The persistent co-circulation of dengue and malaria in tropical regions poses a significant epidemiological challenge, particularly because classical integer-order models fail to capture the memory-driven reinfection, relapse, and recrudescence mechanisms that sustain long-term disease transmission. To overcome these limitations, this study develops a high-dimensional nonlinear co-infection model formulated using the Atangana–Baleanu–Caputo (ABC) fractional derivative, which incorporates nonsingular and nonlocal kernels to realistically represent hereditary effects in host–vector dynamics. The model integrates primary and secondary dengue infections, recurrent malaria pathways, and interactions across two mosquito species within a unified fractional-order framework. Analytical results establish positivity, boundedness, and existence–uniqueness of solutions, and the basic reproduction number R0 is rigorously derived via the next-generation matrix method. Numerical simulations reveal that decreasing the fractional order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\eta\)</EquationSource> </InlineEquation> substantially prolongs transient dynamics, increases infection peaks, and strengthens disease persistence relative to the classical <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\eta = 1\)</EquationSource> </InlineEquation> system; in particular, when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(R_{0} > 1\)</EquationSource> </InlineEquation>, both pathogens exhibit sustained endemicity amplified under fractional dynamics. These findings demonstrate that memory effects encoded by the ABC operator play a critical role in shaping reinfection outcomes, cross-immunity decay, and recurrent malaria episodes. The proposed framework provides a mathematically rigorous and epidemiologically insightful foundation for understanding nonlinear co-infection dynamics and underscores the importance of fractional calculus in improving predictive modeling and informing long-term vector-borne disease control strategies. Graphical Abstract The graphical abstract provides an integrated visual summary of the study titled Fractional ABC Dynamics and Nonlinear Transmission Analysis of Dengue–Malaria Co-infection with Reinfection, highlighting the key components that structure the research. The Data panel contextualizes the epidemiological setting by illustrating Thailand as the study region and depicting the two mosquito vectors, Aedes aegypti and Anopheles, which drive dengue and malaria transmission, respectively. The Model section presents the full compartmental framework that incorporates primary and secondary dengue infections, recurrent malaria mechanisms, and interactions across both mosquito species, all governed by the Atangana–Baleanu–Caputo (ABC) fractional operator. This high-dimensional model (17 state variables) captures memory-dependent reinfection, relapse, and recrudescence—biological processes not adequately represented by classical differential equations. The Analyses panel highlights the core methodological components: proofs of existence and uniqueness of solutions under the ABC operator, sensitivity analysis of the reproduction number R0 with respect to key transmission parameters, and implementation of fractional numerical schemes tailored to nonsingular, nonlocal kernels. The Results section visually summarizes the model’s dynamic behavior by displaying time-series trajectories and 3-D fractional-order simulations demonstrating how decreasing the fractional order intensifies infection persistence and amplifies reinfection pathways. Collectively, this graphical abstract emphasizes the importance of fractional calculus in accurately representing memory-driven co-infection dynamics and underscores the study’s contribution to improving long-term predictive modeling for vector-borne diseases.

Fractional ABC Dynamics and Nonlinear Transmission Analysis of Dengue–Malaria Co-infection with Reinfection

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发表日期：

2026年6月18日

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Jiraporn Lamwong,
Puntani Pongsumpun

摘要

The persistent co-circulation of dengue and malaria in tropical regions poses a significant epidemiological challenge, particularly because classical integer-order models fail to capture the memory-driven reinfection, relapse, and recrudescence mechanisms that sustain long-term disease transmission. To overcome these limitations, this study develops a high-dimensional nonlinear co-infection model formulated using the Atangana–Baleanu–Caputo (ABC) fractional derivative, which incorporates nonsingular and nonlocal kernels to realistically represent hereditary effects in host–vector dynamics. The model integrates primary and secondary dengue infections, recurrent malaria pathways, and interactions across two mosquito species within a unified fractional-order framework. Analytical results establish positivity, boundedness, and existence–uniqueness of solutions, and the basic reproduction number R₀ is rigorously derived via the next-generation matrix method. Numerical simulations reveal that decreasing the fractional order \(\eta\) substantially prolongs transient dynamics, increases infection peaks, and strengthens disease persistence relative to the classical \(\eta = 1\) system; in particular, when \(R_{0} > 1\) , both pathogens exhibit sustained endemicity amplified under fractional dynamics. These findings demonstrate that memory effects encoded by the ABC operator play a critical role in shaping reinfection outcomes, cross-immunity decay, and recurrent malaria episodes. The proposed framework provides a mathematically rigorous and epidemiologically insightful foundation for understanding nonlinear co-infection dynamics and underscores the importance of fractional calculus in improving predictive modeling and informing long-term vector-borne disease control strategies.

Graphical Abstract

The graphical abstract provides an integrated visual summary of the study titled Fractional ABC Dynamics and Nonlinear Transmission Analysis of Dengue–Malaria Co-infection with Reinfection, highlighting the key components that structure the research. The Data panel contextualizes the epidemiological setting by illustrating Thailand as the study region and depicting the two mosquito vectors, Aedes aegypti and Anopheles, which drive dengue and malaria transmission, respectively. The Model section presents the full compartmental framework that incorporates primary and secondary dengue infections, recurrent malaria mechanisms, and interactions across both mosquito species, all governed by the Atangana–Baleanu–Caputo (ABC) fractional operator. This high-dimensional model (17 state variables) captures memory-dependent reinfection, relapse, and recrudescence—biological processes not adequately represented by classical differential equations. The Analyses panel highlights the core methodological components: proofs of existence and uniqueness of solutions under the ABC operator, sensitivity analysis of the reproduction number R₀ with respect to key transmission parameters, and implementation of fractional numerical schemes tailored to nonsingular, nonlocal kernels. The Results section visually summarizes the model’s dynamic behavior by displaying time-series trajectories and 3-D fractional-order simulations demonstrating how decreasing the fractional order intensifies infection persistence and amplifies reinfection pathways. Collectively, this graphical abstract emphasizes the importance of fractional calculus in accurately representing memory-driven co-infection dynamics and underscores the study’s contribution to improving long-term predictive modeling for vector-borne diseases.