Alqahtani Hagag 2023a - Revision history

Rimni at 08:41, 21 July 2023

2023-07-21T08:41:27Z

Rimni at 08:40, 21 July 2023

2023-07-21T08:40:38Z

Rimni: /* 4. Achieve existence and uniqueness */

2023-07-21T08:39:50Z

‎4. Achieve existence and uniqueness

Rimni at 08:35, 21 July 2023

2023-07-21T08:35:35Z

Rimni: /* 3.1.2 Endemic equilibrium (EE) point */

2023-07-21T08:33:38Z

‎3.1.2 Endemic equilibrium (EE) point

Rimni at 08:29, 21 July 2023

2023-07-21T08:29:40Z

Ahmedshehata at 16:04, 20 July 2023

2023-07-20T16:04:30Z

Rimni: /* References */

2023-07-20T12:32:36Z

‎References

Rimni at 12:18, 20 July 2023

2023-07-20T12:18:09Z

Show changes

Rimni at 11:01, 20 July 2023

2023-07-20T11:01:32Z

@@ Line 714: / Line 714: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align}\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]\\
+| style="text-align: center;" | <math>\begin{align}\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]\\
 & \quad +\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{.}\end{align} </math>
 |}

@@ Line 714: / Line 714: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align}\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]\\
+| style="text-align: center;" | <math>\begin{align}\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]\\
 & \quad +\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{.}\end{align} </math>
 |}

@@ Line 668: / Line 668: @@
 |}
-'''Proof<math>\;</math>(i)''' Suppose that <math>S_{1}\left(t\right)</math> is bounded function, then
+'''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;"
@@ Line 680: / Line 680: @@
 |}
-'''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
+'''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;"
@@ Line 714: / Line 714: @@
 {| 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]</math><math>+\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="text-align: center;" | <math>\begin{align}\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]\\
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (45)

@@ Line 475: / Line 475: @@
 ===3.3 <span id='lb-3.3'></span>Sensibility analysis of reproduction number (<math>\mathrm{\mathcal{R}}_{0}</math> ) without control===
-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>
+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-1|Table 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>
 <span id="eq-28"></span>
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 158: / Line 158: @@
 ====3.1.2 Endemic equilibrium (EE) point====
-The endemic equilibrium solution of the system (1) is
+The endemic equilibrium solution of the system (Eq.[[#eq-1|(1)]]) is
 {| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 1,299: / Line 1,299: @@
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+[33]  Driessche P.,  Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29-48, 2002.
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@@ Line 136: / Line 136: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| 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>
+| style="text-align: center;" | <math>\begin{align} \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)\\
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5)