<p>This study analyzes childhood infectious disease dynamics using a general fractional-order model, focusing on measles transmission in the Netherlands in 2025. The model integrates memory effects using a Caputo-type fractional derivative with a general kernel to better represent disease progression and host immunity. We first show that the model is well-posed by establishing that its solutions exist and are unique, positive, and bounded. This makes sure that the model is both mathematically correct and epidemiologically valid. We then identify disease-free and endemic equilibria and analyze their local stability, showing that the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> determines their existence and stability. We also develop a predictor-corrector numerical scheme for general fractional differential equations, achieving a convergence order of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O}(h^{\min \{1+\alpha ,2\}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for accurate simulation of model dynamics, where <i>h</i> is the time step size, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> represents the fractional order. The model is calibrated using real epidemic data through a nonlinear least-squares parameter estimation framework, and uncertainty is quantified via bootstrap-based confidence intervals. As a result, the optimal fractional order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha = 0.95$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </math></EquationSource> </InlineEquation> yields the best fit compared to integer-order models, underscoring the importance of memory in disease transmission. Moreover, public health interventions can be targeted through sensitivity analysis, which depicts that recruitment and transmission rates are the most important factors affecting the basic reproduction number and the endemic infection level. Additionally, preventive and therapeutic interventions are combined in a general fractional optimal control framework to reduce disease. To demonstrate this concept, we develop the optimality system and numerically compute optimal control strategies using Pontryagin’s Maximum Principle. Simulation results indicate that integrated interventions reduce infection prevalence and accelerate recovery better than single interventions, suggesting that a combination of strategies is more effective than isolated approaches in managing disease outbreaks. In conclusion, this study demonstrates that general fractional-order modeling can predict epidemic dynamics, inform public health policy, and help design cost-effective control strategies for childhood infectious diseases.</p>

Stability, sensitivity, and optimal control of dimensionally consistent fractional dynamics for childhood epidemics

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发表日期：

2026年6月1日

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Amin Jajarmi

摘要

This study analyzes childhood infectious disease dynamics using a general fractional-order model, focusing on measles transmission in the Netherlands in 2025. The model integrates memory effects using a Caputo-type fractional derivative with a general kernel to better represent disease progression and host immunity. We first show that the model is well-posed by establishing that its solutions exist and are unique, positive, and bounded. This makes sure that the model is both mathematically correct and epidemiologically valid. We then identify disease-free and endemic equilibria and analyze their local stability, showing that the basic reproduction number $R_0$ $R_{0}$ determines their existence and stability. We also develop a predictor-corrector numerical scheme for general fractional differential equations, achieving a convergence order of $\mathcal {O}(h^{\min \{1+\alpha ,2\}})$ $O (h^{min {1 + α, 2}})$ for accurate simulation of model dynamics, where h is the time step size, and $\alpha $ $α$ represents the fractional order. The model is calibrated using real epidemic data through a nonlinear least-squares parameter estimation framework, and uncertainty is quantified via bootstrap-based confidence intervals. As a result, the optimal fractional order $\alpha = 0.95$ $α = 0.95$ yields the best fit compared to integer-order models, underscoring the importance of memory in disease transmission. Moreover, public health interventions can be targeted through sensitivity analysis, which depicts that recruitment and transmission rates are the most important factors affecting the basic reproduction number and the endemic infection level. Additionally, preventive and therapeutic interventions are combined in a general fractional optimal control framework to reduce disease. To demonstrate this concept, we develop the optimality system and numerically compute optimal control strategies using Pontryagin’s Maximum Principle. Simulation results indicate that integrated interventions reduce infection prevalence and accelerate recovery better than single interventions, suggesting that a combination of strategies is more effective than isolated approaches in managing disease outbreaks. In conclusion, this study demonstrates that general fractional-order modeling can predict epidemic dynamics, inform public health policy, and help design cost-effective control strategies for childhood infectious diseases.