<p>Chickenpox (varicella) is a highly transmissible infection primarily caused by a herpes virus called Varicella Zoster and is a commonly reported childhood disease. In this paper, a deterministic nonlinear model was adapted to investigate the dynamics of chickenpox that incorporates intervention in the form of isolation and treatment. In the mathematical analysis part, the positivity and boundedness of the solution have been ascertained, existence of disease equilibria has also been ascertained, which shows that the model consists of two equilibriums, the DFE (disease-free equilibrium point) and EE (endemic equilibrium point). The effective reproduction number has been computed using the next-generation operator method and the basic reproduction number has been computed by setting all the associated effective reproduction number to zero. The DFE was found to be both (globally and locally asymptotically stable) if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2219_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_c < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>c</mi> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and the global stability of disease-free equilibrium point has been proven using the comparison method. Global asymptotic stability of EEP(endemic equilibrium point) has been ascertained using the nonlinear Lyapunov function of the Go–Volterra type, which reveals that, the EE is globally asymptotically stable if the basic reproduction number is greater than unity and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2219_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="193" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega =\psi _1=\psi _2=\delta _1=\delta _2=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>=</mo> <msub> <mi>ψ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Sensitivity analysis shows that natural recovery and isolation are very sensitive in reducing the control reproduction number and effective contact rate is very sensitive in increasing the effective reproduction number. In the numerical simulation part, the section confirmed the analytical results and further, shows that the natural recovery booster is a vital parameter that helps greatly in the healing process of chickenpox patients. The proposed model also reveals that the isolation and treatment rate of infected individuals are very important parameters in controlling chickenpox in society. We recommend that immunity boosters and isolation be prioritized in controlling chickenpox in the population.</p>

Mathematical analysis of chickenpox population dynamics unveiling the impact of booster in enhancing recovery of infected individuals

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发表日期：

2024年12月31日

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Kabiru Garba Ibrahim,
James Andrawus,
Abbas Abubakar,
Abdullahi Yusuf,
Stephen Ishaya Maiwa,
Kefas Bitrus,
Muhammad Sani,
Ibrahim Abdullahi,
Sunday Babuba,
Joseph Jonathan

摘要

Chickenpox (varicella) is a highly transmissible infection primarily caused by a herpes virus called Varicella Zoster and is a commonly reported childhood disease. In this paper, a deterministic nonlinear model was adapted to investigate the dynamics of chickenpox that incorporates intervention in the form of isolation and treatment. In the mathematical analysis part, the positivity and boundedness of the solution have been ascertained, existence of disease equilibria has also been ascertained, which shows that the model consists of two equilibriums, the DFE (disease-free equilibrium point) and EE (endemic equilibrium point). The effective reproduction number has been computed using the next-generation operator method and the basic reproduction number has been computed by setting all the associated effective reproduction number to zero. The DFE was found to be both (globally and locally asymptotically stable) if $R_c < 1$ $R_{c} < 1$ and the global stability of disease-free equilibrium point has been proven using the comparison method. Global asymptotic stability of EEP(endemic equilibrium point) has been ascertained using the nonlinear Lyapunov function of the Go–Volterra type, which reveals that, the EE is globally asymptotically stable if the basic reproduction number is greater than unity and $\omega =\psi _1=\psi _2=\delta _1=\delta _2=0$ $ω = ψ_{1} = ψ_{2} = δ_{1} = δ_{2} = 0$ . Sensitivity analysis shows that natural recovery and isolation are very sensitive in reducing the control reproduction number and effective contact rate is very sensitive in increasing the effective reproduction number. In the numerical simulation part, the section confirmed the analytical results and further, shows that the natural recovery booster is a vital parameter that helps greatly in the healing process of chickenpox patients. The proposed model also reveals that the isolation and treatment rate of infected individuals are very important parameters in controlling chickenpox in society. We recommend that immunity boosters and isolation be prioritized in controlling chickenpox in the population.