<p>The COVID-19 pandemic has introduced unparalleled challenges worldwide, highlighting the need for sophisticated models to understand its dynamics. In this study, we present a comprehensive analysis of a stochastic model incorporating dual doses of vaccine and the logarithmic mean-reverting Ornstein-Uhlenbeck process to analyze the dynamics of SARS-CoV-2 (the virus responsible for COVID-19) infection in India. The study begins with constructing a deterministic model and then it is extended to a stochastic framework. Firstly, we perform a thorough analysis of the deterministic model to confirm the existence and uniqueness of a globally positive solution, ensuring that it remains bounded. Additionally, the basic reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10917_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$ R_0^d $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mn>0</mn> <mi>d</mi> </msubsup> </math></EquationSource> </InlineEquation> is derived and the local asymptotic stability of the disease-free equilibrium is evaluated under the condition <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10917_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0^d < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>0</mn> <mi>d</mi> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Next, we analyze the formulated stochastic model and identify two critical threshold parameters, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10917_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0^e$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mn>0</mn> <mi>e</mi> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10917_Article_IEq4.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0^s$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mn>0</mn> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation>. Specifically, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10917_Article_IEq5.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0^e < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>0</mn> <mi>e</mi> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> indicates exponential extinction of the infection, whereas <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10917_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_0^s > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> suggests the potential persistence of the infection due to the existence of at least one stationary distribution. Finally, the model is fitted to the actual COVID-19 case data from India during the peak of the second epidemic wave. The insights derived from these projections, particularly regarding the timing of extinction and the magnitude of the infection peak under various scenarios are crucial for public health policymakers for future outbreaks similar to COVID-19.</p>

Stochastic modeling of COVID-19 spread in India: an approach incorporating logarithmic mean-reverting Ornstein-Uhlenbeck process with two-dose vaccine impact analysis

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发表日期：

2025年1月28日

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T. A. Midhun,
Sayooj Aby Jose,
K. Murugesan,
Anuwat Jirawattanapanit

摘要

The COVID-19 pandemic has introduced unparalleled challenges worldwide, highlighting the need for sophisticated models to understand its dynamics. In this study, we present a comprehensive analysis of a stochastic model incorporating dual doses of vaccine and the logarithmic mean-reverting Ornstein-Uhlenbeck process to analyze the dynamics of SARS-CoV-2 (the virus responsible for COVID-19) infection in India. The study begins with constructing a deterministic model and then it is extended to a stochastic framework. Firstly, we perform a thorough analysis of the deterministic model to confirm the existence and uniqueness of a globally positive solution, ensuring that it remains bounded. Additionally, the basic reproduction number $ R_0^d $ $R_{0}^{d}$ is derived and the local asymptotic stability of the disease-free equilibrium is evaluated under the condition $R_0^d < 1$ $R_{0}^{d} < 1$ . Next, we analyze the formulated stochastic model and identify two critical threshold parameters, $R_0^e$ $R_{0}^{e}$ and $R_0^s$ $R_{0}^{s}$ . Specifically, $R_0^e < 1$ $R_{0}^{e} < 1$ indicates exponential extinction of the infection, whereas $R_0^s > 1$ $R_{0}^{s} > 1$ suggests the potential persistence of the infection due to the existence of at least one stationary distribution. Finally, the model is fitted to the actual COVID-19 case data from India during the peak of the second epidemic wave. The insights derived from these projections, particularly regarding the timing of extinction and the magnitude of the infection peak under various scenarios are crucial for public health policymakers for future outbreaks similar to COVID-19.