Fractional dynamics and sensitivity analysis of measles epidemic model through vaccination

Figures & data

Table 1. List of top five nations with worldwide measles outbreaks (7)(7) $$\mathcal{L}{\{}^{\mathit{\text{ABC}}}{D}_0^{\vartheta }\mathbf{U}(t)\}(s)=[\frac{s^{\vartheta }\mathcal{L}\{\mathbf{U}(t)\}(s)-s^{\vartheta -1}\mathbf{U}(0)}{s^{\vartheta }(1-\vartheta )+\vartheta }]\times \mathcal{M}(\vartheta ).$$(7) .

Country	Rank	Number of cases
India	1	12,271
Yemen	2	7,538
Somalia	3	6,261
Zimbabwe	4	4,623
Pakistan	5	2,586

Figure 1. Schematic illustration of the measles transmission in the population.

Figure 2. PRCC statistics regarding the significance of factors associated with ${\mathcal{R}}_0^M.$

Figure 3. The behaviours of ${\mathcal{R}}_0^M$ under different biological parameters.

Figure 4. The behaviours of ${\mathcal{R}}_0^M$ under different biological parameters.

Figure 5. The behaviours of ${\mathcal{R}}_0^M$ under different biological parameters.

Figure 6. Population behaviour of the measles fractional model at the different fractional parameter values.

Figure 7. Simulations for the suggested model (2)(2) $$\{\begin{array}{ccc}{}_0^{\mathit{\text{ABC}}}{D}_t^{\vartheta }\mathbb{S}(t)\hfill & =\hfill & \Lambda -{\alpha }_1\mathbb{S}\mathbb{I}+{\sigma }_1\mathbb{V}-({\mu }_1+{\sigma }_2)\mathbb{S},\hfill \\ {}_0^{\mathit{\text{ABC}}}{D}_t^{\vartheta }\mathbb{V}(t)\hfill & =\hfill & {\sigma }_2\mathbb{S}-({\mu }_1+{\sigma }_1)\mathbb{V},\hfill \\ {}_0^{\mathit{\text{ABC}}}{D}_t^{\vartheta }\mathbb{E}(t)\hfill & =\hfill & {\alpha }_1\mathbb{S}\mathbb{I}-({\alpha }_2+\kappa +{\mu }_1)\mathbb{E},\hfill \\ {}_0^{\mathit{\text{ABC}}}{D}_t^{\vartheta }\mathbb{I}(t)\hfill & =\hfill & {\alpha }_2\mathbb{E}-({\mu }_2+{\alpha }_3+{\mu }_1)\mathbb{I},\hfill \\ {}_0^{\mathit{\text{ABC}}}{D}_t^{\vartheta }\mathbb{R}(t)\hfill & =\hfill & \kappa \mathbb{E}+{\alpha }_3\mathbb{I}-{\mu }_1\mathbb{R},\hfill \end{array}$$(2) using Atangan-Toufik scheme for the different values of fractional order $\vartheta .$

Table 2. Interpretation of involved symbols and their numeric values (47)(47) $$\mathbb{S}(t)-\mathbb{S}(0)=\frac{1-\vartheta }{\mathcal{M}(\vartheta )}{\mathbb{G}}_1\left(t,\mathbb{S}(t)\right)+\frac{\vartheta }{\mathcal{M}(\vartheta )\Gamma (\vartheta )}{\int }_0^t{\mathbb{G}}_1(\rho ,\mathbb{S}(\rho )){(t-\rho )}^{\vartheta -1}d\rho .$$(47) .

Symbol	Description	Value
$\Lambda$	Recruitment or natality rate of the population	7.27892$\times {10}^1$
${\alpha }_1$	Rate of effective contact of susceptible people	0.001
${\alpha }_2$	Rate at which exposed humans become infected	2.83562$\times {10}^{-5}$
${\alpha }_3$	Rate of natural recovery from infection	0.096
$\kappa$	Rate at which exposed humans who have gone through screening and treatment	1.45362
${\mu }_1$	Natural death rate of the population	$\frac{1}{(365\times 57.6)}$
${\mu }_2$	Death rate of infected humans due to measles	0.123
${\sigma }_1$	Rate of vaccination of susceptible humans	0.169
${\sigma }_2$	Rate of vaccine wane of the susceptible people	1.81656$\times {10}^{-1}$

Table 3. Sensitivity indices of ${\mathcal{R}}_0^M$ against the parameters.

Sensitivity index	Value	Sensitivity index	Value
${\Gamma }_{\Lambda }$	$+1.0000$	${\Gamma }_{{\mu }_2}$	$-0.5615$
${\Gamma }_{{\alpha }_1}$	$+1.0000$	${\Gamma }_{{\sigma }_1}$	$-0.4819$
${\Gamma }_{{\alpha }_2}$	$+1.0000$	${\Gamma }_{{\sigma }_2}$	$+0.4818$
${\Gamma }_{{\alpha }_3}$	$-0.4383$	${\Gamma }_{\kappa }$	$-0.9999$
${\Gamma }_{{\mu }_1}$	$-1.0001$	–	–