(t\right)-E_{1}\left(0\right) =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{3}\left(t,E_{1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{3}\left(\theta ,E_{1}\right)d\theta ,</math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (29)

|-

| style="text-align: center;" | <math> I_{1}\left(t\right)-I_{1}\left(0\right) =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{4}\left(t,I_{1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{4}\left(\theta ,I_{1}\right)d\theta ,</math>

|-

| style="text-align: center;" | <math> R_{1}\left(t\right)-R_{1}\left(0\right) =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{5}\left(t,R_{1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{5}\left(\theta ,R_{1}\right)d\theta , </math>

|}

|}

​

where

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\begin{cases}\varrho _{1}\left(t,S_{1}\right)=r(M-N)-\gamma S_{1}\left(t\right)-\frac{a_{1}}{M}S_{1}\left(t\right)I_{1}\left(t\right)-\alpha S_{1}\left(t\right)+\beta P_{1}\left(t\right),\\ \varrho _{2}\left(t,P_{1}\right)=\alpha S_{1}\left(t\right)+\beta P_{1}\left(t\right)-\gamma P_{1}\left(t\right),\\ \varrho _{3}\left(t,E_{1}\right)=\frac{a_{1}}{M}S_{1}\left(t\right)I_{1}\left(t\right)-\left(\gamma{+}a_{2}+b_{1}\right)E_{1}\left(t\right),\\ \varrho _{4}\left(t,I_{1}\right)=a_{2}E_{1}\left(t\right)-\left(\gamma{+}a_{3}+b_{2}+\eta \right)I_{1}\left(t\right),\\ \varrho _{5}\left(t,R_{1}\right)=a_{3}I_{1}\left(t\right)-\left(\gamma{+}b_{3}\right)R_{1}\left(t\right). \end{cases} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (30)

|}

​

When <math>S_{1}\left(t\right)</math> has an upper limit, the Lipschitz condition for the <math>\varrho _{1}\left(t,S_{1}\right)</math> will be fulfilled. So, if <math>S_{1}\left(t\right)</math> has an upper limit, we find that

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\begin{array}{l} \left\Vert \varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\tilde{S_{1}}\right)\right\Vert  & = & \left\Vert -\gamma \left(S_{1}-\tilde{S_{1}}\right)-\frac{a_{1}I_{1}}{M}\left(S_{1}-\tilde{S_{1}}\right)-\alpha \left(S_{1}-\tilde{S_{1}}\right)\right\Vert \\  & \leq & \gamma \left\Vert S_{1}-\tilde{S_{1}}\right\Vert +\frac{a_{1}\phi }{M}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert +\alpha \left\Vert S_{1}-\tilde{S_{1}}\right\Vert \\  & \leq & \left\{\gamma +\frac{a_{1}\phi }{M}+\alpha \right\}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert , \end{array}</math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (31)

|}

​

that is <math>\left\Vert S_{1}\right\Vert \le c_{1}</math> and <math>\left\Vert \tilde{S_{1}}\right\Vert \le c_{2}</math>, where <math>S_{1}</math> and <math>\tilde{S_{1}}</math> are bounded functions and <math>\phi =max\left\Vert I_{1}\right\Vert </math>. We have that,

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math> \left\Vert \varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\tilde{S_{1}}\right)\right\Vert   \leq  X_{S_{1}}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert , </math>

|}

|}

​

similarly, we obtain the other kernels as following

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left\Vert \varrho _{2}\left(t,P_{1}\right)-\varrho _{2}\left(t,\tilde{P_{1}}\right)\right\Vert   \leq  X_{P_{1}}\left\Vert P_{1}-\tilde{P_{1}}\right\Vert ,</math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (32)

|-

| style="text-align: center;" | <math> \left\Vert \varrho _{3}\left(t,E_{1}\right)-\varrho _{3}\left(t,\tilde{E_{1}}\right)\right\Vert   \leq  X_{E_{1}}\left\Vert E_{1}-\tilde{E_{1}}\right\Vert ,</math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (33)

|-

| style="text-align: center;" | <math> \left\Vert \varrho _{4}\left(t,I_{1}\right)-\varrho _{4}\left(t,\tilde{I_{1}}\right)\right\Vert   \leq  X_{I_{1}}\left\Vert I_{1}-\tilde{I_{1}}\right\Vert ,</math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (34)

|-

| style="text-align: center;" | <math> \left\Vert \varrho _{5}\left(t,R_{1}\right)-\varrho _{5}\left(t,\tilde{R_{1}}\right)\right\Vert   \leq  X_{R_{1}}\left\Vert R_{1}-\tilde{R_{1}}\right\Vert , </math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (35)

|}

|}

​

where

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>X_{S_{1}}=\gamma +\frac{a_{1}\phi }{M}+\alpha ,\quad X_{P_{1}}=\beta{+\gamma},\quad X_{E_{1}}=\gamma{+}a_{2}+b_{1},\quad X_{I_{1}}=\gamma{+}a_{3}+b_{2}+\eta ,\quad X_{R_{1}}=\gamma{+}b_{3}, </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (36)

|}

​

and <math>\left\Vert P_{1}\right\Vert \le c_{3}</math>, <math>\left\Vert \tilde{P_{1}}\right\Vert \le c_{4}</math>, <math>\left\Vert E_{1}\right\Vert \le c_{5}</math>, <math>\left\Vert \tilde{E_{1}}\right\Vert \le c_{6}</math>, <math>\left\Vert I_{1}\right\Vert \le c_{7}</math>, <math>\left\Vert \tilde{I_{1}}\right\Vert \le c_{8}</math>, <math>\left\Vert R_{1}\right\Vert \le c_{9}</math>, <math>\left\Vert \tilde{R_{1}}\right\Vert \le c_{10}</math>. Hence, for the kernels <math>\varrho _{1}\left(t,S_{1}\right)</math>, <math>\varrho _{2}\left(t,P_{1}\right)</math>, <math>\varrho _{3}\left(t,E_{1}\right)</math>, <math>\varrho _{4}\left(t,I_{1}\right)</math> and <math>\varrho _{5}\left(t,R_{1}\right)</math>, the Lipschitz condition is justified.

​

'''Theorem4.1''' Presume that <math display="inline">S_{1}\left(t\right)</math> is obliged, then the operator <math display="inline">\xi \left\{S_{1}\left(t\right)\right\}</math> is supplied by

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\xi \left\{S_{1}\left(t\right)\right\}  =  S_{1}\left(0\right)+\frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{1}\left(t,S_{1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{1}\left(\theta ,S_{1}\right)d\theta , </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (37)

|}

​

satisfies the Lipschitz condition.

​

'''Proof''' Assume that <math>S_{1}\left(t\right)</math> and <math>\tilde{S_{1}}\left(t\right)</math> are bounded functions with <math display="inline">S_{1}\left(0\right)=\tilde{S_{1}}\left(0\right)</math>, then we have

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left\Vert \xi \left\{S_{1}\left(t\right)\right\}-\xi \left\{\tilde{S_{1}}\left(t\right)\right\}\right\Vert   =  \left\Vert \frac{1-\upsilon }{\chi \left(\upsilon \right)}\left(\varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\tilde{S_{1}}\right)\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left(\varrho _{1}\left(\theta ,S_{1}\right)-\varrho _{1}\left(\theta ,\tilde{S_{1}}\right)\right)d\theta \right\Vert </math>

|-

| style="text-align: center;" | <math>   \leq  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\left\Vert \varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\tilde{S_{1}}\right)\right\Vert +\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left\Vert \varrho _{1}\left(\theta ,S_{1}\right)-\varrho _{1}\left(\theta ,\tilde{S_{1}}\right)\right\Vert d\theta </math>

|-

| style="text-align: center;" | <math>   \leq  \left(\frac{1-\upsilon }{\chi \left(\upsilon \right)}X_{S_{1}}-\frac{X_{S_{1}}t^{\upsilon }}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\right)\left\Vert S_{1}-\tilde{S_{1}}\right\Vert . </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (38)

|}

​

This completes the proof. The same process can be applied to <math display="inline">P_{1}\left(t\right)</math>, <math display="inline">E_{1}\left(t\right)</math>, <math display="inline">I_{1}\left(t\right)</math> and <math display="inline">R_{1}\left(t\right)</math>.

​

'''Theorem4.2''' If <math display="inline">S_{1}\left(t\right)</math> is a bounded, then the operator

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\xi _{1}\left\{S_{1}\left(t\right)\right\}=r(M-N)-\gamma S_{1}\left(t\right)-\frac{a_{1}}{M}S_{1}\left(t\right)I_{1}\left(t\right)-\alpha S_{1}\left(t\right)+\beta P_{1}\left(t\right), </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (39)

|}

​

satisfies

​

'''(i)'''  

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left|\left\langle \xi _{1}\left(S_{1}\right)-\xi _{1}\left(\tilde{S_{1}}\right),S_{1}-\tilde{S_{1}}\right\rangle \right| \leq  X_{S_{1}}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert ^{2}, </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (40)

|}

​

where <math display="inline">\langle{.},.\rangle </math> is the inner product space bounded in <math display="inline">\mathcal{L}^{2}</math>.

​

'''(ii)''' 

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math> \left|\left\langle \xi _{1}\left(S_{1}\right)-\xi _{1}\left(\tilde{S_{1}}\right),D\right\rangle \right| \leq  X_{S_{1}}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert D\right\Vert ,0<\left\Vert D\right\Vert <\infty{.} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (41)

|}

​

'''Proof<math>\;</math>(i)''' Suppose that <math>S_{1}\left(t\right)</math> is bounded function, then

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\begin{array}{l} \left|\left\langle \xi _{1}\left(S_{1}\right)-\xi _{1}\left(\tilde{S_{1}}\right),S_{1}-\tilde{S_{1}}\right\rangle \right|& = & \left|\left\langle -\gamma \left(S_{1}-\tilde{S_{1}}\right)-\frac{a_{1}I_{1}}{M}\left(S_{1}-\tilde{S_{1}}\right)-\alpha \left(S_{1}-\tilde{S_{1}}\right),S_{1}-\tilde{S_{1}}\right\rangle \right|\\  & \leq & \left|\left\langle \gamma \left(S_{1}-\tilde{S_{1}}\right),S_{1}-\tilde{S_{1}}\right\rangle \right|+\left|\left\langle \frac{a_{1}I_{1}}{M}\left(S_{1}-\tilde{S_{1}}\right),S_{1}-\tilde{S_{1}}\right\rangle \right|+\left|\left\langle \alpha \left(S_{1}-\tilde{S_{1}}\right),S_{1}-\tilde{S_{1}}\right\rangle \right|\\  & \leq & \gamma \left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert S_{1}-\tilde{S_{1}}\right\Vert +\frac{a_{1}\phi }{M}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert S_{1}-\tilde{S_{1}}\right\Vert +\alpha \left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert S_{1}-\tilde{S_{1}}\right\Vert \\  & \leq & \left\{\gamma +\frac{a_{1}\phi }{M}+\alpha \right\}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert ^{2}\\  & \leq & X_{S_{1}}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert ^{2}. \end{array}</math>

|}

|}

​

'''Proof<math>\;</math>(ii)''' Suppose that <math display="inline">0<\left\Vert D\right\Vert <\infty </math>, since <math>S_{1}\left(t\right)</math> is bounded function, so

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\begin{array}{l} \left|\left\langle \xi _{1}\left(S_{1}\right)-\xi _{1}\left(\tilde{S_{1}}\right),D\right\rangle \right|& = & \left|\left\langle -\gamma \left(S_{1}-\tilde{S_{1}}\right)-\frac{a_{1}I_{1}}{M}\left(S_{1}-\tilde{S_{1}}\right)-\alpha \left(S_{1}-\tilde{S_{1}}\right),D\right\rangle \right|\\  & \leq & \left|\left\langle \gamma \left(S_{1}-\tilde{S_{1}}\right),D\right\rangle \right|+\left|\left\langle \frac{a_{1}I_{1}}{M}\left(S_{1}-\tilde{S_{1}}\right),D\right\rangle \right|+\left|\left\langle \alpha \left(S_{1}-\tilde{S_{1}}\right),D\right\rangle \right|\\  & \leq & \gamma \left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert D\right\Vert +\frac{a_{1}\phi }{M}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert D\right\Vert +\alpha \left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert D\right\Vert \\  & \leq & \left\{\gamma +\frac{a_{1}\phi }{M}+\alpha \right\}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert D\right\Vert \\  & \leq & X_{S_{1}}\left\Vert S_{1}-\tilde{S_{1}}\right\Vert \left\Vert D\right\Vert . \end{array}</math>

|}

|}

​

This completes the proof. The same process can be applied to <math display="inline">P_{1}\left(t\right)</math>, <math display="inline">E_{1}\left(t\right)</math>, <math display="inline">I_{1}\left(t\right)</math> and <math display="inline">R_{1}\left(t\right)</math>.

​

An inquiry into the existence and uniqueness of Eq. ([[#eq-1|1]]) will be discussed in the following. From Eq. ([[#eq-29|29]]) we can write

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left(S_{1}\right)_{n}  =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\varrho _{1}\left(t,\left(S_{1}\right)_{n-1}\right)+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n-1}\right)d\theta ,\;n=1,2,3,\cdots{.} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (44)

|}

​

The difference of the successive term can be written as

​

<span id="eq-45"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\Upsilon _{n+1}\left(t\right)=\left(S_{1}\right)_{n+1}-\left(S_{1}\right)_{n}=\frac{1-\upsilon }{\chi \left(\upsilon \right)}\left[\varrho _{1}\left(t,\left(S_{1}\right)_{n}\right)-\varrho _{1}\left(t,\left(S_{1}\right)_{n-1}\right)\right]+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left[\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n}\right)-\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n-1}\right)\right]d\theta{.} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (45)

|}

​

According to [34], it would be easy to write

​

<span id="eq-46"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left(S_{1}\right)_{n+1}  =  \stackrel={\scriptscriptstyle I_{1}=1}\Upsilon _{{\scriptscriptstyle I_{1}-1}}. </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (46)

|}

​

Then, from Eq. ([[#eq-45|45]]), we get

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left\Vert \Upsilon _{n+1}\left(t\right)\right\Vert =\left\Vert \left(S_{1}\right)_{n+1}-\left(S_{1}\right)_{n}\right\Vert =\left\Vert \frac{1-\upsilon }{\chi \left(\upsilon \right)}\left[\varrho _{1}\left(t,\left(S_{1}\right)_{n}\right)-\varrho _{1}\left(t,\left(S_{1}\right)_{n-1}\right)\right]+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left[\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n}\right)-\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n-1}\right)\right]d\theta \right\Vert . </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (47)

|}

​

By triangular inequality, the above Eq. turn into

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left\Vert \Upsilon _{n+1}\left(t\right)\right\Vert \leq \frac{1-\upsilon }{\chi \left(\upsilon \right)}\left\Vert \varrho _{1}\left(t,\left(S_{1}\right)_{n}\right)-\varrho _{1}\left(t,\left(S_{1}\right)_{n-1}\right)\right\Vert +\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left\Vert \varrho _{1}\left(\theta ,\left(S_{1}\right)_{n}\right)-\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n-1}\right)\right\Vert d\theta{.} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (48)

|}

​

Whereas, the kernel fulfilled the Lipschitz condition. we have

​

<span id="eq-49"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\left\Vert \Upsilon _{n+1}\left(t\right)\right\Vert \leq \frac{1-\upsilon }{\chi \left(\upsilon \right)}X_{S_{1}}\left\Vert \left(S_{1}\right)_{n}-\left(S_{1}\right)_{n-1}\right\Vert +\frac{\upsilon X_{S_{1}}}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left\Vert \left(S_{1}\right)_{n}-\left(S_{1}\right)_{n-1}\right\Vert d\theta{.} </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (49)

|}

​

'''Theorem4.3''' If <math display="inline">t_{0}</math> fits the following condition

​

{| class="formulaSCP" style="width: 100%; text-align: left;" 

|-

| 

{| style="text-align: left; margin:auto;width: 100%;" 

|-

| style="text-align: center;" | <math>\frac{1-\upsilon }{\chi \left(\upsilon \right)}X_{S_{1}}+\frac{X_{S_{1}}t_{0}^{\upsilon }}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\leq{1}, </math>

|}

| style="width: 5px;text-align: right;white-space: nowrap;" | (50)

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then, model ([[#eq-1|1]]) has a unique solution.

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'''Proof''' Suppose that <math display="inline">S_{1}(t)</math> is bounded. As the Lipschitz condition is fulfilled by the kernel, therefore, utilizing the recursive process of Eq. ([[#eq-49|49]]), we acquire

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math> \left\Vert \Upsilon _{n+1}\left(t\right)\right\Vert \leq \left\{\frac{1-\upsilon }{\chi \left(\upsilon \right)}X_{S_{1}}+\frac{\upsilon X_{S_{1}}t_{0}^{\upsilon }}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\right\}^{n+1}\left\Vert S_{1}\left(0\right)\right\Vert . </math>

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Therefore, the <math display="inline">S_{1}(t)</math> function offered by Eq. ([[#eq-46|46]]) exists and is smooth as well. Currently, we wish to illustrate that the above-mentioned functions are actually a solution to englishmodel ([[#eq-1|1]]). Presume that

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>S_{1}(t)-S_{1}(0)  =  \left(S_{1}\right)_{n}-\left(\overline{S_{1}}\right)_{n} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (51)

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So that,

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\left\Vert \left(\overline{S_{1}}\right)_{n}\right\Vert   =  \left\Vert \frac{1-\upsilon }{\chi \left(\upsilon \right)}\left[\varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\left(S_{1}\right)_{n}\right)\right]+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left[\varrho _{1}\left(\theta ,S_{1}\right)-\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n}\right)\right]d\theta \right\Vert .</math>

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| style="text-align: center;" | <math>   \leq  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\Bigl\Vert \varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\left(S_{1}\right)_{n}\right)\Bigr\Vert +\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\Bigl\Vert \varrho _{1}\left(\theta ,S_{1}\right)-\varrho _{1}\left(\theta ,\left(S_{1}\right)_{n}\right)\Bigr\Vert d\theta </math>

| style="width: 5px;text-align: right;white-space: nowrap;" | (52)

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| style="text-align: center;" | <math>   \leq  \frac{1-\upsilon }{\chi \left(\upsilon \right)}X_{S_{1}}\left\Vert S_{1}-\left(S_{1}\right)_{n}\right\Vert +\frac{\upsilon X_{S_{1}}t^{\upsilon }}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\left\Vert S_{1}-\left(S_{1}\right)_{n}\right\Vert . </math>

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Using the recursive approach once more, we get

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{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\left\Vert \left(\overline{S_{1}}\right)_{n}\right\Vert   \leq  \left\{\frac{1-\upsilon }{\chi \left(\upsilon \right)}+\frac{\upsilon t^{\upsilon }}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\right\}^{n+2}X_{S_{1}}^{n+2}. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (53)

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If <math display="inline">t=t_{0}</math>, we get

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<span id="eq-54"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>\left\Vert \left(\overline{S_{1}}\right)_{n}\right\Vert   \leq  \left\{\frac{1-\upsilon }{\chi \left(\upsilon \right)}+\frac{\upsilon t_{0}^{\upsilon }}{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\right\}^{n+2}X_{S_{1}}^{n+2}. </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (54)

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Using the limit in Eq. ([[#eq-54|54]]) as <math display="inline">n</math> gets closer to <math display="inline">\infty </math> , we arrive at <math display="inline">\left\Vert \left(\overline{S_{1}}\right)_{n}\right\Vert \rightarrow{0}</math>. Thus, the existence is demonstrated. It is still necessary to demonstrate the uniqueness of the model english([[#eq-1|1]]). Assume that <math display="inline">\tilde{S_{1}}\left(t\right)</math> is another solution of model ([[#eq-1|1]]), then

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<span id="eq-55"></span>

{| class="formulaSCP" style="width: 100%; text-align: left;" 

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{| style="text-align: left; margin:auto;width: 100%;" 

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| style="text-align: center;" | <math>S_{1}\left(t\right)-\tilde{S_{1}}\left(t\right) =  \frac{1-\upsilon }{\chi \left(\upsilon \right)}\left[\varrho _{1}\left(t,S_{1}\right)-\varrho _{1}\left(t,\tilde{S_{1}}\right)\right]+\frac{\upsilon }{\chi \left(\upsilon \right)\lambda \left(\upsilon \right)}\stackrel=0\left(t-\theta \right)^{\upsilon{-1}}\left[\varrho _{1}\left(\theta ,S_{1}\right)-\varrho _{1}\left(\theta ,\tilde{S_{1}}\right)\right]d\theta{.} </math>

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| style="width: 5px;text-align: right;white-space: nowrap;" | (55)

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