<p>In our current article, we have considered a fractional-order SIR-type epidemic model with media coverage and vaccination control to discuss the disease dynamics. We have thoroughly checked the existence, uniqueness, non-negativity, and boundedness criteria of the solution of the suggested framework. In addition, we have computed the basic reproduction number (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1800_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>) and equilibrium points (disease-free and endemic). Based on the values of the basic reproduction number, we have discussed that disease-free equilibrium is asymptotically stable for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1800_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and endemic equilibrium is asymptotically stable for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1800_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Also, we have established a transcritical bifurcation at the threshold <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40435_2025_1800_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\Re _0=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ℜ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we have applied fractional-order optimal control to form an objective functional to minimize the effect of the disease and used Pontryagin’s maximum principle to solve it. We have also executed a sensitivity analysis to check the influence of the system parameters on the numerical value of the basic reproduction number. Additionally, some computer simulation work has been done to verify the theoretical studies.</p>

Impact of vaccination and media on a Caputo derivative-based fractional-order epidemic model with PRCC analysis

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发表日期：

2025年8月16日

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Snehasis Barman,
Soovoojeet Jana,
Suvankar Majee,
Sathi Patra,
Tapan Kumar Kar

摘要

In our current article, we have considered a fractional-order SIR-type epidemic model with media coverage and vaccination control to discuss the disease dynamics. We have thoroughly checked the existence, uniqueness, non-negativity, and boundedness criteria of the solution of the suggested framework. In addition, we have computed the basic reproduction number ( $\Re _0$ $ℜ_{0}$ ) and equilibrium points (disease-free and endemic). Based on the values of the basic reproduction number, we have discussed that disease-free equilibrium is asymptotically stable for $\Re _0<1$ $ℜ_{0} < 1$ and endemic equilibrium is asymptotically stable for $\Re _0>1$ $ℜ_{0} > 1$ . Also, we have established a transcritical bifurcation at the threshold $\Re _0=1$ $ℜ_{0} = 1$ . Furthermore, we have applied fractional-order optimal control to form an objective functional to minimize the effect of the disease and used Pontryagin’s maximum principle to solve it. We have also executed a sensitivity analysis to check the influence of the system parameters on the numerical value of the basic reproduction number. Additionally, some computer simulation work has been done to verify the theoretical studies.