A Hybrid Fractional-Order Epidemic Model Incorporating Memory-Dependent Transmission Dynamics

1. INTRODUCTION

Mathematical models have long served as essential tools for understanding infectious disease transmission and for supporting evidence-based public health decision-making. Classical compartmental frameworks, including the SIR and SEIR models, have been widely employed to describe epidemic dynamics and to identify key thresholds governing disease persistence or elimination [1]. These models are typically formulated using integer-order differential equations, implicitly assuming that the evolution of each epidemiological compartment depends exclusively on its current state [2]. While such assumptions facilitate mathematical analysis and interpretation, they overlook an important feature of real-world epidemics: the influence of historical exposure and cumulative transmission processes.

In many infectious diseases, particularly those characterized by long incubation periods, delayed immune responses, or prolonged infectiousness, the risk of infection cannot be adequately represented as an instantaneous interaction [3]. Instead, it is shaped by past contacts and exposure history accumulated over time. This temporal dependence, commonly referred to as a memory effect, represents a fundamental limitation of integer-order models, which are inherently memoryless and thus unable to capture long-range temporal correlations in disease transmission.

The limitations of memory-free modeling approaches became particularly evident during the COVID-19 pandemic. Forecasts based solely on present-state dynamics often underestimate epidemic persistence, fail to reproduce delayed peaks, and inadequately represented the lagged impact of public health interventions [4]. In many regions, infection levels remained elevated despite stringent control measures, highlighting the need for epidemiological models that explicitly incorporate historical dependence. Fractional-order models have emerged as a natural extension of classical approaches to address this shortcoming. By employing derivatives of non-integer order, fractional calculus intrinsically embeds memory effects into dynamical systems, as the rate of change depends on the entire past trajectory of the system, weighted by a suitable kernel function [5]. This property makes fractional-order formulations particularly well suited for epidemic modeling, where transmission pressure and recovery dynamics often reflect cumulative processes.

The inclusion of memory through fractional derivatives enriches the modeling framework and enables a more faithful representation of epidemic phenomena. For instance, exponential-type kernels describe short-memory processes dominated by recent exposures, whereas power-law kernels account for long-term memory effects in which distant past events continue to influence present dynamics [6]. Such flexibility has proven valuable for capturing prolonged epidemic tails, recurrent waves, and persistent infection patterns observed in real data. However, epidemic transmission is rarely governed by smooth dynamics alone. In practice, disease spread is frequently disrupted by abrupt changes resulting from interventions such as lockdowns, vaccination campaigns, travel restrictions, or behavioral adaptations within populations [7]. These measures induce discontinuous changes in transmission parameters, leading to hybrid dynamics in which multiple regimes of disease spread coexist.

Motivated by these observations, hybrid epidemic models incorporating regime switching have been proposed to represent intervention-driven changes. Meanwhile, fractional-order epidemic models have demonstrated improved performance in fitting real-world data for diseases such as COVID-19 and influenza [8–10]. Empirical studies have shown that fractional formulations more accurately reproduce delayed epidemic peaks, long-tailed decay, and intervention lag effects [11]. Applications to influenza dynamics further suggest that fractional derivatives effectively capture seasonal recurrence and memory embedded in contact networks [12, 13]. Despite these advances, most existing studies have considered either fractional-order models with fixed transmission structures or hybrid integer-order systems without memory. A unified framework combining fractional memory effects with hybrid switching mechanisms remains largely unexplored. Moreover, many studies emphasize numerical simulations while providing limited rigorous analysis. In particular, the explicit influence of memory kernel properties on epidemic thresholds has not been systematically examined, and stability results are often derived heuristically rather than through formal fractional stability theory [14].

This study addresses these gaps by proposing a generalized hybrid fractional-order SEIR model with memory-dependent incidence and regime-switching transmission rates. The model explicitly incorporates admissible memory kernels and demonstrates how their total mass directly influences the basic reproduction number, offering a quantitative characterization of memory effects on epidemic thresholds. Using tools from fractional calculus, including fixed-point arguments and Matignon’s stability criterion, the analysis rigorously establishes positivity, boundedness, and well-posedness of solutions. An explicit expression for the basic reproduction number is derived, clarifying its dependence on epidemiological parameters and kernel properties while revealing independence from the fractional order itself.

Beyond threshold analysis, the stability of both disease-free and endemic equilibria is investigated within the fractional-order setting. The results show that the fractional order predominantly affects transient behavior and convergence rates, whereas the memory kernel mass determines whether disease persistence or eradication occurs. The hybrid switching mechanism further enables the model to capture intervention-induced parameter shifts, allowing stability regions to be characterized symbolically and supported by qualitative numerical illustrations. Sensitivity analyses and symbolic–numerical examples demonstrate how memory effects and interventions interact to shape epidemic outcomes.

The contribution of this study is intentionally positioned as an analytical extension rather than a replacement of existing epidemic modeling frameworks. The novelty lies in deriving an explicit reproduction threshold for a memory-dependent fractional SEIR system, showing that the basic reproduction number depends on the total memory-kernel mass , while the fractional order governs transient growth, decay, and persistence patterns. This separation between threshold dynamics and transient dynamics provides a useful interpretive distinction that is often implicit but not clearly formalized in fractional epidemic models. The hybrid switching component further enables intervention regimes to be represented through regime-specific transmission rates and dwell-time assumptions, thereby linking memory-dependent transmission to intervention timing.

This work bridges rigorous mathematical analysis with epidemiological relevance by providing a unified hybrid fractional-order framework for memory-driven epidemics. The proposed model extends classical epidemic theory and offers insights into the timing and effectiveness of interventions in systems where historical dependence plays a dominant role. The remainder of the paper is organized as follows. Section 2 introduces the necessary mathematical preliminaries. Section 3 presents the formulation of the hybrid fractional-order SEIR model and establishes well-posedness. Section 4 examines positivity, boundedness, and equilibrium structure. Section 5 derives the basic reproduction number, while Section 6 analyzes stability using spectral methods and fractional stability theory. Section 7 provides sensitivity analysis and illustrative examples. Section 8 discusses theoretical implications, policy relevance, and limitations, and Section 9 concludes the paper with a summary and directions for future research.

2. MATHEMATICAL PRELIMINARIES

2.1. Fractional Calculus: Caputo and Riemann–Liouville Derivatives

Fractional calculus generalizes the concept of differentiation and integration from integer orders to arbitrary real or even complex orders. Among several possible definitions, the Caputo and Riemann–Liouville derivatives are the most widely used in modelling physical and biological processes [15]. For a sufficiently smooth function , the Caputo derivative of order  is defined as (Eq. 2.1):

 (2.1)

where  is the Euler gamma function [16]. This definition has the advantage of requiring initial conditions in the same form as classical integer-order derivatives, making it particularly suitable for epidemiological modelling. In contrast, the Riemann–Liouville derivative is expressed as:

The Riemann–Liouville derivative of order , where , is defined as (Eq. 2.2):

 (2.2)

Unlike the Caputo derivative, the Riemann–Liouville formulation often leads to initial conditions expressed in terms of fractional integrals, which are less direct for epidemiological interpretation. The Laplace transform expressions are valid for functions of exponential order whose Caputo derivatives and convolution integrals exist on the considered time interval.

which often leads to initial conditions expressed in fractional integrals, a feature that complicates biological interpretation [17]. The Caputo definition is therefore preferred in the current study, as it allows us to specify initial numbers of susceptible, exposed, infected, and recovered individuals, consistent with epidemiological practice.

The key feature of both derivatives is their non-locality. Unlike integer-order differentiation, where the derivative at a point depends solely on local behavior, fractional derivatives depend on the entire history of the function. This historical dependence introduces memory into the system, which is central to the motivation of this research.

2.2. Convolution Operators and Memory Kernels

The non-local property of fractional derivatives can be formally expressed through convolution operators. Let  be a non-negative kernel function parameterized by . The convolution of  with a function  is given by (Eq. 2.3):

 (2.3)

In epidemic models, this convolution plays the role of memory-dependent incidence, as it weights past infectious individuals by the memory kernel. The total mass of the kernel, defined as (Eq. 2.4):

 (2.4)

quantifies the cumulative influence of past infections. Although kernel mass summarizes cumulative historical infectiousness, it should not be interpreted as identical to all memory effects. Its direct mathematical role is to scale the integrated transmission contribution and therefore influence the reproduction threshold. In contrast, the kernel shape determines temporal characteristics such as peak timing, persistence, and decay behavior. The interpretation of  depends on the kernel family. For an exponential kernel, the mass is finite and reflects short-term memory with rapid decay, while a power-law kernel yields heavier tails, indicating long-lasting memory where even distant past exposures affect current transmission. This flexibility enables models to capture fast-decaying and persistent memory effects, aligning the mathematics more closely with observed epidemiological data.

Admissible kernels in fractional epidemic models include exponential functions of the form , power-law kernels , and Mittag–Leffler kernels that interpolate between exponential and power-law behaviours. Each family corresponds to a different epidemiological interpretation. Exponential kernels approximate diseases where recent contacts dominate infection probability [18], while power-law kernels describe diseases with long incubation periods or recurrent exposures [19]. The Mittag–Leffler kernel provides a flexible framework for capturing both intermediate and complex memory behaviors, making it particularly relevant for hybrid fractional models [14].

2.3. Laplace Transform Properties and Analytical Tools

Fractional calculus analysis often relies on the Laplace transform, as it provides tractable forms for both Caputo derivatives and convolution operators. The Laplace transform of the Caputo derivative of a function  with initial condition  is (Eq. 2.5):

 (2.5)

where  denotes the Laplace transform of  [20]. This relationship enables the transformation of fractional differential equations into algebraic equations in the Laplace domain, which are often more amenable to stability and spectral analysis.

Another essential tool is the convolution theorem, which states that the Laplace transform of a convolution is simply the product of the Laplace transforms of its components. For a kernel  and an incidence function , (Eq. 2.6):

 (2.6)

This property simplifies the analysis of memory-incidence terms and allows explicit dependence of system stability on the kernel mass and its spectral properties. In addition, Grönwall-type inequalities for fractional systems are frequently employed to establish bounds on solutions and to prove positivity and boundedness [21].

2.4. Stability Criteria in Fractional Systems

The analysis of stability in fractional-order systems requires specialized tools because the classical criteria used in integer-order systems do not directly apply. A key result is Matignon’s theorem, which provides a criterion for the asymptotic stability of linear fractional-order systems [22]. According to this theorem, consider a linear system  with . The system is asymptotically stable if and only if the eigenvalues  of the matrix  satisfy (Eq. 2.7):

 (2.7)

This criterion highlights the effect of fractional order on stability regions. Unlike integer-order systems, where stability depends on whether eigenvalues lie in the left half-plane, fractional systems require eigenvalues to lie outside a sector whose angle depends on . This dependence reveals how memory effects influence the resilience of equilibrium states, making fractional stability analysis more nuanced than classical methods.

2.5. Functional Spaces and Well-Posedness

A rigorous analysis of fractional-order epidemic models necessitates the precise specification of the functional spaces in which solutions are defined. Typically, solutions are sought in appropriate Banach spaces of continuous or integrable functions over non-negative time intervals, where the associated norms are designed to capture the inherent nonlocality of fractional differential operators. The existence and uniqueness of solutions are commonly established by applying fixed-point techniques, most notably Banach’s contraction principle, within these functional settings [23].

Moreover, ensuring the epidemiological validity of the model requires that all state variables remain biologically meaningful throughout the evolution process. In particular, the non-negativity and boundedness of the susceptible, exposed, infectious, and recovered populations must be rigorously guaranteed. By employing fractional comparison principles together with suitably chosen admissible kernels, it can be shown that the model solutions remain confined to a positively invariant and bounded region for all time. These mathematical properties are not only essential from a theoretical standpoint but also fundamental for preserving the biological interpretability and reliability of the proposed epidemic model.

2.6. Kernel Families and Epidemiological Interpretation

Different classes of memory kernels influence not only the analytical behavior of fractional-order epidemic systems but also their epidemiological interpretation. Exponential-type kernels, characterized by a finite memory mass, are well-suited for modeling diseases such as seasonal influenza, where transmission dynamics are primarily governed by recent exposure events [24]. In contrast, power-law kernels with heavy tails are capable of capturing long-term memory and persistence effects, which are particularly relevant for diseases with complex immune responses, such as tuberculosis or viral infections exhibiting prolonged latency periods. Mittag–Leffler kernels provide an intermediate description between these two extremes, allowing for a gradual decay of memory and accommodating more intricate temporal dependencies observed in real-world epidemiological data.

The ability to flexibly select the kernel function within the fractional-order framework constitutes a major advantage of this modeling approach. It enables the formulation of disease-specific models that faithfully represent diverse memory mechanisms while preserving analytical tractability and mathematical generality.

3. MODEL FORMULATION

3.1. Compartmental Design

The proposed framework is built upon a classical compartmental structure that partitions the total population into four mutually exclusive classes: susceptible , exposed , infectious , and recovered . This SEIR configuration is particularly appropriate for infectious diseases characterized by a non-negligible incubation period, during which individuals are infected but not yet capable of transmitting the disease. Throughout the considered time horizon, the total population is assumed to remain constant, implying that births and natural deaths are either balanced or negligible relative to disease-driven dynamics. Although this assumption simplifies the model formulation, it is standard in mathematical epidemiology, especially when the primary objective is to isolate and analyze the impact of memory effects on transmission processes rather than demographic variations. Extensions incorporating vital dynamics can be accommodated within the proposed framework; however, such additions would complicate the analysis without fundamentally altering the core mathematical insights related to memory-dependent transmission [25]. This fractional formulation inherently incorporates memory effects, ensuring that the rates of new infections, disease progression, and recovery at time depend not only on the current state of the system but also on the cumulative history of the epidemic up to time .

The use of both Caputo fractional derivatives and an explicit convolution kernel do not imply a duplication of the same memory mechanism. In the proposed framework, the Caputo derivative represents system-level memory, meaning that the rates of change of epidemiological compartments depend on the past trajectory of the state variables. This captures inertia in disease progression, recovery, and relaxation toward equilibrium. By contrast, the convolution kernel  represents transmission-specific memory, meaning that present infection pressure depends on the weighted history of infectious individuals. Epidemiologically, this may reflect delayed infectiousness, environmental persistence, repeated exposure, or reporting delay. Thus,  controls the temporal response of the compartmental system, whereas  and its mass  control the accumulated force of infection. This distinction explains why enters the reproduction threshold, while primarily affects transient dynamics.

3.2. Memory-Dependent Incidence

The key novelty of the present model lies in the formulation of incidence, that is, the rate at which susceptible individuals become exposed due to contact with infectious individuals. In classical models, the incidence is typically given by , where  denotes the transmission coefficient. This formulation assumes that the force of infection depends solely on the current number of infectious individuals. In contrast, the proposed model incorporates memory by defining the incidence function as a convolution of the infectious population with a kernel . Specifically, the infection pressure at the time  is given by (Eq. 3.1):

 (3.1)

Where,  is the total population, and  represents an admissible memory kernel that weights past infections. The biological interpretation is that the infection risk faced by susceptible individuals is not simply proportional to the number of currently infectious persons but instead reflects a weighted accumulation of all individuals who were infectious in the past, with the weighting determined by the kernel.

This formulation captures a wide range of epidemiological realities. For example, in diseases where recent exposure dominates, an exponential kernel ensures that the influence of past infections decays rapidly. Conversely, in diseases where prolonged contact or environmental persistence of pathogens is relevant, a power-law kernel reflects the enduring effect of older infections [26]. The inclusion of kernel mass  provides a measure of the cumulative influence of past exposures on present dynamics and plays a crucial role in determining the reproduction number and stability conditions.

For consistency, the notation used throughout the paper is fixed as follows. The parameter  denotes only the biological progression rate from the exposed class to the infectious class. The switching signal is denoted by , not by . The transmission rate under regime  is denoted by , while the time-varying transmission rate is denoted by . Thus, when the system operates in regime , the corresponding reproduction number is written as . This notation is used consistently in the model equations, stability analysis and sensitivity discussion.

The hybrid structure is described through a switching signal . Let , where  indicates that the epidemic system is operating under an intervention regime . Each regime is associated with a transmission coefficient , so that the time-dependent transmission rate is written as . The symbol  is reserved exclusively for the exposed-to-infectious progression rate and is not used to denote switching. The dwell time of a regime is the duration for which  remains active before switching to another regime. Accordingly, all regime-specific threshold quantities are written as , computed using .

3.3. Hybrid Fractional Formulation

While fractional-order derivatives and convolution kernels embed memory into the epidemic system, real-world disease spread is also subject to abrupt changes arising from interventions. Public health measures such as lockdowns, social distancing mandates, vaccination campaigns, or behavioral changes in the population effectively alter the transmission rate over time [27]. To capture this feature, the present model allows the transmission coefficient  to vary in a piecewise constant manner, representing different regimes of disease spread. For example, a high value of  may correspond to periods of unrestricted contact, while a lower value corresponds to phases when strict measures are enforced.

This hybrid formulation combines two dimensions of realism: fractional-order derivatives for memory effects and hybrid switching structures for intervention-induced dynamics. Formally, the state equations are expressed as (Eq. 3.2, 3.3, 3.4, 3.5):

 (3.2)

 (3.3)

 (3.4)

 (3.5)

where  represents the rate at which exposed individuals become infectious, and  is the recovery rate. The incidence  is defined as above, incorporating both the fractional structure and the kernel convolution. The inclusion  of as a piecewise-constant function introduces hybrid behavior into the system and represents sudden changes in transmission conditions caused by interventions, policy shifts, or population-level behavioural responses.

3.4. Conditions for Well-Posedness

A critical requirement for any epidemic model is that it be biologically and mathematically well-posed. This means that solutions should exist, be unique, and remain within epidemiologically meaningful bounds forever. In the present context, this translates into three conditions: positivity, boundedness, and admissibility of kernels.

Positivity ensures that population compartments cannot assume negative values, which would be biologically meaningless. This property can be established using fractional comparison principles, which extend classical comparison results to fractional differential equations. These principles guarantee that if initial conditions are non-negative, the solutions remain non-negative for all subsequent times.

Boundedness requires that the total population not grow unboundedly. Since the system is closed and no new individuals are introduced beyond the initial total population, construction satisfies boundedness. However, the rigorous proof employs Grönwall-type inequalities adapted to fractional systems, which show that the sum  is preserved under the dynamics.

The admissibility of kernels requires that  is non-negative, integrable, and possesses a finite or controlled mass . These properties ensure that the memory-incidence term is mathematically well-defined and biologically interpretable. In particular, the integrability of  guarantees that the convolution  is finite for bounded . The choice of admissible kernels is thus central to establishing the well-posedness of the model.

3.5. Interpretation and Relevance

The proposed formulation extends the classical SEIR framework in two fundamental directions. First, it introduces fractional-order dynamics, thereby embedding memory effects directly into the epidemic evolution equations. This formulation reflects the realistic observation that infection and recovery processes are influenced not only by the current epidemiological state but also by accumulated past exposures. Second, the model incorporates hybrid regime switching to capture the discontinuous and policy-driven nature of real-world public health interventions. Together, these features yield a flexible and realistic modeling framework that surpasses the descriptive capabilities of both conventional integer-order models and fractional-order models lacking regime-switching mechanisms.

From a theoretical perspective, the model provides a systematic platform for examining the interaction between memory effects and intervention strategies in determining epidemic thresholds, stability properties, and long-term behavior. From an applied standpoint, it offers insights into how public health policies may interact with intrinsic memory mechanisms in disease transmission, thereby modulating intervention effectiveness. For instance, the introduction of a lockdown during periods when the memory kernel exhibits substantial cumulative mass, indicating strong persistence of past infections, may lead to markedly different epidemic outcomes than an intervention implemented under weaker memory conditions. Such observations underscore the importance of accounting for memory effects when designing timely and effective epidemic control strategies. To clarify the computational and analytical sequence of the proposed hybrid fractional-order memory-dependent SEIR framework, Fig. (1) presents the stepwise workflow of the model. The algorithm begins with the input of epidemiological parameters and the selection of an admissible memory kernel. The kernel mass  is then computed to quantify cumulative transmission memory. Next, the hybrid switching signal is defined to represent intervention regimes, followed by the construction of the memory-dependent incidence function. The fractional SEIR system is then solved numerically, and regime-specific reproduction numbers are calculated. Finally, stability conditions, sensitivity analysis, and epidemiological outputs are generated.

As shown in Fig. (1), the proposed framework links kernel selection, hybrid switching, fractional numerical solution, reproduction-number computation, stability assessment, and sensitivity analysis within a single structured workflow.