<p>This study introduces a Caputo fractional-order epidemic model to analyze the transmission dynamics of waterborne diseases, such as cholera, with vaccination as a control measure. The population is divided into five compartments alongside the pathogen concentration in water. Using the mass-action incidence rate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\beta S(t)B(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, a fractional-order system is formulated to capture memory-dependent transmission dynamics. We first establish the existence, uniqueness, positivity, and boundedness of solutions within a positively invariant region. The basic reproduction number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {R}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is derived, and two equilibrium points–disease-free and endemic–are identified. Through fractional Routh–Hurwitz criteria and a direct Lyapunov method adapted to the Caputo derivative, we prove that the DFE is globally asymptotically stable when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {R}_0 < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, while the EE becomes stable when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {R}_0 > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Numerical simulations, implemented via generalized hat functions, illustrate the influence of the fractional-order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> on convergence rates. Results indicate that fractional models lead to slower, more prolonged outbreaks compared to classical integer-order models, emphasizing the role of memory effects in epidemic persistence. The findings highlight the combined importance of vaccination and water sanitation in disease control and demonstrate the utility of fractional calculus in modeling complex biological systems with hereditary dynamics.</p>

A caputo fractional-order model for An aquatic diseases with control and its stability analysis

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发表日期：

2026年5月7日

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S. Kosari,
M. H. Heydari,
M. Bayram

摘要

This study introduces a Caputo fractional-order epidemic model to analyze the transmission dynamics of waterborne diseases, such as cholera, with vaccination as a control measure. The population is divided into five compartments alongside the pathogen concentration in water. Using the mass-action incidence rate $\beta S(t)B(t)$ $β S (t) B (t)$ , a fractional-order system is formulated to capture memory-dependent transmission dynamics. We first establish the existence, uniqueness, positivity, and boundedness of solutions within a positively invariant region. The basic reproduction number $\mathcal {R}_0$ $R_{0}$ is derived, and two equilibrium points–disease-free and endemic–are identified. Through fractional Routh–Hurwitz criteria and a direct Lyapunov method adapted to the Caputo derivative, we prove that the DFE is globally asymptotically stable when $\mathcal {R}_0 < 1$ $R_{0} < 1$ , while the EE becomes stable when $\mathcal {R}_0 > 1$ $R_{0} > 1$ . Numerical simulations, implemented via generalized hat functions, illustrate the influence of the fractional-order $\alpha $ $α$ on convergence rates. Results indicate that fractional models lead to slower, more prolonged outbreaks compared to classical integer-order models, emphasizing the role of memory effects in epidemic persistence. The findings highlight the combined importance of vaccination and water sanitation in disease control and demonstrate the utility of fractional calculus in modeling complex biological systems with hereditary dynamics.